Nonlinearity

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8 papers in the last four weeks, up 60% on the four weeks before. 0.1% of all new papers.

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Latest papers 104

Oct 1, 2026cs.LG

Beyond Linear Concepts: Discovering and Aligning Non-Linear Concept Manifolds in Large Language Models

Understanding information processing in large language models (LLMs) requires dissecting the geometric organization of their internal token representations. While existing mechanistic interpretability (MI) methods seek to extract concepts, they are constrained by a strong linearity assumption challenged by evidence of non-linear feature manifolds. We move beyond linear concepts by adapting Non-Linear Multi-Dimensional Concept Discovery (NLMCD) from computer vision to token-level LLM activations, modeling concepts as low-dimensional manifolds. To compare concept manifolds across layers and models, we introduce a concept-based alignment (CBA) score, a generalized Rand index that measures geometric proximity without explicit feature matching. Our analysis yields six key findings: (i) a neighboring-layer sanity check shows CBA is more sensitive than PCA- or CKA-based linear baselines; (ii) layer-by-layer alignment matrices reveal two block structures in intermediate and late layers, consistent across models and obscured by linear metrics; (iii) concept composition remains syntax-dominated through most of the network before giving way to increasingly mixed syntactic-semantic concepts in later layers, with increasing output-orientation toward the final layers; (iv) multilingual concept sharing between English and Mandarin is training-dependent rather than universal, strongest in Qwen, weaker in Llama, and absent in GPT-2; (v) inter-model alignment mirrors this structure, with strong correspondence between same-family Qwen models of different scale but weak alignment across model families; and (vi) across Tulu-3 training stages, alignment is highest between adjacent stages, with the largest shift between the base model and SFT, while subsequent preference-alignment stages (DPO, RLVR) leave early layers largely unchanged and RLVR mostly preserves DPO's concepts in late layers.

Tido Specht, Elias Benedict Krey, Nils Neukirch +1

University of Groningen · Carl von Ossietzky University of Oldenburg

Large Language Model Alignment Multimodal Large Language Models Mechanistic Interpretability Syntactic Structure Nonlinearity Data Manifold

Oct 1, 2026cs.LG

From Redundancy to Minimality: Fixed-Point-Guided Hierarchical Reduction of Learned Piecewise-Linear Dynamics

Understanding a nonlinear dynamical system from time series requires not only reproducing its trajectories, but also identifying a simple representation that preserves its essential dynamical structure. Almost-linear recurrent neural networks (AL-RNNs) are piecewise-linear RNNs in which only a subset of units use ReLU nonlinearities, so that nonlinear capacity is explicitly controlled by the number of ReLU units. Their activation patterns define linear regions, represented as symbols, whose observed transitions form a symbolic transition graph. However, directly training AL-RNNs with few ReLU units to realize minimal dynamical representations can be unreliable. We ask whether an AL-RNN with more ReLU units can instead be trained first and systematically reduced to a minimal dynamical representation. We introduce a fixed-point-guided hierarchical reduction procedure that progressively linearizes selected ReLU units, merging neighboring linear regions and graph nodes while preserving distinct symbols containing fixed points (FPs). The resulting reduction tree defines a hierarchy of progressively simpler candidates. Each reduced candidate is initialized from the parent parameters and retrained under guidance from the parent dynamics. We also prove that reproducing

Q

distinct fixed points requires at least

Q

FP-containing symbols, providing a certificate of symbol-level minimality when this bound is attained. On the 3-scroll Chua system, direct training with the theoretical minimum of three ReLU units achieves high-fidelity minimal realizations in only 20% of seeds, whereas our learn-reduce-retrain strategy increases the seed-macro success rate to approximately 71% at the same final nonlinear capacity. These results show that redundant nonlinear capacity can serve as a scaffold for discovering and realizing minimal dynamical representations.

Hiroto Tamura, Gouhei Tanaka

Graduate School of Informatics, Osaka Metropolitan University, Osaka, Japan · International Research Center for Neurointelligence (IRCN), The University of Tokyo, Tokyo, Japan · Graduate School of Engineering, Nagoya Institute of Technology, Nagoya, Japan

Nonlinear Dynamics Nonlinearity Rectified Linear Unit Recurrent Neural Networks Inference Dynamics

Sep 30, 2026cs.LG

On Parameters of Nonlinear Scalar Dynamics from Video: Invariants, Calibration, and Identifiability

Physical parameter estimation from video aims to recover the parameters of a known family of governing dynamical equations from pixel observations. Existing identifiability theory for this setting has focused on linear time-invariant (LTI) second-order systems, leaving open what can be identified for nonlinear scalar dynamics. We develop an identifiability theory for nonlinear scalar second-order ODEs, organized by how their velocity dependence interacts with changes of the learned state coordinate. Under a shared non-collapsed state map and explicit same-state velocity-coverage conditions, we show that parameter identifiability depends on the ODE family: some parameters are uniquely identifiable, while in other families only invariant parameter combinations are identifiable or external physical calibration is required. For laws that are at most linear in velocity, compatibility forces affine coordinate alignment, yielding explicit parameter relations, invariants, and calibration conditions. This affine conclusion extends to broader finite velocity-feature families when coordinate curvature can be separated from the declared velocity dependence. For families admitting a squared-velocity term, nonlinear coordinate ambiguity can remain; a law-derived normalization instead enables affine comparison between canonical laws. Experiments on synthetic systems and real pendulum and free-fall videos support the predicted parameter relations, coverage effects, and calibration requirements.

Wenjie Wang, Yuanyuan Wang, Zixiang Jiang +2

University of Melbourne · Mohamed bin Zayed University of Artificial Intelligence · Carnegie Mellon University

System Identification Identifiability Nonlinear Dynamics Conditional Velocity Nonlinearity

Sep 27, 2026cs.LG

From Distributions to Stochastic Processes: Neural Approximation of Measure-Valued Maps

Learning mappings between probability distributions arises naturally when inputs and outputs are represented by populations of samples rather than individual observations. We develop an approximation-theoretic framework for distribution-to-distribution learning and extend it to mappings between stochastic processes. For continuous operators on

W_2

-compact families of finite-dimensional probability laws, we establish uniform neural approximation in the 2-Wasserstein metric using finite law statistics, a simplex-valued neural map, and a shared atomic output support that guarantees valid probability measures. We further extend this principle to probability laws on separable Hilbert spaces through finite-rank orthogonal projections. These results establish the representational feasibility of learning transformations between probability laws rather than deterministic vectors or functions. To demonstrate practical relevance, we study two problems naturally defined at the distribution level: prediction of first-passage-time distributions for an Ornstein--Uhlenbeck process and nonlinear response-path laws of a Duffing oscillator. Because the theory is model-agnostic and broader than any single practical architecture, the experiments use task-adapted neural models rather than reproducing the theoretical construction exactly. In both problems, the proposed models outperform a fixed-feature MLP baseline and distribution-space kernel regression. These experiments complement the theory by demonstrating the practical learnability of distribution-to-distribution transformations in random systems.

Yichen Wang, Ziyi Wang, Wenlian Lu +5

Shanghai Center for Mathematical Sciences, Shanghai, China · Department of Mathematics, Fudan University, Shanghai, China · Center for Applied Mathematics, Fudan University, Shanghai, China +2

Probability Measures Stochastic Processes Distributions Neural Approximations Distributional Reinforcement Learning Nonlinearity

Sep 24, 2026cs.CL

Your Transformer Can Hold Two Thoughts at Once: Evidence of Linear Superposition in LLMs

While Large Language Models (LLMs) rely on highly non-linear components, in this work we demonstrate that they exhibit fundamental linearity: when inputs from distinct text streams are linearly combined, the model outputs a superposition of the individual next-token distributions. We term this the \textit{Superposition Linearity Hypothesis}. We provide evidence that superposition is an intrinsic property of the Transformer architecture rather than an emergent consequence of training; in fact, we observe that it tends to diminish as pretraining progresses. However, we demonstrate that linearity can be substantially restored through lightweight fine-tuning, significantly reducing the divergence between the predicted next-token distribution and the average of the individual next-token distributions. Finally, we introduce a guided decoding procedure that disentangles superposed outputs, enabling the simultaneous generation of two coherent continuations from a single forward pass.

Pavel Tikhonov, Anton Korznikov, Matvey Mikhalchuk +6

Transformer Architectures Next-Token Distribution Thoughts Nonlinearity

Sep 23, 2026cs.CL

Computation Over Geometry: Meaning Identity Is Computed, Not Shipped in the Embeddings

Meaning identity (whether two sentences say the same thing after wording changes) is treated in retrieval and RAG as a geometric fact about independently encoded sentence vectors. We show that, for frozen off-the-shelf encoders and language models, it is not: identity is computed when both sentences share one forward pass, and is not a property of the embedding geometry those systems ship. On overlap-matched PAWS-X, purpose-built encoders (BGE, E5, GTE, MiniLM, E5-Mistral-7B) reach English confirm AUC only 0.55-0.65 (dense peak 0.70). Independently encoded last-token states of Llama 3, Mistral, and Qwen do no better; late fusion of the two vectors stays near chance. The same probe on a joint forward pass reaches 0.90-0.96 from 1.5B to 32B, collapses under partner shuffle, is mid-depth, saturates near 0.94 by 3B, and appears more weakly in GPT-2 XL (0.76). The gap holds beyond Llama-style models on other causal LMs, bidirectional encoders (DeBERTa, RoBERTa), and encoder-decoders (Flan-T5, T5, BART). Fixed or linear readers over frozen independent encodings never unlock identity; nonlinear pair readers recover part of it only on the full 49k-pair PAWS train split (0.68-0.87). Off-the-shelf rerankers split: BGE-reranker-large reaches 0.94, while MS-MARCO and Jina stay at 0.55-0.64. Independently trained families compute the same relation and a 1.5B joint reader can distill it from unlabelled teacher scores, while no linear function of the teachers own independent vectors can. Bi-encoders can be fine-tuned to fit PAWS (0.87-0.93), but transfer and STS-B suffer. Cosine compares wording neighbourhoods; identity is a cheap computed operator, not a property of either sentence vector.

Jiaqi Deng

Independent Researcher

Token Embeddings Nonlinearity

Sep 14, 2026physics.flu-dyn

Computer-assisted global regularity across nonlinear families of three-dimensional periodic Navier-Stokes flows

Numerical simulations reveal how vortices stretch and transfer energy, but establishing smooth evolution requires bounds that remain valid beyond the simulated resolution. Here I develop a computer-assisted framework that establishes global regularity for continuous families of three-dimensional periodic Navier-Stokes flows. Its central construction combines finite reference trajectories with a common error bound that covers an interval of centre fields and infinitely many smooth perturbation modes. The method retains the complete nonlinear residual before spectral truncation and controls the evolution until viscous decay guarantees regularity for all subsequent times. Applications to cyclic-shear, Arnold-Beltrami-Childress and three-component Taylor-Green fields yield explicit perturbation radii and include initial conditions outside the direct Fourier-Wiener smallness criterion. A parameter-uniform extension covers a connected family of non-Beltrami Taylor-Green centres without repeating the proof for individual parameter values. An ensemble of 4,096 configurations, supplemented by 1,600 refinement trajectories and public turbulence data, connects the mathematical observables to spectral transfer and vortex geometry. Matched neural-operator experiments show that physics-informed training improves physical prediction, while also revealing that these gains do not necessarily improve the discovery of proof-limiting initial conditions. Together, these results provide a reusable method for establishing regularity across prescribed flow families and a quantitative setting for evaluating how learned predictions can assist rigorous computation.

