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44 papers in the last four weeks, up 159% on the four weeks before. 0.4% of all new papers.

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Latest papers 319

Jun 11, 2026math.NA

Neural Architectures as Functional Priors in Physics-Informed Control Problems

In this work we investigate the role of neural architectures as implicit functional priors in control problems governed by ordinary differential equations. Rather than focusing on highly complex problems, our objective is to investigate architecture-dependent effects in controlled dynamical systems within the simplest physically interpretable settings possible. In particular, we study a controlled linear RLC electrical circuit and a nonlinear Duffing-type dynamical system. Both systems are analyzed first through classical optimal-control formulations and later through PINN-based approaches. We compare different combinations of multilayer perceptrons (MLPs) and Fourier-based KAN-like architectures, and analyze their influence on the resulting controls. The numerical experiments suggest that different architectural choices systematically generate qualitatively distinct controls, even under identical governing equations, loss functionals, initial and target states, training parameters and physical constraints. Significant differences appear in the spectral structure, smoothness, energy distribution, and phase-space behavior of the learned solutions. A central observation of this work is the emergence of a functional specialization phenomenon when the neural architectures are allowed sufficient freedom to shape the structure of the learned controls. More specifically, in the systems considered here, Fourier-based architectures tend to produce trajectories with richer oscillatory content, whereas smoother low-frequency-biased architectures tend to generate more regular and energetically efficient controls. This suggests that different functional components of the control problem may be handled more efficiently by different neural architectures, leading to an implicit specialization between state representation and control generation.

Sonia Rubio Herranz, Fernando Carlos López Hernández, Antonio López Montes

Department of Statistics and Operations Research (EIO), Universidad Complutense de Madrid, 28040 Madrid, Spain · Department of Software Engineering and Artificial Intelligence, Universidad Complutense de Madrid, 28040 Madrid, Spain · Department of Analysis and Applied Mathematics, Universidad Complutense de Madrid, 28040 Madrid, Spain

Dynamical Systems Physics-Informed ML Optimal Control

Jun 11, 2026cs.LG

Order Is Not Control: Driven-Dissipative Response Laws Across Artificial and Biological Systems

AI alignment, interpretability, steering, and neural perturbation studies identify order-inducing objects. We argue that order is not control. Control requires a receiver-gated response law: a denominator-indexed operator mapping material state, action/drive, bath, and receiver state to response displacement, sinks, effort, and basin projection. We identify it across biological, LLM, adapter, and stochastic-operator panels. The laws are local: an intervention can be admitted, saturated, sign-changing, leaky, or overdriven depending on medium, bath, receiver state, action port, and comparator. Control is assigned when finite effort moves a target or outcome-readout class under the same denominator while damage, null/evasive, invalid format, overdrive, and unnecessary effort stay bounded. Mouse ALM, C. elegans, and zebrafish panels provide physical response-operator evidence while excluding coordinate identity and controller conclusions. LLM panels show generated-output response laws: across four material conditions, response vectors are predictable at 72.8-73.7% component-sign accuracy, rising to 84.3-84.8% on nonzero components; held-out observers predict system-effect and target/oracle families at 93.6% and 91.7% accuracy. Constitution-conditioned adapters reshape susceptibility as prepared media, and stochastic-operator panels separate measured opportunity from deployable action policies. This gives a driven-dissipative response-system account at the mesoscopic control level: drives act through prepared media, baths, and receivers, producing admitted movement, impedance, sinks, or overdrive. The evidence supports local admitted control and measurable stochastic response operators, while leaving deployable pre-generation control, hidden/logit causal sufficiency, biological-to-LLM coordinate identity, and literal thermodynamic quantities outside scope.