Jose Luis Lima de Jesus Silva

Federal University of Bahia, Department of Geophysics Salvador, BA 40170-115, Brazil · Grupo de Estudos e Aplicação de Inteligência2026 Artificial em Geofísica (GAIA) Federal University of Bahia, Salvador, BA 40170-115, Brazil

Vortex Reynolds-Averaged Navier-Stokes Regularity Nonlinearity

Sep 14, 2026math.AP

Linearized PINN with pretrained nonlinear layers

We propose a linearized Physics-Informed Neural Network (lPINN), a reduced-order neural basis method for forward and inverse differential equations. In an offline stage, lPINN learns operator-compatible continuous neural basis functions from an ensemble of numerical solutions. The basis functions are differentiable through automatic differentiation and are pretrained using solution data together with either derivative information or physics residuals. For each new problem instance, the basis functions are frozen and the solution is obtained by minimizing the governing-equation residual together with applicable initial, boundary, regularization, and observational terms. Unlike surrogate and operator-learning methods, the training data define the trial space offline, while the instance-specific solution is computed online by enforcing the governing physics. Relative to vanilla PINNs, lPINN pretrains the nonlinear hidden-layer representation offline and performs online inference only in the final linear layer. We evaluate lPINN on forward and inverse problems for the advection-diffusion equation, Burgers' equation, and the nonlinear pendulum equation. Compared with vanilla PINNs, lPINN achieves lower solution and parameter errors while reducing online inference times by approximately one to more than three orders of magnitude, with the largest gains generally observed for limited residual or measurement data. Cross-resolution experiments show that the learned continuous representation can be evaluated on finer meshes without retraining and with nearly unchanged accuracy.

Wenhao Chen, Alexandre M. Tartakovsky

Civil and Environmental Engineering Department University of Illinois Urbana-Champaign Urbana, IL 61801, USA · Pacific Northwest National Laboratory Richland, WA 99352, USA

Parametric Physics-Informed Neural Network Nonlinearity Reduced-Order Models

Sep 7, 2026stat.ML

Distributed Lag Neural Additive Models

We introduce Distributed Lag Neural Additive Models (DLNAMs), neural-additive analogues of Distributed Lag Non-linear Models (DLNMs) for learning nonlinear effects distributed over lags. DLNAMs replace a prespecified spline cross-basis with neural components that learn exposure--lag response surfaces, avoiding choices of basis family, dimension, and knot placement while preserving additive interpretability and familiar distributed-lag summaries. Exp-centered input layers, smooth activations, and learned subnetwork mixtures produce smooth, locally adaptive representations; pointwise uncertainty combines a conditional last-layer Laplace approximation with between-member ensemble variation. In simulations, DLNAMs generally outperformed DLNM comparators, including penalized and treed variants, in recovering known response functions, with lower bias, stronger boundary recovery, and better-calibrated cumulative intervals; gains were largest for more demanding functions. The architecture performed consistently across sample sizes, outcome families, lag horizons, and jointly fitted multi-exposure settings, retaining recovery performance as exposures were added; fit-specific changes were largely confined to optimization, and applications recovered established empirical patterns.

Calle Helmersson, Shivang Pandey, Leonardo Olivetti +1

Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden · Department of Global Public Health, Karolinska Institutet, Stockholm, Sweden · Swedish Centre for Impacts of Climate Extremes (climes), Uppsala University, Uppsala, Sweden +4

Generalized Linear Model Nonlinearity

Sep 3, 2026cs.LG

Neural-Network Maxent: a general extension with learned nonlinearity, applied to time-series for Desert Locust distribution modelling

Species Distribution Modelling (SDM) is essential for understanding how environmental conditions shape biodiversity, particularly for destructive pests such as the Desert Locust (Schistocerca gregaria), whose breeding dynamics are tightly coupled to rapidly evolving environmental conditions. Maxent has become the dominant method for presence-only data, but its reliance on a linear combination of hand chosen feature transforms limits its ability to capture the nonlinear, temporal relationships common in ecological monitoring, where covariates such as precipitation, soil moisture, and vegetation indices evolve meaningfully over time. Standard implementations flatten time-series covariates into independent features, discarding sequential structure that carries critical signal. We introduce RNN Maxent, an extension of the Maxent framework that replaces the fixed feature dictionary with a neural network, specifically a Gated Recurrent Unit (GRU), trained end to end via backpropagation. The approach preserves Maxent's presence only statistical foundations, background normalization, and probability calibration, differing only in that the nonlinearity is learned from data rather than fixed in advance. We apply RNN Maxent to map suitable habitat for the Desert Locust using 50 day environmental time series derived from ERA5 Land, MODIS, and Sentinel 3, maintaining a 7 day gap between covariates and presence records to yield forecasting behavior. Compared against standard Maxent, RNN Maxent improves performance across metrics (ROC AUC 0.862 std 0.036 vs. 0.792; F1 0.671 std 0.056 vs. 0.590).

Alessandro Grassi, Edoardo Kimani Bellotto, Wassim El Azami +2

SISTEMA GmbH, Vienna, Austria · Beyond EO, Paris, France

Time Series Cnn-Lstm Recurrent Neural Networks Nonlinearity

Sep 1, 2026cs.LG

DeSyR: A Decoupled Symbolic Recovery Framework with PINN-Guided Structure Search and Physics-Informed Coefficient Refinement

Recovering compact explicit solutions from neural approximations is challenging when imperfect teacher data guide symbolic topology search and coefficient estimation. We present DeSyR, a decoupled symbolic recovery framework for differential equations. A physics-informed neural network guides repeated searches to construct candidate topologies with provisional constants. Once a topology is fixed, its coefficients are refined solely from the governing equation and prescribed constraints, followed by gated selection and verification. For linear fixed-topology parameterizations, we characterize teacher-error inheritance and show that finite-weight mixed data--physics fitting retains an

O(β^{-1})

teacher-dependent contribution when the teacher error projects onto the model space. Under well-posedness, representability, zero-residual attainment, and discrete determinacy, physics-only refinement conditionally recovers exact coefficients; for nonlinear parameterizations, the corresponding guarantees are local. DeSyR is evaluated on 15 differential-equation problems across 18 configurations covering high-order, space--time, multidimensional, nonlinear, and coupled systems. A candidate-level audit yields a 99.23% convergence rate among free-parameter refits, while every selected refinement involving free coefficients converges. Configuration-level median refined relative

L_2

errors are

2.31\times10^{-14}

or lower. In same-topology comparisons, refinement reduces error by eight to fourteen orders of magnitude. These results show that an approximate neural teacher can guide topology discovery without imposing its error scale on final recovered coefficients, provided a target-capable topology is retained and physics-only refinement converges.

Pancheng Niu, Jun Guo, Qiaolin He +2

Chengdu University of Information Technology, College of Applied Mathematics, Chengdu, 610225, China · Sichuan University, School of Mathematics, Chengdu, 610065, China · Hong Kong Polytechnic University, Department of Computing, Hong Kong, 999077, China +1

Ordinary Differential Equations Physics-Aware Models Nonlinearity Topology

Aug 11, 2026cs.LG

$β$ -VAEs as Effective Theories: Tolerance-Dependent Dimension

In a

β

-VAE, increasing the regularization strength acts as a spectral cutoff by collapsing low-utility latent coordinates. In the linear Gaussian VAE, the collapse order matches the ranking of reconstruction utilities exactly, because both are set by the PCA spectrum. We ask which parts of this picture survive in fully connected nonlinear VAEs trained on WorldClim. We find that nonlinear interactions shift and broaden collapse onsets, so thresholds no longer coincide exactly with utilities. However, the common ordering is preserved over the resolved ranks, so the spectral cutoff still acts as a utility cutoff and the effective-description logic carries through. The resulting effective-dimension curves reveal a head--tail tradeoff: increasing depth concentrates utility into the first few coordinates but worsens tail fidelity.

Johannes Hirn

Image Processing Laboratory (IPL), Universitat de València, Paterna, València 46980, Spain

Representation Collapse Nonlinearity Conditional Variational Autoencoder Latent Variable Regularization Spectral

Aug 11, 2026cs.RO

Nonlinear Model Predictive Control via Sequential Convex Programming for Drone-to-Drone Docking

Autonomous mid-air docking of multi-rotor vehicles under disturbance-driven target motion poses a constrained non-linear trajectory optimization challenge. This work formulates the docking task as a finite-horizon optimal control problem based on a reduced-order nonlinear model augmented with disturbance states. The resulting problem is solved using sequential convex programming within a receding-horizon framework to generate dynamically feasible docking trajectories. State estimation with noisy measurements is incorporated to enable robust relative motion prediction, while trajectory execution is validated in a high-fidelity rigid-body MuJoCo simulation environment. The proposed framework is evaluated for stationary and constant-velocity target motions, demonstrating reliable convergence to the docking interface while satisfying geometric capture constraints. Quantitatively, the method maintains negligible docking-cone violations and terminal state errors within prescribed tolerances, and achieves consistent, safe docking performance for cone half-angles as low as 10 degrees. Robust operation is observed for wind disturbance levels up to a standard deviation of 0.5, while preserving bounded approach velocities and stable control effort. These results demonstrate the effectiveness of the SCP-based trajectory optimization framework for disturbance-robust aerial docking under estimation uncertainty.

Neeraj Balachandar, Shriram Hari, Vishnu R. Unni

Department of Mechanical and Aerospace Engineering, Indian Institute of Technology Hyderabad, Telangana, India

Model Predictive Control Robust Trajectory Convex Optimization Nonlinearity

Aug 11, 2026cond-mat.mtrl-sci

On the Importance of Geometric Nonlinearity and Temperature-Dependent Properties in Multi-Material Thermo-Mechanical Topology Optimization

Thermo-mechanical compliant devices are commonly designed with small-strain linear elasticity and temperature-independent material properties, even though they might operate hundreds of kelvin above ambient where both assumptions are questionable. In this work, we quantify the effect and cost of each assumption in multi-material topology optimization of thermally actuated compliant devices. To this end, we introduce a physics-informed, simultaneous analysis-and-design framework with (i) a finite-strain quadratic-Hencky (logarithmic-strain) constitutive model whose isotropic thermal eigenstrain admits an exact additive split in log-strain space, and (ii) temperature-dependent conductivity, thermal expansion, and elastic moduli for a titanium--copper--steel material system. We optimize a thermal actuator and a thermal gripper at three design temperatures under both a baseline model and the full physics, subject to mass and manufacturability constraints. Every converged design is re-evaluated by verified nonlinear finite element solvers in the full factorial of constitutive law and property model. The comparison between the two factors reveals that the constitutive law is the decisive modeling choice: These devices work as linkages where linear kinematics mistakes rotation for compressive strain; its error therefore grows with the design temperature and concentrates on the very layouts that exploit rotation best. Because a linear optimizer also steers away from the rotation-rich mechanisms that would expose this bias, the model can deceptively appear trustworthy when validated against its own designs. Designing with the full physics yields consistently stronger and more temperature-robust devices at a modest increase in design-time cost.

Shirin Hosseinmardi, Xiangyu Sun, Ramin Bostanabad

Department of Mechanical and Aerospace Engineering, University of California, Irvine · Department of Civil and Environmental Engineering, University of California, Irvine

Topology Optimization Nonlinearity Importance

Aug 8, 2026cs.RO

Diminishing Returns of Intelligence: The Non-Linear Relationship Between LLM Scale and User Perception in Short-Duration Open-Ended Social Human-Robot Interactions

Large Language Models (LLMs) are increasingly used to drive embodied social agents, yet it remains unclear whether larger models improve user perception during brief human-robot encounters. This paper examines the effect of LLM parameter size on short-duration, open-ended social interactions with a robot interface. In a within-subjects study, 19 participants interacted with robot faces driven by Qwen3-VL models at 4B, 8B, and 30B parameters. Participants evaluated the interactions in terms of perceived intelligence, naturalness, enjoyment, and humor. Results showed no significant overall preference for the 30B model over the smaller variants, including no significant advantage over the 4B model in perceived naturalness or intelligence. A significant relationship between AI interaction frequency and intelligence rankings for the 30B model suggests that more experienced users may be more sensitive to differences in model capability. Overall, the findings indicate diminishing returns from model scaling in brief open-ended social HRI, where conversational flow, responsiveness, and socially appropriate behavior potentially matter as much as raw parameter count.

Amanda Rasille Røn Volf, Morten Roed Frederiksen

Computer Science Department of The IT-University of Copenhagen

Human-Ai Interaction Model Size Intelligence Human-Robot Interaction Nonlinearity

Aug 5, 2026cs.CV

Splat-Based Metal Artifact Reduction in Cone-Beam CT via Compact Attenuation Modeling

X-ray computed tomography (CT) suffers from severe metal artifacts when high-attenuation objects such as dental fillings or orthopedic implants are present. These artifacts originate from the polychromatic nature of X-rays, where attenuation varies strongly with photon energy and material composition, breaking the monochromatic assumption used by conventional reconstruction algorithms. Recent neural rendering approaches attempt to address this mismatch through differentiable polychromatic projection models, but they still struggle with smoothness bias, loss of fine structures, and prohibitive computation when extended to large-scale cone-beam CT. We introduce a splat-based metal artifact reduction framework that incorporates a physically grounded polychromatic forward model into a continuous Gaussian representation for cone-beam CT. Each Gaussian encodes the energy-dependent attenuation of the underlying material using a compact material parameterization, which enables efficient joint optimization of geometric and material properties without relying on a metal mask. This compact attenuation formulation captures the essential variation across biological tissues and metallic implants, allowing our model to explain metal-induced nonlinearity while preserving high-frequency structure. Experiments on simulated and real cone-beam CT scans show that our method converges significantly faster and suppresses metal artifacts more effectively than existing reconstruction and neural field-based approaches.

Kiseok Choi, Jaemin Cho, Inchul Kim +1

KAIST

Cone-Beam Computed Tomography Nonlinearity

Aug 1, 2026math.CV

Exact Contraction Rates via the Berkson--Porta Representation: A Sharp Threshold and Its Herglotz-Kernel Obstruction

Semigroups of holomorphic self-maps of the unit disc with an interior fixed point are, by the classical Berkson--Porta representation, entirely determined by a single holomorphic function constrained only by a positivity condition on its real part. This paper uses that representation to determine exactly when the associated flow contracts the Kobayashi metric of the disc at its best possible rate --- the rate dictated by linearization at the fixed point --- rather than at some smaller, conservative rate of the kind ordinarily obtained through auxiliary metric constructions. The question is reduced to a single pointwise inequality on the representing function, and this inequality is resolved completely for a natural one-parameter family of nonlinearities, yielding an exact threshold rather than a sufficient condition of undetermined tightness. Beyond this family, an explicit representing function is exhibited for which the inequality fails almost everywhere on the disc, and the Herglotz integral representation underlying the associated Carathéodory class is used to trace this failure to concentration of the representing measure, explaining rather than merely documenting why no threshold-free general theorem is available. The results are illustrated by direct numerical verification of the sharp threshold and of the explicit obstruction, and the paper closes by identifying the precise class of representing measures --- point masses and their neighborhoods --- that any future general sufficient condition would need to exclude.

Soumic Sarkar

Institute of Technology, University of Tartu, Nooruse 1, Tartu, 50411, Estonia.

Convergence Decision Thresholds Nonlinearity

Jul 27, 2026quant-ph

Variational Quantum Conditional Boltzmann Machines for Time-Series Forecasting: Architectures, Symmetric Hyperparameter Evaluation, and a Nonlinear Benchmark

In this study, we developed and evaluated four conditional energy-based forecasting architectures: a classical Gaussian-Bernoulli CRBM, a hybrid quantum-classical QCRBM, a full-register QQRBM, and a lag-feature QFeatureQRBM with complete derivations of their conditional distributions, Contrastive-Divergence gradients, and hybrid training, bridging the energy-based formulation and the implementation-level quantum computation. Unlike prior comparisons, our evaluation enforces symmetric hyperparameter optimisation: classical and quantum-specific hyperparameters receive an equally thorough grid search across thirteen structured experiments. We test on two data classes, a Gaussian-process dataset (GP) generated with real financial data and the input-driven NARMA-10 nonlinear benchmark. Across both regimes we find no systematic evidence of a quantum advantage at the available sample size: no quantum architecture improves on the best classical baseline. The fully quantum QQRBM and QFeatureQRBM are significantly worse, whereas the hybrid QCRBM is statistically indistinguishable from the strongest classical CRBM on both datasets. A power analysis bounds this null result: at n = 12 only medium-to-large effects are detectable, so small advantages cannot be excluded. An iso-parameter (matched-budget) comparison reaches the same conclusion: the classical CRBM is lowest at three of the four budgets and no CRBM-vs-QCRBM difference is significant at any budget.

Gerhard Hellstern, Danyal Maheshwari, Martin Zaefferer +2

Centre of Finance DHBW Stuttgart Stuttgart, Germany · Zentrum für Digitale Innovationen DHBW Ravensburg Ravensburg, Germany · DATEV eG Nürnberg, Germany

Hybrid Quantum-Classical Pipeline Time Series Forecasting Hyperparameter Gaussian Mixture Models Nonlinearity

Jul 27, 2026cs.RO

Moving-Horizon Estimation and Nonlinear Model Predictive Control of Cable-Driven Soft Manipulators

Precise control of soft manipulators remains challenging due to the difficulty of developing accurate yet computationally tractable models for model-based estimation and control. Reduced Cosserat-rod models provide a physics-based and control-oriented description of soft-robot dynamics, offering an explicit alternative to purely data-driven input-output representations. In this paper, we propose a moving-horizon estimation (MHE) and nonlinear model predictive control (NMPC) framework for cable-driven soft manipulators based on reduced Cosserat dynamics. A smooth cable-length-driven modeling formulation is developed by approximating the complementarity relationship between cable tension and cable slackness, enabling cable-length control without direct tension sensing. Based on this formulation, an MHE method is introduced to estimate the reduced state and reconstruct the manipulator configuration from end-effector pose measurements and cable-length information. An NMPC controller is then formulated to achieve task-space control under cable-length and cable-rate constraints. The proposed framework is validated through numerical simulations and experiments. Simulation results demonstrate the effectiveness of the estimator and controller for pose and strain-related regulation on a multi-cable soft manipulator. Experimental results on a four-cable prototype further show that the proposed MHE-NMPC scheme can be implemented in real time and enables accurate end-effector position tracking through cable-length control.

Lingxiao Xun, Haihong Li, Gang Zheng

Cosserat Rod Theory Robotic Manipulator Model Predictive Control End-Effector Pose Nonlinearity

Jul 20, 2026cs.NE

Organization of computation in reservoir computing

Reservoir computing exploits nonlinear dynamical systems to encode temporal inputs into high-dimensional state space representations. Although reservoir performance is often characterized through memory, nonlinearity, and their tradeoff, such aggregate measures do not reveal how task-relevant information is organized within the state space. Here, we introduce an eigen-spectral decomposition framework linking the degree-wise information processing capacity to the corresponding state space modes. As a result, we are able to quantify the degree-wise representation energy, and show that in some cases, substantial amounts of information processing capacity may reside in low-energy modes that are vulnerable to experimental noise. These results suggest that useful reservoir computation depends not only on dimensionality expansion, but also on the geometric organization of task-relevant information, with direct implications on physical reservoir computers.

Mohab Abdalla, Damien Rontani

CentraleSupélec, Université de Lorraine, LMOPS EA-4423 Laboratory, 57070 Metz, France

Reservoir Computing Nonlinearity Spectral Decomposition State Space Cognitive Science

Jul 20, 2026cs.RO

Predicting Grasping Compliance in Robotic Hands through Analytical-Model-Informed Neural Networks

In robotic manipulation studies, grasping is often treated as a binary success or failure problem, usually defined by whether the object simply stays in the hand. For forceful tool use, however, this view is insufficient because grasp compliance becomes a critical factor governing how the hand and tool behave under load. Compliance arises from coupled kinematics, grasp configuration, passive mechanics, and contact conditions, producing nonlinear behavior in which deformation and interaction forces influence each other. Understanding this relationship is essential for predictive models of how a grasped tool and a compliant hand jointly respond to external loading. In underactuated hands, these effects are amplified: such designs offer low cost and adaptive grasping, but make compliance behavior more difficult to model and predict. Our goal is therefore to develop a predictive model for grasped tool behavior during forceful interactions. To address this challenge, we introduce an analytical model informed neural network (AMINN), a hybrid predictive model that combines an analytical mechanics layer with data driven learning to estimate grasp stability and in hand tool displacement under external loading. The model is evaluated on a three finger underactuated robotic hand and shows strong predictive capability with mechanically meaningful outputs across diverse loading conditions. Compared with a black box multilayer perceptron baseline, AMINN also achieves better energy based physical consistency. Beyond prediction accuracy alone, this framework advances physically interpretable learning for robotic manipulation and supports more reliable, safer, and more trustworthy autonomous tool use in safety critical settings during forceful interaction.

Qianwen Zhao, Long Wang

Department of Mechanical Engineering, Stevens Institute of Technology, Hoboken, NJ, USA

Robotic Hands Robotic Manipulation Neural Network Controlling Tool Use Compliance Nonlinearity

Jul 17, 2026cs.LG

Spatio-Temporal Prediction of Unsteady Airfoil Aerodynamics Using Augmented Graph Neural Ordinary Differential Equations with Exogenous Controls

Unsteady aerodynamic phenomena, such as gusts, turbulence, and fluid-structure interactions affect an aircraft during flight. For design, optimisation and certification, it is indispensable to quantify such unsteady aerodynamic effects. Industry-standard computational fluid dynamics methods, such as solving the unsteady Reynolds-averaged Navier-Stokes equations or the linearized frequency domain method, are either computationally expensive or restricted by assumptions like linearity. Once trained, machine learning methods are capable of computing non-linear relationships very fast, making them suitable as surrogate models. By autoregressively applying graph neural networks (GNNs), operating on a discretised spatial domain, spatio-temporal predictions can be made. However, autoregressive GNNs suffer from error accumulation leading to unstable rollouts over time. Here we show that combining GNNs with augmented Neural Ordinary Differential Equations yields temporally stable predictions of the surface forces on a pitching airfoil. We found that our approach, called GNODE, based on Graph Neural Ordinary Differential Equations, provides temporally more stable, spatially smoother, and overall more accurate results than an autoregressive GNN baseline. Tests are conducted on a dataset consisting of a simulations of a pitching airfoil, including transonic shocks, transient behaviour and dynamic non-linearities. Augmenting GNODEs with additional latent dimensions improves the expressivity and accuracy by capturing underlying history effects. The developed method demonstrates an approach that is suitable to model non-linear spatio-temporal systems with exogenous inputs.