Gareth Seneque, Lap-Hang Ho, Nafise Erfanian Saeedi +2

Australian Broadcasting Corporation

Dynamical Systems Language Model Steering

Jun 10, 2026cs.LG

Two-Layer Linear Auto-Regressive Models Estimate Latent States

Auto-regressive models have emerged as powerful tools for sequential data, from language to video. Understanding how and why these models learn latent representations remains an open theoretical question. In this work, we demonstrate that when trained by empirical risk minimization on data from partially observed linear dynamical systems, two-layer linear auto-regressive models naturally learn to approximate Kalman filtering. In particular, we show that the learned hidden representation coincides, up to a similarity transformation, with the state estimates produced by the optimal (Kalman) filter, even though the model has no explicit knowledge of the underlying dynamics or state. The result follows from three main insights. First, we establish that the Kalman filter is well approximated by an auto-regressive model with bounded truncation error. Second, we show that despite non-convexity, the two-layer optimization landscape is benign, i.e., all stationary points are either strict saddles or global minima. Finally, as our main contributions, we provide finite-sample guarantees on prediction error, parameter estimation error, and latent state recovery. Numerical simulations support the theoretical results and demonstrate that the latent representations of auto-regressive models recover state estimates.

Yahya Sattar, Sunmook Choi, Leo Maynard-Zhang +3

Cornell · U Washington · Imperial College London +1

Dynamical Systems Kalman Filtering State Representation Learning Statistical Learning Theory Autoregressive Modeling

Jun 10, 2026cs.LG

Physics-Informed Neural Networks for Chemotherapy Pharmacokinetics: Benchmarking the Clinical Estimator and Exposing Parameter Identifiability

Physics-Informed Neural Networks (PINNs) are an attractive tool for partial-observation problems in biology, where the governing dynamics are known but some compartments cannot be measured. Chemotherapy pharmacokinetics (PK) is a clean instance: drug concentration in plasma is routinely measured, but concentration in tissue -- which determines tumour kill and off-target toxicity -- is not. We benchmark a PINN against the standard clinical baseline (nonlinear least-squares on the analytical biexponential plasma solution, hereafter NLS) and a physics-agnostic neural baseline (a data-only MLP) on two PK problems. On the linear two-compartment problem, NLS is near-optimal; the PINN matches it to within a small constant factor while also producing the tissue curve in a single training pass, whereas the data-only MLP fails on tissue by roughly 10x. On a Michaelis-Menten extension (saturable elimination), the biexponential closed form no longer exists, so NLS is mis-specified and silently returns meaningless rate constants. The PINN instead exposes a deeper fact: the Michaelis-Menten two-compartment model is non-identifiable from plasma alone, and the PINN reports this honestly by converging to a basin with k12 -> 0. Adding two sparse tissue observations largely resolves identifiability: across five seeds the PINN recovers k21 to within 1% of truth and Vmax, Km to within one standard-deviation bar, while k12 moves in the correct direction (0.02 -> 0.82) but remains ~2 sigma below truth -- a recovery the closed-form NLS estimator cannot attempt at all, because its biexponential ansatz describes only plasma. Our claim is not that PINNs beat NLS. It is that PINNs offer a uniform recipe that ties the textbook estimator on the textbook problem, exposes structural identifiability that the textbook estimator hides, and absorbs heterogeneous measurements within a single loss.

Riya Bisht, Dhruv Agarwal

Dynamical Systems Parameter Estimation Parameter Identifiability Physics-Informed ML

Jun 10, 2026cs.LG

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds

Physics-informed neural networks (PINNs) combine machine learning with physical laws to solve differential equations. While existing results provide rigorous \emph{a posteriori} upper bounds for PINN prediction errors, complete certification also requires complementary lower information in order to obtain computable two-sided error enclosures. In this paper, we derive computable \emph{a posteriori} lower bounds for PINN errors in ordinary differential equations on suitable certified state-space domains under a localized strong monotonicity condition. We combine these estimates with complementary localized upper bounds under a one-sided Lipschitz condition, which is weaker than the global Lipschitz assumption used in previous work and can yield sharper upper error bands. The resulting bounds depend only on the neural-network approximation, the ODE residual, and local monotonicity and growth constants, and therefore do not require access to the exact solution. For linear time-invariant and time-varying systems, we further derive explicit formulas in terms of the minimal and maximal eigenvalues of the symmetric part of the system matrix. We also discuss the distinction between soft and hard enforcement of initial conditions in PINNs and explain why exact enforcement can make the scalar lower certificate uninformative. To recover nontrivial lower information in the linear setting, we use a signed-residual finite-probe certificate based on coordinate unit vectors. We also formulate a certificate-informed training strategy in which the propagated upper certificate is used as an auxiliary regularizer, while lower certificates remain post-training diagnostics. Altogether, the proposed framework provides rigorous and practically computable error certificates for PINN approximations of ODEs, while making explicit the domains and model classes for which the assumptions can be verified.