Henrik Lange, Reik Thormann, Philipp Bekemeyer

German Aerospace Center (DLR), Lilienthalpl. 7, 38108 Brunswick, Germany · Airbus Operations, Airbus Allee 1, 28199 Bremen, Germany

Aerodynamics Computational Fluid Dynamic Spatiotemporal Prediction Neural Ordinary Differential Equations Graph Neural Networks Nonlinearity

Jul 16, 2026cond-mat.stat-mech

Moment-Resolved Readout and Reservoir Diversity in Nonequilibrium Langevin Computing

Nonlinear thermodynamic computers based on Langevin dynamics exploit thermal fluctuations as a physical substrate for computation. Recent work has shown that quartic-confined fluctuating degrees of freedom can act as thermodynamic neurons capable of nonlinear function approximation at finite observation times. Here we extend this paradigm from mean-only readout to moment-resolved readout. Instead of representing each driven reservoir solely by its first moment, we construct a response vector from the elementwise raw polynomial moments

\mathbb{E}[\bm{x}]

,

\mathbb{E}[\bm{x}^{\odot 2}]

, and

\mathbb{E}[\bm{x}^{\odot 4}]

. These observables combine displacement and central-shape contributions and are naturally aligned with the linear, quadratic, and quartic terms of the local driven dynamics. We further introduce a heterogeneous multi-reservoir architecture in which three reservoirs with distinct initialization and training histories form a joint

2304

-dimensional response representation. Under the fixed MNIST

60000/10000

reproduction protocol, feature-level fusion achieves the best observed accuracy of

9695/10000=96.95\%

, compared with

9682/10000=96.82\%

for the strongest single-reservoir model and

9684/10000=96.84\%

for equal-weight logit averaging. An exact paired McNemar test does not establish a statistically significant improvement over the strongest single reservoir, but the ablation and wrong-set overlap results provide suggestive evidence of complementary classification errors. These results motivate higher-order polynomial-moment readout and reservoir heterogeneity as candidate design principles for finite-time Langevin computing.

JiZheng Duan, MingYang Zhao, YanWei Chen +1

Institute of Modern Physics, Chinese Academy of Sciences

Reservoir Computing Langevin Dynamics Thermodynamics Readout Nonlinearity Neurons+1

Jul 14, 2026cs.LG

Quantum Port-Hamiltonian Neural Networks: Learning Conservative and Dissipative Dynamics via Measurement-Induced Nonlinearity

We introduce Quantum Port-Hamiltonian Neural Networks (Q-pHNNs), a family of parameterised quantum circuits that learn classical dynamics in a structure-preserving manner. The framework relies on the Isomorphic Hamiltonian Mapping (IHM): the skew-symmetric interconnection matrix

\mathbf{J}

corresponds to unitary gate evolution, and the positive-semidefinite dissipation matrix

\mathbf{R}

corresponds to Measurement-Induced NonLinearity (MINL) realised via mid-circuit measurement and classical feedforward. This ensures conservation and passivity are enforced by construction rather than penalty terms. We instantiate the IHM in four architectures: (1) a Quantum HNN that learns conservative energy manifolds and extracts Hamilton's equations exactly via the Parameter-Shift Rule; (2) a Q-pHNN using Born-rule measurement for dissipation; (3) a Q-pHNN jointly learning the energy ansatz and damping coefficient; and (4) a topology-entangled Quantum Graph Neural Network for

N

-node coupled-phasor networks. Experiments on the nonlinear pendulum and damped harmonic oscillator demonstrate: (i) ~~$1.35\%$ relative energy drift with a symplectic integrator and scale correction; (ii)~~

100\%

energy monotonicity for the MINL circuit; and (iii)~

12.1\%

error in damping-coefficient identification from vector-field snapshots with no direct supervision on the damping coefficient.

Dibakar Sigdel

1Mindverse Computing LLC, Lynnwood, WA 98087

Quantum Neural Networks Hamiltonian Quantum Circuits Nonlinearity Chaos System Identification

Jul 12, 2026stat.ML

Demixing Sparse Signals from Nonlinear Observations using Generalized Non-convex Regularization

We consider the recovery of a pair of sparse vectors from a limited number of nonlinear observations of their superposition:

y_i=g(\inner{\ba_i}{\bPhi\bw^\ast+\bPsi\bz^\ast})+e_i

,

i=1,\dots,m

, with

m\ll n

, incoherent orthonormal bases

\bPhi,\bPsi

, a scalar link

g

, and noise

e_i

that may be heavy-tailed or contaminated. We propose a regularization-based framework combining a Huberized data fidelity with generalized folded-concave penalties (SCAD, MCP), and a two-block proximal alternating algorithm with backtracking (NLD-PALM) whose whole iterate sequence provably converges to critical points under the Kurdyka--Łojasiewicz property, with local linear rates. On the statistical side we establish restricted strong convexity of the Huberized nonlinear loss through an exact sign-definite decomposition, and derive estimation error bounds of order

σ\sqrt{s\log(n)/m}

that hold at \emph{every} localized stationary point, an oracle rate

σ\sqrt{s/m}

free of

\log n

and shrinkage bias under a beta-min condition, and a co-equal recovery theorem for \emph{unknown} monotone links via a linear surrogate and a clipped Plan--Vershynin decoupling. The estimator requires no knowledge of the sparsity levels, and its guarantees hold under symmetric noise with only finite variance. Experiments at

n=512

under a frozen data-driven regularization rule show an earlier phase transition than convex

\ell_1

demixing and greedy hard-thresholding baselines, a

35\times

accuracy advantage over squared-loss estimation under

5\%

gross outliers, and successful demixing of spike-plus-background signals observed through a saturating amplifier.

Raziyeh Takbiri

School of Electrical Engineering, Iran University of Science and Technology, Tehran, Iran

Regularization Nonlinearity Signal Reconstruction Time-To-First-Spike

Jul 7, 2026stat.ML

Separation Capacity of Scattering Networks on Low-Dimensional Datasets

We aim to identify scattering network architectures that maximize the separation capacity on data with low intrinsic dimension. The networks we consider employ a fixed monomial nonlinearity and no pooling, so that the only design variable is the frame generated by the network filters. For data modeled as rectifiable sets, we first characterize and bound the separation capacity of general feature extractors in terms of the geometry of the dataset. We then particularize to scattering networks and obtain two design criteria: (i) the filters should meet the data on sufficiently many frequencies, and (ii) the matrices coupling the frame to the geometry of the data should be well-conditioned.

Konstantin Häberle, Helmut Bölcskei

ETH Zurich

Intrinsic Dimensionality Separability Separation Feature Extraction Nonlinearity Optical Networks

Jul 6, 2026cs.LG

Layer-Parallel Inference Reduces Encrypted Nonlinear Depth in Transformers

Fully homomorphic encryption (FHE) enables computation on encrypted data, but practical encrypted Transformer inference is bottlenecked by the sequential composition of many nonlinear blocks. We study whether Structured Newton Layer Parallelism (SNLP) can make this inter-layer composition more FHE-friendly: each Transformer block still requires polynomial approximations for operations such as softmax and RMSNorm, but SNLP reduces the layerwise sequential nonlinear depth from L stages to a small number of solver iterations plus linear structured corrections. Using a simulation framework based on Chebyshev polynomial approximations, we measure error accumulation under sequential versus SNLP inference across 8 models and 4 architecture families. On a 0.5B IDN-trained model, SNLP reduces symbolic bootstraps from 53 to 20 (2.65x) with only +1.2% perplexity degradation, while lowering error amplification (1.36x vs. 1.42x). Across all tested models, SNLP has lower amplification than sequential inference. Ablations show that softmax approximation dominates the error budget and CKKS arithmetic noise is negligible in our setting, suggesting that SNLP is complementary to block-level FHE-friendly operator design rather than a replacement for it.

Ligong Han, Kai Xu, Hao Wang +3

MBZUAI IFM · Red Hat AI Innovation · MIT-IBM Watson AI Lab +3

Homomorphic Encryption Transformer Architectures Layer-Wise Nonlinearity

Jul 3, 2026cs.LG

Rethinking Neural Nonlinearity as Gating

Activation functions are considered an essential primitive for neural nonlinearity, i.e., they enable neural networks to serve as universal approximators. In this paper, we show that this nonlinearity can also be achieved by input-conditioned threshold gating through branches as a universal primitive. We demonstrate that standard activations -- whether piecewise-linear (ReLU, PReLU, Hardtanh) or smooth (SiLU, Sigmoid, Tanh, GELU) -- are in fact instances of a single Threshold Gating (TG) primitive. For softmax, we show that it admits an exact TG conversion via its equivalent per-element Sigmoid form. We then validate these equivalences by converting pretrained networks across CNNs, transformer-based models, and recurrent architectures, preserving model performance without requiring retraining. Threshold Gating also enables training from scratch that goes beyond replacing existing activations, enabling gains in model compression, performance, and shorter training. We also propose a 'Minimal Branch Theorem' which relates the minimum number of required branches in our primitive to the trainability of general deep neural networks. In terms of hardware implementation, TG maps to a unified implementation in the case of analog in-memory systems, addressing the bottleneck of analog-to-digital and digital-to-analog converters (ADC/DAC) that is known to significantly impact power consumption and on-chip area.

Muhammad Sabih, Frank Hannig, Jürgen Teich

Department of Computer Science, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Germany

Activation Functions Discontinuous Gating Nonlinearity Decision Thresholds Analog

Jul 1, 2026eess.SY

GPU-Parallel Linearization Error Bounds for Real-Time Robust Optimal Control of Nonlinear and Neural Network Dynamics

This paper studies real-time robust optimal control for uncertain nonlinear systems, where linear time-varying (LTV) approximations make planning tractable but require sound linearization error bounds (LEBs) to guarantee robust constraint satisfaction. We develop tight, differentiable, GPU-parallel LEBs for LTV approximations of nonlinear and neural network (NN) dynamics. For analytic dynamics, we introduce path-based Hessian bounds that are tighter than standard interval methods. For NN dynamics, we derive certified LEBs using NN verifier-generated affine relaxations and local Jacobian corrections. We adapt a GPU-parallel system-level synthesis LTV-based robust control solver to be compatible with these LEBs by extending it to handle right-invertible disturbance matrices and non-zero-centered disturbance sets for tight zonotopic uncertainty propagation. Our method, GPUSLS-LEO, enables online optimization of robust feedback policies that account for linearization error, producing tight, formally verified reachable tubes. On complex nonlinear and NN dynamics up to 168 state dimensions, our method can compute robust control policies on the GPU at rates up to 67 Hz, reducing solve times and conservativeness relative to baselines while preserving formal guarantees and real-time performance.

Jeffrey Fang, Keyi Shen, Anutam Srinivasan +1

Nonlinear Dynamics Neural Dynamics Nonlinearity Cpu-Gpu Hybrid Designs Hessian Constraint Satisfaction

Jun 30, 2026cs.RO

Improving path-tracking performance of an articulated tractor-trailer system using a non-linear kinematic model

This paper presents a novel non-linear mathematical model of an articulated tractor-trailer system that can be used, in combination with receding horizon techniques, to improve the performance of path tracking tasks of articulated systems. Due to its dual steering mechanisms, this type of vehicle can be very useful in precision agriculture, particularly for seeding, spraying and harvesting in small fields. The articulated tractor-trailer system model was embedded within a non-linear model predictive controller and the trailer position was monitored. When the kinematic of the trailer was considered, the deviation of trailer's position was reduced substantially alongside not only straight paths but also in headland turns. Using the proposed mathematical model, we were able to control the trailer's position itself rather than the tractor's position. The Robot Operating System (ROS) framework and Gazebo simulator were used to perform realistic simulations examples.

Marina Murillo, Guido Sanchez, Nestor Deniz +2

Research Institute for Signals, Systems and Computational Intelligence, sinc(i), FICH-UNL/CONICET, Ciudad Universitaria UNL, 4◦piso FICH, (S3000) Santa Fe, Argentina.

Robust Trajectory Model Predictive Control Kinematics Steering Nonlinearity Precision Agriculture

Jun 30, 2026cs.LG

AETDICE: Unified Framework and Offline Optimization for Nonlinear Multi-Objective RL

Optimizing nonlinear preferences in multi-objective reinforcement learning (MORL) is essential for capturing complex trade-offs like risk aversion or fairness. However, such non-linearity has historically bifurcated nonlinear MORL objectives into two distinct paradigms: Scalarized Expected Return (SER) and Expected Scalarized Return (ESR). While SER requires global-level optimization and ESR requires non-Markovian policies, leading to fragmented optimization strategies, we bridge this divide through the Aggregation-Expectation-Transformation (AET) framework. By unifying both criteria through a tripartite decomposition of scalarization, AET provides a principled foundation for general nonlinear MORL. Building on this framework, we propose AETDICE, a tractable offline RL algorithm for AET objectives. By utilizing DICE-style density-ratio estimation in an augmented state space, AETDICE enables sample-based optimization from static datasets. Our framework resolves long-standing barriers and captures respective trade-offs induced by AET framework, which existing methods fail to address.

Woosung Kim, Youngjun Suh, Jinho Lee +2

1Korea University · 2Yonsei University · 3Gauss Labs Inc.

Multi-Objective Reinforcement Learning Multi-Objective Optimization Utility Maximization Nonlinearity

Jun 28, 2026cs.LG

On the Nonlinearity of Learning Rate Scaling for LLM Training

Learning-rate transfer can reduce the cost of training large language models: instead of sweeping learning rates at target scale, practitioners extrapolate from smaller runs. Existing approaches often assume that the optimal learning rate follows a log-linear scaling law in data scale and model size. We carefully examine and evaluate this scaling law. In our empirical study of GPT-2--style models from 22M to 707M parameters trained on 5B to 100B tokens, the optimal learning rate develops upward curvature at larger scales, leading to inaccurate extrapolation. We find that this curvature largely disappears when learning rates are replaced by effective learning rate (the step size in normalized weight space), and when data

D

extrapolation is used instead of model size

N

extrapolation. Next, we explain nonlinearity in scaling: weight-norm converges to equilibrium slower when optimal learning is small, requiring a larger step size to reduce the transient phase. Experiments with AdamH, which directly controls the effective learning rate, further support this explanation.

Zaiwen Yang, Huaqing Zhang, Jing Xu +1

Tsinghua University · Tsinghua University, Shanghai Qi Zhi Institute

Batch Large Language Model Training Scaling Laws Scaling Transfer Learning Extrapolation+1

Jun 26, 2026cs.CE

Higher-Order Fourier Neural Operator: Explicit Mode Mixer for Nonlinear PDEs

Neural operators provide deep neural networks for learning mappings between function spaces. Among them, the Fourier Neural Operator (FNO) is particularly effective: its spectral convolution relies on low-dimensional Fourier-domain representations and can handle inputs at different resolutions. This design aligns well with settings where the Fourier basis diagonalizes the underlying operator, such as linear, constant-coefficient PDEs on periodic domains, in which Fourier modes evolve independently. However, nonlinear PDEs may benefit from an additional inductive bias, as they exhibit structured interactions between modes, governed by polynomial nonlinearities. To capture this inductive bias, we introduce the Higher-Order Spectral Convolution, a spectral mixer that extends FNO from diagonal modulation to explicit n-linear mode mixing, aligned with the dynamics of nonlinear PDEs. Our experiments on standard benchmarks show that the proposed Higher-Order FNO (HO-FNO) retains the efficiency of FNO-based architectures and consistently improves over other spectral neural operators. HO-FNO also performs on par with or better than state-of-the-art transformers and state-space models on several datasets, with stronger gains in highly nonlinear regimes, such as the Poisson equation with polynomial forcing, where a single HO-FNO layer outperforms FNO models with up to 16 layers. We open-source our code for reproducibility at: https://github.com/AlexColagrande/HO-FNO.

Alex Colagrande, Paul Caillon, Eva Feillet +1

Miles Team, LAMSADE, Université Paris Dauphine-PSL, Paris, France · Université Paris-Saclay, CNRS, LISN, 91400, Orsay, France · ESPCI PSL, Paris, France

Fourier Neural Operators Partial Differential Equations Fourier Nonlinearity Convolutional Mix

Jun 25, 2026cs.LG

Algorithmic Foundations of Deep Learning: Complexity-Theoretic Rates and a Characterization of Universal Approximation

Feedforward neural network (NN) expressivity is typically studied by emulating optimal basis-expansion schemes. While powerful, this perspective is incomplete: it primarily captures complexity through regularity, and therefore does not distinguish intuitively simple and complicated objects with comparable regularity, such as the square-root function and a typical Brownian path. The guiding message is that neural networks should be viewed not only as flexible basis functions, but also as models of computation. If a function is computable by a real-valued circuit over a prescribed elementary gate language, then it can be computed to comparable accuracy by an NN with explicit depth, width, and non-zero-parameter bounds controlled by the depth, width, gate count, and gate structure. Thus, neural-network complexity is not governed by regularity alone, but also by algorithmic complexity. We then show that any definable NN model satisfying a natural parallelization condition, allowing possibly multivariate non-linearities such as attention or layer normalization, is a universal approximator if and only if it contains a non-affine nonlinearity. The scope of our theory is illustrated by deducing universal approximation guarantees for continuous functions, minimax-optimal approximation guarantees for Besov classes, logarithmic-error complexity for holomorphic functions, and by showing that NNs can emulate numerical algorithms such as Newton-Raphson root finding and power iteration without architecture-specific arguments. Its precision is illustrated by shortest-path computation on

k

-vertex graphs: compiling the tropical dynamic-programming circuit yields NNs with O(log(1/ε)) non-zero parameters, exponentially improving in 1/ε over the generic

O(ε^{-c k^2})

Lipschitz-approximation scale, for a constant c>0.

Anastasis Kratsios, Simone Brugiapaglia, Bum Jun Kim +2

McMaster University & Vector Institute, Toronto, ON, Canada · Concordia University, Montréal, QC, Canada · University of Tokyo, Tokyo, Japan +2

Neural Approximations Approximation Feed-Forward Deep Learning Nonlinearity

Jun 23, 2026math.NA

Dirac-Frenkel dynamics with inertia for nonlinearly parametrized solutions of evolution problems

Even when Dirac-Frenkel dynamics determine a well-defined evolution in function space, the corresponding parameter dynamics can be non-unique or ill-conditioned for redundant nonlinear parametrizations such as neural networks or mixture models. We propose to add inertia to the Dirac-Frenkel dynamics and show that this allows useful parameter velocity information to persist from the past trajectory in directions that are weakly informed, while well-informed parameter velocity directions continue to follow the Dirac-Frenkel dynamics. We prove that the inertial formulation yields well-posed parameter dynamics and provide a posteriori error bounds. After time discretization, the method requires the solution of the same type of regularized linear least-squares problem as standard Dirac-Frenkel dynamics, but with the previous velocity appearing as an anchor. Numerical experiments demonstrate the increased robustness obtained with inertia.

Matteo Raviola, Benjamin Peherstorfer

Scientific Computing and Uncertainty Quantification - CADMOS Chair, EPFL · Courant Institute of Mathematical Sciences, New York University

Inertial Nonlinearity Conditional Velocity Evolutionary Search

Jun 18, 2026cs.AI

Leveraging systems' non-linearity to tackle the scarcity of data in the design of Intelligent Fault Diagnosis Systems

Deep Transfer Learning (DTL) allows for the efficient building of Intelligent Fault Diagnosis Systems (IFDS). On the other hand, DTL methods still heavily rely on large amounts of labelled data. Obtaining such an amount of data can be challenging when dealing with machines or structures faults. This document proposes a novel approach to the design of vibration-based IFDS using DTL in condition of strong data scarcity. A periodic multi-excitation level procedure leveraging intrinsic non-linearities of real-world systems is used to produce images that can be conveniently analysed by pre-trained Convolutional Neural Networks (CNNs) to diagnose faults. A new data visualization method and its augmentation technique are proposed in this paper to tackle the typical lack of data encountered during the design of IFDS. Experimental validation on a railway pantograph structure provides effective support for the proposed method.

Giancarlo Santamato, Andrea Mattia Garavagno, Massimiliano Solazzi +1

1*Institute of Mechanical Intelligence, Scuola Superiore Sant’Anna, via Alamanni 13b, Ghezzano, 56010, Pisa, Italy. · Department of Excellence in Robotics & AI, Scuola Superiore Sant’Anna, Pisa, Italy.

Fault Diagnosis Vibration Nonlinearity

Jun 18, 2026cs.IT

Doeblin Curves

Recent research on Doeblin coefficients has shed light on their usefulness as a multi-way generalization of the Dobrushin contraction coefficient for TV distance, in a separate vein from their classic role in the theory of Markov chain ergodicity. However, strong conditions, such as being bounded away from 0, are typically necessary for Doeblin coefficients to establish the existence of information contraction. Building on recently formulated concepts of nonlinear information contraction, we aim to propose a finer-grained Doeblin-based characterization of multi-way contraction behavior which yields non-vacuous contraction guarantees even for channels whose Doeblin coefficient is 0. To this end, we introduce the notion of a Doeblin curve -- a nonlinear function which quantifies the contraction behavior of a Markov kernel on collections of input distributions at specific levels of divergence and power. Through the course of our analysis, we develop a new variational characterization of Doeblin coefficients, present several properties of Doeblin curves, define several versions of power-constrained Doeblin curves, and derive upper and lower bounds using our aforementioned variational characterization. We then utilize these results in diverse areas, including generalization bounds for noisy iterative optimization, error bounds for reliable computation with noisy circuits, and differential privacy guarantees for online iterative algorithms. In particular, we extend results in these areas to broader domains or group settings, leveraging Doeblin curves to reveal finer-grained contraction phenomena than Doeblin coefficients.

Dongmin Lee, William Lu, Anuran Makur +1

Department of Computer Science, Purdue University, West Lafayette, IN 47907, USA · Department of Computer Science and the Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA · Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA

Curves Coefficients Variational Nonlinearity

Jun 17, 2026eess.SP

Explicit Interaction Architectures for Dynamical Learning: A Controlled Study of Structural Inductive Bias

We investigate a structure-first approach to dynamical learning in which the organization of stateful interactions is prescribed explicitly rather than left entirely to a generic recurrent parameterization. We introduce causal recurrent units built from an ordered sequence of local, state-modulated transformations. The construction is motivated by wave-based interaction models, but the units studied here do not impose scattering, passivity, or energy-balance constraints. Because fixed recurrent dynamics, designed reservoir topologies, readout-only learning, and recurrent depth are already well established, the empirical question is deliberately narrower: does the proposed interaction organization provide a useful inductive bias under controlled computational conditions? We compare a one-layer structured model, a two-layer structured model, and a generic echo-state network (ESN), all with 12 recurrent states and the same strictly linear ridge readout. Each model family receives the same random-search budget on calibration data that are disjoint from the final test data, after which the selected hyperparameters are frozen. On a custom nonlinear identification task, the one-layer structured model attains a mean validation NMSE of 2.76 x 10^{-4}, compared with 3.19 x 10^{-4} for the two-layer model and 3.94 x 10^{-4} for the ESN. On NARMA10 the ordering reverses: the ESN attains 0.312, compared with 0.348 and 0.357 for the one- and two-layer structured models. Thus, the proposed organization can be competitive and advantageous on one task, but it is not universally superior; moreover, recurrent depth does not provide a systematic benefit under matched state dimension. The results support a task-dependent interpretation of structural inductive bias and position the present architecture as a controlled precursor to stronger wave- and system-theoretic constructions.