Ismail Huseynov, Arzu Ahmadova, Agamirza Bashirov

Physikalisch-Technische Bundesanstalt (PTB), Germany · Technical University of Berlin, Germany · Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany +1

Dynamical Systems Neural Network Verification Physics-Informed ML

Jun 10, 2026cs.LG

What Uncertainties Do We Need for Dynamical Systems?

The distinction between aleatoric and epistemic uncertainty has received considerable attention in machine learning research, mainly in the context of supervised learning but also in other settings such as generative modeling. In this paper, we offer a machine learning perspective on uncertainty modeling for dynamical systems, which has been studied much less so far. In particular, we ask: what uncertainties do we need for dynamical systems? We discuss sources of uncertainty, clarify their nature (aleatoric or epistemic), and consider how the objectives of representing and quantifying uncertainty vary across different tasks.

Yusuf Sale, Christopher Bülte, Felix Czaja +2

Institute of Computer Science, LMU Munich · Munich · Department of Mathematics, LMU Munich +1

Dynamical Systems Uncertainty Quantification Aleatoric Uncertainty Epistemic Uncertainty

Jun 10, 2026eess.SY

Model-Based and Data-Driven Hierarchical Control and Topology Co-Design for Robust Networked Systems

In this paper, we consider a class of networked systems comprising an interconnected set of linear subsystems, disturbance inputs, and performance outputs. Using dissipativity theory, we first propose a model-based hierarchical control design strategy to ensure the closed-loop networked system is dissipative from its disturbance inputs to performance outputs. This involves designing local controllers for each subsystem to enforce local dissipativity guarantees, which are then exploited to co-design distributed global controllers and the interconnection topology to enforce global dissipativity guarantees while optimizing interconnection topology costs. The overall design process requires only solving a sequence of linear matrix inequality (LMI) problems, thereby retaining compositionality and decentralizability while avoiding non-convex, iterative design processes that are inefficient and centralized. This model-based hierarchical control design strategy assumes the knowledge of the subsystem dynamics, which may not hold in many real-world networked systems. Motivated by this, we also propose a data-driven hierarchical control design strategy that assumes only the availability of rich input-state-output trajectory data from the subsystems. The proposed data-driven design process assumes that the unknown disturbances affecting the subsystem dynamics are bounded by a quadratic matrix inequality (relaxing conventional bounds) and accounts for this by using the matrix S-lemma. Finally, the effectiveness of the proposed model-based and data-driven hierarchical control designs is illustrated for a networked system representing a DC microgrid, with the aim of enforcing robust (dissipative) voltage regulation and current sharing.

Shirantha Welikala, Zihao Song, Hai Lin +1

Department of Electrical and Computer Engineering, School of Engineering and Science, Stevens Institute of Technology, Hoboken, NJ 07030 · Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN, USA