Augusto Sarti

Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB), Politecnico di Milano, Piazza L. Da Vinci 32, 20133 Italy.

Learning Dynamics System Identification Nonlinearity Deep Learning Architectures Complex Wavefield

Jun 16, 2026cs.LG

Measurement noise limits the advantage of nonlinear models over linear models in biomedical prediction

On biomedical tabular data, flexible models such as deep networks, gradient-boosted trees, and kernel methods are repeatedly matched or beaten by linear and logistic regression given the same features. The usual reaction is to treat this as a model-side shortfall, to be fixed with more data, a better architecture, or tuning, on the assumption that the nonlinear structure is there and the model has failed to capture it. We argue that these fixes cannot help when the binding limit is the measurement rather than the model, as it frequently is in biomedicine. Additive noise blurs the population-optimal predictor, and because blurring removes a function's fine, rapidly varying detail before its broad shape, it erases nonlinear structure faster than linear structure. A degree-

k

interaction is attenuated by the

k

-th power of feature reliability, while the linear part is attenuated only once. At the reliabilities typical of biomedical measurement, the nonlinear advantage can vanish even when the underlying biology is strongly nonlinear, and what the noise removes cannot be recovered by a larger cohort or a more flexible model, only by better measurement. The nonlinearity is hidden, not absent, and a tie between linear and flexible models is not by itself a verdict on the biology. These pieces are classical, drawn from measurement-error statistics, psychometrics, and Gaussian analysis, and we assemble them into an exact excess-risk identity. Measurement reliability is one of three conditions, alongside sample size and feature representation, that must align for a flexible model to help, and together they leave only a narrow window that most biomedical tasks fall outside. Across 140 UK Biobank tasks, the gap between flexible and linear models, where it exists, carries the predicted noise signature, and the three conditions can be separated by intervention but not by a benchmark alone.

Marc-Andre Schulz, Kerstin Ritter

Hertie Institute for AI in Brain Health, University of Tübingen, Tübingen, Germany · Tübingen AI Center, University of Tübingen, Tübingen, Germany · Department of Psychiatry and Neurosciences, Charité – Universitätsmedizin Berlin, Berlin, Germany +2

Nonlinearity Noisy Data Deep Network

Jun 16, 2026cs.LG

ReRAM-aware Model Finetuning addressing I-V Non-linearity and Retention Errors

Traditional CPU, GPU, and NPU architectures are increasingly limited by the von Neumann bottleneck. While In-Memory Computing (IMC) using ReRAM crossbar arrays offers a high-density, energy-efficient alternative, its practical deployment is constrained through their non-idealities. Existing hardware-aware training frameworks often require training from scratch, which is computationally prohibitive for modern large-scale models. In this work, we propose a finetuning-based hardware-aware training algorithm that enables robust DNN deployment on ReRAM with minimal training overhead. Our approach mitigates I-V non-linearity by applying a range-shrunk sinh transformation and incorporates retention errors directly into a regularization loss during the finetuning process. We evaluate our framework across models and tasks such as image classification and question-answering (QA). Experimental results demonstrate that our method achieves similar accuracy on large-scale models like ResNet18 and DeiT-Tiny as the base model. In-case of ImageNet for MobileNetV3 families the technique has only less than 2% accuracy degradation. Further, applying the technique on the SQuAD v2 dataset results in only 1 point degradation of F-1 score.

Ching-Yi Lin, Shamik Kundu, Arnab Raha +1

University of Maryland, College Park, MD, USA · Intel Corporation, 2200 Mission College Blvd, Santa Clara, CA 95054, USA

In-Memory Computing Neural Network Inference Reram Crossbar Mobilenetv2 Retention Nonlinearity

Jun 12, 2026cs.LG

Graph Structured Combinatorial Semi-Bandit with Nonlinear Reward Associations through Separable Signals

The identification of optimal structures within vast arrays of interconnected data necessitates significant sampling- and computational effort. Learning and leveraging underlying signal dependencies can improve efficiency and predictive capabilities considerably, but the ubiquity of nonlinear statistical relations amplifies the complexity of such undertakings. In this paper, we develop novel generic and adaptive strategies equipped with routines for graph-based causal reward modeling, analytic reproducing kernel methods, and Taylor approximation of functional processes. We establish theoretical performance guarantees sublinear in time and linear in data volume over time. Our analyses cover robustness to a multitude of uncertainties arising from noise interference, gradual model convergence, and solution space mismatch. The framework's general appeal is substantiated by a minimalistic set of conditions or reliance on prior estimates, while various outlined modifications address specific or extended settings. To demonstrate practical effectiveness, we conduct numerical experiments using both benchmarked synthetic and real-world transportation datasets.

Christoph Bauschmann, Setareh Maghsudi

Department of Electrical Engineering and Information Technology Ruhr University Bochum Bochum, Germany

Nonlinearity Convergence

Jun 12, 2026cs.IT

Nonlinear Two-Time-Scale Stochastic Approximation: A Sharp Phase Transition and How to Beat It

Recent finite-time analyses of nonlinear two-time-scale stochastic approximation show that under contractive assumptions the slow iterate

Y_k

with stepsizes

β_k=Θ(k^{-1})

and

α_k=Θ(k^{-a})

,

a\in(1/2,1)

, generally satisfies a mean-square rate of order

k^{-a}

; decoupled

k^{-1}

rates require strong local linearity. We identify a sharp regularity-dependent boundary. In a rate-determining normal form where the slow drift contains a locally linear leakage and a nonlinear remainder of order

1+ρ

(

ρ\in[0,1]

), the uncorrected recursion satisfies

\mathbb{E}\|Y_k\|^2 \le C\bigl(k^{-1}+k^{-a(1+ρ)}\bigr),

and a matching scalar Gaussian lower bound shows that the slower term is unavoidable without modifying the update. Thus the decoupled

k^{-1}

rate is guaranteed for the uncorrected recursion exactly when

a(1+ρ)\ge 1

. This lower bound concerns only the naive update; it is not an information-theoretic obstruction. We demonstrate this by equipping the normal-form recursion with an auxiliary online bias estimator

M_{k+1}=M_k+γ_k(R(X_k)-M_k),\qquad β_k\llγ_k\llα_k,

and subtracting

M_k

from the slow update. Under the same stability, moment, and remainder assumptions, the corrected recursion achieves

\mathbb{E}\|\widetilde Y_k\|^2=O(k^{-1})

for every

ρ\in[0,1]

, including regimes where the uncorrected update provably suffers the slower rate. Finally, we prove localized transfer theorems that extend the phase-transition mechanism to general nonlinear TTSA in fast-manifold coordinates. The proofs are non-asymptotic and rely on two Abel-transform cancellations: one for the locally linear fast-error leakage, and one for the tracked nonlinear bias.

Dhruv Sarkar, Vaneet Aggarwal

1Indian Institute of Technology Kharagpur · 2Mohamed bin Zayed University of Artificial Intelligence · 3Purdue University

Stochastic Approximation Finite Time Analysis Nonlinearity Phase Transitions Transitions Data Leakage

Jun 12, 2026cs.LG

How Linear Is a Transformer Feed-Forward Block? Per-Block Linear Recoverability Is Learned, Not Architectural

Transformer feed-forward networks (FFNs) are often treated as nonlinear stores of computation, yet how nonlinear a trained FFN block actually is has rarely been measured. We treat each FFN as a position-wise input-to-output map and split it into the exact least-squares linear approximation plus a residual. The held-out variance the closed-form linear map explains defines a block's linear recoverability (R^2_lin), an optimiser-free measure of its linearity. Across all twelve blocks of GPT-2, Pythia-160m, and llama-160m, R^2_lin is highly heterogeneous and non-monotone with depth, ranging from near-linear (>0.99) to strongly nonlinear (<0.3) between adjacent blocks, and is not set by the activation function: same-width GELU models GPT-2 and Pythia-160m have sharply different profiles, so recoverability is a learned property of individual trained blocks, not an architectural one. A low-rank bilinear probe of the residual recovers only a few points of R^2, with gain uncorrelated with residual nonlinearity: the unrecovered computation is not a single position-wise product but higher-order or distributed structure. The measurement also serves as a targeted compression signal: recoverable blocks admit large single-layer replacements (GPT-2's early FFN at 8x fewer parameters for +0.77 perplexity), while low-recoverability blocks flag where this is unsafe. It further exposes a methodological pitfall: trained linear baselines can badly under-converge on ill-conditioned transformer activations, so we report the exact closed-form least-squares ceiling throughout.

Stuart Whipp

Independent Research

Feed-Forward Transformer Architectures Nonlinearity

Jun 9, 2026cs.CE

Non-linear mechanical field reconstruction coupling recurrent neural networks with physics-informed graph neural networks

Reconstructing local stress fields in heterogeneous microstructures under non-linear, history-dependent loading remains a major computational bottleneck in multi-scale simulations. We propose a coupled LSTM-GNN framework that links the temporal and spatial aspects of local stress field reconstruction. A Long Short-Term Memory network encodes macroscopic stress-strain sequences into a compact hidden state that captures the path-dependent constitutive response, while a physics-informed Graph Neural Network reconstructs the spatially-resolved stress field at each time step. We introduce a relative weighting strategy with linear warm-up to balance the data-driven reconstruction loss and a discrete divergence-based equilibrium penalty. This resolves the scale mismatch that prevents fixed-weight formulations from converging in the elasto-plastic regime. The model is trained on 10,000 non-proportional loading paths applied to a periodic plate-with-a-hole microstructure and von Mises elasto-plasticity. The model achieves three orders of magnitude speedup over finite element simulations and generalizes to loading sequences twice the training length, with 1.9% cumulative error. Because the graph relies on mesh connectivity instead of the specific element type, one trained surrogate can be applied directly without retraining to meshes with different element types and to both coarser and finer resolutions, while in all cases reproducing the high-fidelity quad-element FE field used during training. Indeed, the message passing characteristics inherent to GNN and MeshGraphNet architecture render the model mesh-agnostic. Analysis of the LSTM hidden states suggests a low-dimensional structure related to the internal state variables of the constitutive model.

Manuel Ricardo Guevara Garban, Yves Chemisky, Étienne Prulière +3

Univ. Bordeaux, CNRS, Bordeaux INP, I2M, UMR 5295, Talence, France · Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, Talence, France · Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Freiberg, 09599, Germany +2

Finite Element Method Porous Media Parametric Physics-Informed Neural Network Recurrent Neural Networks Nonlinearity

Jun 9, 2026cs.RO

Model-based Optimization of Anguilliform Swimming Gaits for Soft Robotic Applications

In this paper, we introduce the Soft Lamprey-Inspired Dual Environment Robot (SLIDER) and a proper modeling and optimization procedure employed to design the robot. We represent the primary fluid environment actions - inertial effects, vortex forces, and viscous dissipation - using Lighthill's theory for large-amplitude elongated bodies. For structural design parameters such as internal pressure, tail size, and body stiffness, a fast, geometrically and materially nonlinear model is developed and validated. The fluid-structure interaction equations are solved implicitly with an efficient second-order box method. A pneumatic manifold robotic system is employed to actuate SLIDER in a quiescent water tank environment, allowing cross-comparison of computational and experimental results. We find that low-frequency swimming is dominated by resistant environmental forces, whereas higher-frequency swimming is primarily affected by inertial fluid forces. Using our efficient model alongside a genetic algorithm, we co-optimize a swimming control pattern and caudal fin design (subject to SLIDER's climbing morphology) to achieve a tethered swimming speed of 21.7 +/- 0.4 cm/s (0.59 Bl/s). Furthermore, we investigate the optimization procedure for a multimodal robot performing both swimming and climbing tasks.

Brian Van Stratum, James Gallentine, Caleb Rucker +3

FAMU/FSU College of Engineering, Tallahassee, FL 323109 · Mechanical Engineering, Vanderbilt University, Nashville Tennessee 37235 · Mechanical Engineering, The University of Tennessee, Knoxville, Tennessee 37996

Soft Robotics Fluid Dynamics Soft Nonlinearity

Jun 7, 2026stat.ML

Generalization in Nonlinear Least Squares via Learned Feature Geometry

We study the generalization of ridge-regularized nonlinear least-squares models via on-average algorithmic stability, deriving error bounds for local minimizers in terms of a data-dependent effective dimension that reflects the geometry of the gradient model at the trained parameters, through the empirical Jacobian Gram matrix and a residual-curvature term. In the linear case, where the curvature term vanishes, this recovers the classical effective dimension of the Jacobian kernel covariance, but evaluated at the trained model rather than at initialization as is typical in neural tangent kernel analyses. We further bound this effective dimension via covering complexity of the gradient features, leading to guarantees that depend on learned geometry rather than parameter count. In particular, for manifold-supported data and piecewise Lipschitz Jacobians, the bounds scale with intrinsic dimension, while for one-hidden-layer ReLU networks, the mechanism can be made explicit through counts of activation-stable regions. Experiments on synthetic manifolds, clustered distributions, and benchmark datasets illustrate trained-Jacobian compression, the tightness of the residual-curvature linearization, and agreement between the stability bound and observed generalization gaps. A key feature of our bounds is the simplicity of their derivation, which follows from first principles using the Brascamp-Lieb inequality under strongly log-concave noise.

Ayub Kharel, Ilja Kuzborskij, Patrick Rebeschini +1

University of Oxford · Google DeepMind · Sapient Intelligence

Generalization Bounds Least Squares Jacobian Neural Tangent Kernel Gradient Descent Nonlinearity+1

Jun 2, 2026cs.LG

PE-MHL: Physics-Encoded Modular Hybrid Layers for Scalable Learning of Complex Systems

Hybrid models that combine physics-based and data-driven components have shown strong potential for achieving accuracy and interpretability in control applications. While recent methods have made progress in incorporating physical consistency, challenges remain in scalability, robustness to noise, and control of model complexity. This paper proposes a Physics-Encoded Modular Hybrid Layer (PE-MHL) framework, in which a baseline physics-based model is incrementally refined through the addition of new sub-models, where each new component adds complexity while preserving what previous components have already learned. We establish a theoretical guarantee for this construction: with a least-squares initialization of each new sub-model, the training error is monotonically non-increasing in the number of sub-models and provably converges. Empirical evaluations on a nonlinear NARX benchmark and the Quanser Aero 2 platform demonstrate that PE-MHL outperforms equivalently sized monolithic networks in both accuracy and generalization, while also providing more stable training dynamics and better preservation of underlying data structures.

Ismail Hassaballa, Mircea Lazar

Eindhoven University of Technology Eindhoven, Netherlands

Multilayer Perceptrons Nonlinearity

Jun 1, 2026cs.AI

S3TS: Stochastic Scenario-Structured Tree Search for Advanced Planning Under Uncertainty

Effective scheduling in the energy sector is essential to ensure the reliable operation of electrical grids and their connected assets by, for instance, optimizing the dispatch of generation units and storage systems. An effective planning strategy must (a) accommodate advanced and potentially non-linear system models -- exploiting the increasing data availability of modern grids, and (b) explicitly handle uncertainties arising, for instance, from the integration of renewable energy sources. While existing approaches can address either non-linearity (e.g., Monte Carlo Tree Search) or uncertainty (e.g., stochastic mathematical optimization), there is a lack of planning techniques capable of addressing both challenges simultaneously. To bridge this gap, we propose a Stochastic Scenario-Structured Tree Search (S3TS) algorithm that explicitly represents uncertainty through scenario trees while enabling the integration of advanced non-linear models. We evaluate S3TS on a simulated demand response signal publication problem, largely mimicking the imbalance settlement mechanism in Belgium. The results demonstrate near-optimal performance in linear, analytically tractable settings, with costs within 14% of the mathematically optimal solution conditioned to the scenario trees. In highly non-linear scenarios, S3TS significantly outperforms baseline methods, achieving cost reductions of up to 51% and 5.4% compared to a myopic algorithm and deterministic MCTS, respectively.

Fabio Pavirani, Bert Claessens, Pierre Pinson +1

IDLab Ghent university – imec, Technologiepark Zwijnaarde 126, 9052 Gent, Belgium · Beebop.ai, Belgium · Imperial College London, London, UK

Demand Response Power Systems Monte Carlo Tree Search Stochastic Optimization Tree Search Stochastic+2

May 31, 2026cs.LG

Learning-based Directed Graph Abstraction of Combinatorial Spaces for Order-Preserving Search in Mixed-Combinatorial Nonlinear Optimization

Mixed-combinatorial nonlinear programming (MCNLP) problems arise in many engineering design and planning applications, e.g., due to categorical, component, and geometric design choices, as well as joint task and motion planning. Traditional representations of combinatorial spaces, such as integer or binary encoding, often introduce spurious relations, increase dimensionality, and require additional compatibility constraints. Instead, this paper draws on recent developments in robot planning and vehicle/network routing domains that aim to learn search heuristics over combinatorial spaces using graph neural networks (GNNs). More specifically, this paper presents a first-of-its-kind structured abstraction of the combinatorial space by learning a mapping from an undirected fully connected graph of combinations to a directed graph indicating improvement directions using an Edge Field Graph Network (EFGN). To demonstrate the utility of this new way of abstracting the combinatorial space in solving MCNLPs, we adopt a recent optimization framework that purely searches over the non-combinatorial (e.g., continuous) variables and retrieves the best-suited combination for each candidate design by using the abstraction model, akin to a recommender system. The presented direction-aware abstraction model provides a potentially more scalable and interpretable retrieval of combinations compared to the original recommendation system in that framework. For evaluation, the proposed method is integrated with a well-known particle swarm optimization and genetic algorithm solvers on three benchmark nonlinear problems with varying numbers of combinations and variables. Compared to baseline solvers using indexified combinations, the GNN-based recommender consistently achieves better mean optimum values and robustness across multiple runs.

Gishnu Madhu, Feng Liu, Souma Chowdhury

University at Buffalo Buffalo, New York, 14260

Particle Swarm Optimization Robot Planning Nonlinearity

May 31, 2026physics.optics

Breaking the Cascade: Compact Nonlinear Optical Computing with Single-Layer Encoder-Decoder Co-Localization

We demonstrate that nonlinear computing can be achieved with a single linear diffractive surface under coherent illumination. We introduce a compact encoder-decoder co-localization (E+D) architecture in which an input-dependent dynamic encoder and a static optimized decoder are integrated within the same phase-only diffractive plane. Following free-space propagation, coherent interference between the encoder and decoder fields, combined with intensity detection, generates programmable nonlinear input-output mappings without requiring nonlinear optical materials or multiple diffractive layers. We prove that the proposed E+D optical processor is a universal approximator for arbitrary real-valued band-limited nonlinear functions and identify the physical factors governing its approximation fidelity, including the decoder degrees-of-freedom, detector aperture, and axial propagation distance. Crucially, we demonstrate that introducing a trained, frozen phase bias to the encoder region systematically enhances functional expressivity, providing robustness against coarse phase quantization on spatial light modulators. Using this framework, we accurately synthesize diverse nonlinear functions, including commonly used neural network activation functions and complex-valued nonlinear functions. Finally, we experimentally validate the proposed approach using a visible-light optical set-up trained through in situ learning, demonstrating the parallel approximation of 9 nonlinear functions in a single optical forward pass. By collapsing nonlinear optical computation into a single diffractive surface, the E+D architecture substantially reduces hardware and alignment complexity while preserving powerful function-approximation capabilities, providing a compact and scalable framework for analog information processing.

Yuntian Wang, Alexander Chen, Md Sadman Sakib Rahman +1

1Electrical and Computer Engineering Department, University of California, Los Angeles, CA, 90095, USA

Photonic Transformer Accelerators Nonlinearity Spatial Encoding

May 29, 2026cs.RO

Series-Parallel Integrated Nonlinear Elastic Actuator applied to the lean motion of a bicycle simulator

Designing robots for high-torque, high-fidelity haptic interaction is challenging. Parallel Elastic Actuators (PEAs) use elastic elements in parallel to smaller motors to complement torques, and Series Elastic Actuators (SEAs) use elastic elements in series to decouple motor impedance and improve force control. Recent work combines SEAs and PEAs to obtain both benefits but requires separate elastic elements or clutching. This paper presents the Series Parallel Integrated Nonlinear Elastic Actuator (SPINEA), which merges SEA and PEA such that a single elastic element takes on dual roles simultaneously, parallel and series. This is achieved by a nonlinear transmission in which the motor and load have misaligned rotation axes and are elastically connected. This geometry enables both high peak torque and precise torque tracking. We apply SPINEA to actuate lean of a haptic bicycle simulator, which requires high moments and precise rendering for safe and realistic rider interactions. We realized a prototype and performed experiments, both with an external excitation setup and with riders cycling. Our results confirm SPINEA's low impedance and precise torque tracking, up to 4.25 Hz with the bicycle frame fixed and up to 4 Hz with riders. The benefits may transfer to other applications requiring compact, high-performance actuation.

Christina Kohler, Michiel Plooij, Nuria Peña-Perez +2

Institute of Automatic Control, RWTH Aachen University, 52074 Aachen, Germany · Demcon Life Sciences & Health, 2628 Delft, The Netherlands · Hapticlink Technologies, 2611 Delft, The Netherlands +2

Soft Pneumatic Actuators Haptic Feedback Nonlinearity

May 29, 2026cs.LG

Flow map learning in nonlinear vector autoregressive models: influence of the feature-library structure on the training error

Time series forecasting often requires learning nonlinear and time-delayed dependencies. A paradigmatic class of forecasting models are nonlinear vector autoregressive processes (NVAR), also known as next-generation reservoir computers (NG-RCs). These models approximate the Koopman operator on the space spanned by their explicit feature library. We consider the identifiability problem for learning Markovian nonlinear dynamical systems and show that the training error as a function of time resolution follows characteristic (pre-)asymptotic scaling laws. These laws depend on whether the feature library can represent the early Lie-series coefficients of the flow map (propagator) exactly or merely approximately. For dynamical systems governed by polynomial vector fields, we demonstrate the mechanism for NVAR/NG-RC models with monomial and Fourier feature libraries. We determine the dependence of the training error on the temporal resolution, the involved nonlinear degree, and the number of delay terms. While delay terms reduce the optimal one-step training error, they improve long-horizon forecasts only when the library provides sufficient nonlinearity. Thus, small training error coexists with weak generalization as the model class is mismatched to the true data-generating process. Numerical experiments on various chaotic dynamical systems confirm the theoretical predictions.

Markus Gross

Institute for AI Safety and Security, German Aerospace Center (DLR), Sankt Augustin and Ulm, Germany

Nonlinear Dynamics Nonlinearity Reservoir Computing Autoregressive Model Flow Map Markov

May 26, 2026physics.flu-dyn

Sparse POD Mode Selection and Manifold Dimensionality Reduction with Neural Networks

Linear dimensionality reduction methods such as proper orthogonal decomposition (POD) make high-dimensional data amenable to analysis by identifying the principal components, or modes, that capture the most variance, or energy, in the data and constructing a low-dimensional representation in the subspace they span. Such linear methods struggle, however, for data with slowly decaying Kolmogorov

n

-widths, such as advection-dominated and turbulent flows, which require many modes for accurate reconstruction; moreover, energy-based truncation can discard low-energy modes needed to capture small-scale features. Recent nonlinear manifold methods using polynomial mappings with alternating or greedy mode selection achieve better reconstruction with fewer modes, but fix the form of the nonlinear mapping a priori, limiting expressivity. In contrast, neural network (NN) manifolds offer greater expressivity yet employ energy-based selection. We present SparseModesNet, a dimensionality reduction framework that employs linear encoding and nonlinear NN decoding. The decoder leverages LassoNet, a method enforcing hierarchical sparsity through a residual connection with a linear skip layer, to simultaneously select informative modes and learn a nonlinear mapping that minimizes reconstruction error. On benchmark advection-dominated and chaotic flows, SparseModesNet matches or exceeds state-of-the-art performance. For turbulent channel flow at friction Reynolds number

Re_τ= 5200

, our method reduces reconstruction error by 51-78% compared to existing polynomial manifold methods while maintaining interpretability through physically meaningful mode selection.

Tomoki Koike, Prakash Mohan, Marc T. Henry de Frahan +2

School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA · Computational Science Center, National Laboratory of the Rockies (NLR), Golden, CO

Dimensionality Reduction Reynolds-Averaged Navier-Stokes Low-Dimensional Structure Data Manifold Nonlinearity Neural Network

May 26, 2026math.PR

On the Subgaussianity of Quantized Linear Maps: An AI-Assisted Note

This short note presents a dimension-independent subgaussian concentration bound for Gaussian vectors under coordinate-wise nonlinear mappings. Discovered by Gemini 3.5 Flash, this result applies to any bounded function under a well-conditioned covariance. We apply this tool to answer a question of Simone Bombari on sign-quantized linear maps

Y = \text{sgn}(Wx)

.

Guangyi Zou, Roman Vershynin

Department of Mathematics, University of California, Irvine

Non-Gaussianity Gaussian Primitives Concentration Bounds Covariance Nonlinearity

May 26, 2026cs.LG

Nonlinear Data Integration via Kernel Methods for Data Collaboration Analysis

Collaborative analysis of decentralized confidential datasets is important, but direct sharing of original datasets is often restricted by privacy and institutional constraints. Data collaboration (DC) analysis transforms each dataset into privacy-preserving intermediate representations via party-specific obfuscation functions and integrates them into common collaboration representations using an anchor dataset. However, many existing DC analysis methods rely on linear transformations for data obfuscation and integration, which may increase reconstruction risk. Although nonlinear dimensionality reduction can mitigate this risk, conventional linear integration methods cannot accurately align intermediate representations produced by nonlinear transformations. Moreover, existing integration methods mainly minimize discrepancies among parties and do not explicitly incorporate geometric or target-variable information useful for downstream analysis. To overcome these limitations, we first formulate linear kernel integration (LKI) as a linear integration method and then kernelize it to obtain nonlinear kernel integration (NKI). NKI admits a globally optimal solution via kernel ridge regression and an eigenvalue problem. We also introduce graph regularization and a centering constraint so that the target representation can capture geometric and target-variable information useful for downstream analysis. Experiments on image classification tasks demonstrate that NKI improves classification accuracy over existing linear integration methods under nonlinear dimensionality reduction, with further gains from target-variable-aware graph regularization and centering. The results also show that dimensionality reduction choices substantially affect both classification accuracy and reconstruction risk.

Yamato Suetake, Yuta Kawakami, Shunnosuke Ikeda +1

Graduate School of Science and Technology, University of Tsukuba, Tsukuba, Ibaraki, Japan · Institute of Systems and Information Engineering, University of Tsukuba, Tsukuba, Ibaraki, Japan

Kernel Method Dimensionality Reduction Privacy-Preserving Machine Learning Nonlinearity Collaboration

May 26, 2026cs.CV

Chaos-SSL: An Attention-Based Self-Supervised Learning Framework with Chaotic Transformation for Medical Image Classification

Self-Supervised Learning (SSL) has emerged as a powerful paradigm to mitigate the reliance on large, annotated datasets, a common bottleneck in medical image analysis. However, standard SSL methods, which rely on simple geometric and color augmentations, may fail to capture the fine-grained, complex textural details necessary for classifying subtle pathologies. This paper introduces Chaos-SSL, a novel two-stage framework for medical image classification. In the first stage, we propose a new self-supervised pre-training strategy that leverages 1D chaotic maps (Logistic, Tent, and Sine) as a complex, non-linear augmentation for contrastive learning. We hypothesize that these chaotic transformations create ``harder'' and more semantically-rich views, forcing a network to learn robust representations of fine-grained medical textures. In the second stage, we introduce an attention-based fusion model that dynamically combines the specialized features from our Chaos-SSL model with the general-purpose features of a larger, ImageNet-pre-trained model. We validate our method on two public datasets: ISIC 2018 (skin lesions) and APTOS 2019 (diabetic retinopathy). Our results demonstrate that the Chaos-SSL model pre-trained with a Tent map for 30 epochs, followed by attention fusion, achieves performance fully competitive with the state-of-the-art, yielding an accuracy of 0.9261 on ISIC 2018 and 0.8726 on APTOS 2019. This significantly outperforms existing SSL methods, including several recent approaches.

Joao Batista Florindo

Institute of Mathematics, Statistics and Scientific Computing, University of Campinas, Rua Sergio Buarque de Holanda, 651, Campinas, Brazil

Medical Image Classification Self-Supervised Learning Texture Chaos Nonlinearity

May 25, 2026stat.ML

Learning Nonlinear Factor Models with Unknown Monotone Links from Incomplete and Noisy Data

We study a nonlinear factor model in which observed responses depend on low-rank latent factors through an unknown monotone link function. This setting is challenging and largely underexplored due to severe nonconvexity and identifiability issues. The link function is assumed to lie in a reproducing kernel Hilbert space (RKHS), enabling flexible nonparametric modeling while preserving identifiability. We formulate the problem as the joint recovery of the low-rank factors, loadings, and the nonlinear link function from possibly incomplete and noisy observations and propose a projected block coordinate descent (BCD) algorithm with explicit regularization to address scale and rotational ambiguities. Under mild incoherence of factors and standard sampling conditions, we establish convergence guarantees in both noiseless and noisy regimes, along with sublinear regret bounds for the link-function updates. Our results extend classical linear factor models to a broad nonlinear regime and provide a principled framework for learning nonlinear latent structures. We evaluate the proposed approach using controlled synthetic experiments, indicating promising performance.

Yutong Chao, Resat Gökhan, Jalal Etesami +1

School of Computation, Information and Technology, Technical University of Munich, Germany · Munich Institute of Robotics and Machine Intelligence · Department of Economics, Virginia Tech, USA

Exploratory Factor Analysis Nonlinearity Low-Rank Structure Monotonicity Noisy Data Incomplete

May 22, 2026cs.LG

Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions

Classical archetypal analysis is appealing for its interpretability, but its linear geometry can limit performance on data with strongly non-linear structure; at the same time, existing neural extensions improve flexibility while often weakening the geometric meaning of archetypes and interpolations. In this work, we develop a Riemannian version of archetypal analysis based on data-driven pullback geometry for real-valued data, with the goal of combining the interpretability of classical archetypal analysis with the expressive power of modern non-linear models. We introduce a class of deformed star distributions together with associated pullback Riemannian geometry to provide a statistical interpretation of the resulting manifold mappings, define the Riemannian archetypal mapping (RAM) as a projection onto the manifold of geodesically convex combinations of archetypes, and propose a practical optimization scheme based on convex relaxation followed by non-convex refinement. We further propose a learning scheme that yields reasonable, albeit generally suboptimal, deformed star distributions from data. Experiments on synthetic examples and MNIST show that the resulting framework produces meaningful geodesics, useful denoising projections, and geometry-aware classifications, while also clarifying where current optimization limitations remain.

Willem Diepeveen, Deanna Needell

Department of Mathematics University of California, Los Angeles Los Angeles, CA 90095, USA

Riemannian Manifolds Nonlinearity Distributions Interpretability

May 22, 2026cs.LG

Coupling-Robust Accuracy in Multiphysics Physics Informed Neural Networks via Kronecker-Preconditioned Optimization

Physics-informed neural networks (PINNs) for coupled multiphysics systems suffer systematic accuracy degradation as inter-equation coupling strengthens. We provide a theoretical explanation through neural tangent kernel (NTK) analysis: for linearly coupled systems, the standard NTK's spectral radius grows as

Ω(γ^2)

with coupling strength

γ

, shrinking the stable learning rate, while block-diagonal Gauss--Newton (GN) preconditioning yields a preconditioned NTK

K_P = JH^{+}J^\top

whose spectral radius is bounded by

S

(number of networks), independent of

γ

. Adam's diagonal preconditioning destroys this projector structure -- inflating

λ_{\max}

far above

S

for any coupling type -- and its residual-dynamics kernel grows as

Θ(γ)

, placing its stable learning rate strictly between gradient descent and GN. For one-way coupling the limitation is class-wide: no diagonal preconditioner, fixed or adaptive, halves the driving residual in fewer than

Ω(γ)

iterations (

Ω(γ^2)

if fixed), whereas block-diagonal GN requires

O(1)

. We verify

Ω(γ^2)

growth across linearly coupled benchmarks and confirm

λ_{\max}(K_P) = S

in all three 1D systems, including nonlinearly coupled NP+P. Combining the Kronecker-preconditioned optimizer SOAP with inverse-gradient-norm loss balancing (SOAP+GradNorm) yields coupling-robust accuracy: across 222 experiments spanning three 1D systems and a 2D electroosmotic flow benchmark, SOAP+GradNorm maintains final-epoch

L_2

accuracy across coupling strengths, with

\leq 2.3\times

degradation in nonlinear NP+P while Adam+GradNorm fails (

L_2 > 0.1

). SOAP+GradNorm further scales to a 2D, 6-PDE electroosmotic flow at EDL-resolved conditions down to

\varepsilon = 0.01

-- a regime all prior PINN electrokinetics studies have avoided -- where Adam+GradNorm fails entirely (

L_2 > 0.3

).

Youngjae Park, Jaemin Kim, Junghwa Hong

Dept. of Control and Instrumentation Engineering, Korea University, Sejong, South Korea · BK21 FOUR Smart Mobility Education and Research Team, Korea University, Sejong, South Korea

Parametric Physics-Informed Neural Network Spectral Preconditioning Neural Network Nonlinearity

May 19, 2026cs.LG

B-cos GNNs: Faithful Explanations through Dynamic Linearity

We introduce B-cos GNNs, an inherently explainable class of graph neural networks whose predictions decompose exactly into per-node, per-feature contributions via a single input-dependent linear map. B-cos GNNs use linear (sum-based) aggregation and replace non-linear message and update functions with B-cos transforms. This induces meaningful, task-specific weight-input alignment that is directly accessible through the model's dynamic linearity. Instance-level explanations follow from a single forward and backward pass, requiring no auxiliary explainer, modified learning objective, or perturbation procedure. Instantiated as a GIN, our approach trades small losses in predictive accuracy for state-of-the-art explainability across diverse synthetic and real-world benchmarks, producing explanations orders of magnitude faster than post-hoc baselines.

Joschka Groß, Mohammad Shaique Solanki, Verena Wolf

Saarland Informatics Campus, Saarland University · DFKI

Explainability Graph Neural Networks Nonlinearity Faithful Explanation