Dynamical Systems Robust Control Data-Driven Control Feedback Control

Jun 10, 2026cs.LG

Kuramoto Attention: Synchronizing Self-Attention on the Torus

Transformer models are increasingly used as computational models of cognition and neural representation, so the mechanism implemented by self-attention is of interest beyond engineering performance. A complementary tradition in cognitive science models coordination, binding, and memory through dynamical interactions such as oscillator synchrony; we bring this mechanism into self-attention by introducing the Kuramoto Attention layer, whose value update is a synchronization step. Each token carries a bank of phase oscillators, so its hidden state lives on a high-dimensional torus. The attention weights form an adaptive coupling graph, and using the raw phase states as values makes the value update exactly the Kuramoto coupling direction for fixed attention weights. The softmax selects which oscillators couple, while the value path moves each token toward the attention-weighted circular mean of the tokens it selects. We train Kuramoto Attention on enwiki8 and CodeParrot against parameter-matched RoPE and SwiGLU transformers. At 5M parameters on CodeParrot, it improves on the transformer by both median and mean, with mean gaps of 0.012 validation and 0.010 test bits per byte. At 5M on enwiki8, all six runs have lower validation/test medians than the transformer and all-seed means within 0.01 BPC; five of six also form a tight lower-mean cluster. At 1M, it trails by about 0.02 BPC on enwiki8 and by 0.013-0.015 bits per byte on CodeParrot. Ablations and phase diagnostics show how the layer's synchronization and geometry-motivated components shape model performance. The result is a self-attention mechanism whose learned computation can be read directly as adaptive synchronization on phase states.

Joshua Nunley

Department of Informatics, Luddy School of Informatics, Computing, and Engineering Cognitive Science Program Indiana University Bloomington

Dynamical Systems Self-Attention

Jun 10, 2026cs.RO

Learning Object Manipulation from Scratch via Contrastive Interaction

Contrastive Reinforcement Learning (CRL) has seen recent success in a wide variety of goal-conditioned robotics tasks by learning structured representations of the dynamics. However, despite its success in locomotion and simpler control domains, CRL often struggles in interaction-rich manipulation. We argue that a key source of this difficulty is object-centric interaction, such as contact or grasping, that induces distinct changes in the underlying dynamic modes. In this work, we formulate manipulation dynamics as a piecewise-smooth Markov process and show that interaction-induced mode changes create piecewise nonlinear reachability structures that are difficult for standard CRL energy functions to represent and plan over. Based on this analysis, we introduce Interaction-weighted Resampling (IWR). IWR performs interaction-aware resampling around phases before, during, and after interactions, encouraging the learned representation to preserve the mode boundaries that determine future reachability to capture multi-modal and piecewise nonlinear reachability. Across interaction-centric environments, including 2D dynamic control, robotic manipulation, and robot air hockey, IWR improves both sample efficiency and overall performance over prior CRL methods, with 19.8% average improvement in simulation. Finally, using a sim-to-real pipeline with policies trained by IWR, we demonstrate the first real-world goal-conditioned robot air hockey agent capable of hitting goals, improving success from 25% to 60%. Project Page: IWR-arxiv.github.io.

Tongle Shen, Caleb Chuck, Fan Feng +1

1UC San Diego · 2UT Austin

Dynamical Systems Reinforcement Learning Robot Skill Learning Contact-Rich Robotic Manipulation Contrastive RL Goal-Conditioned RL

Jun 9, 2026cs.LG

First-Order Trajectory Matching: Fast Ensemble Predictions of Chaotic, Turbulent, Stochastic Systems

We introduce First-Order Trajectory Matching (FTM), a surrogate-modeling method that learns the first-order local transport of probability mass from trajectories of stochastic systems. By matching the symmetric first-order motion of trajectories, FTM learns the probability current velocity, whose flow preserves time marginals to match ensemble averages, while also capturing current-like trajectory quantities such as fluxes, circulations, and barrier-crossing currents. FTM learns the current velocity directly from trajectories, avoiding drift, diffusion, and score estimation. Our stability analysis separates discretization error from sampling variance and shows that the one-step simulation-free FTM loss is stable when temporal resolution and sample size are properly balanced. Across stochastic dynamical systems and PDE examples, we empirically demonstrate that FTM provides trajectory-aware ensemble predictions at low, deterministic-rollout cost.

Shreya Jha, Timo Schorlepp, Nicholas Geissler +2

Courant Institute of Mathematical Sciences, New York University

Dynamical Systems Surrogate Modeling Stochastic Linear Dynamical Systems PDE Surrogate Modeling Probabilistic Forecasting

Jun 9, 2026cs.LG

Physics-Constrained Neural Surrogate for Domain Growth Prediction in Systems with Conserved Kinetics

The spatiotemporal evolution of many physical, chemical, and biological systems is described by nonlinear partial differential equations (PDEs). Recently, deep neural network-based surrogate models have emerged as efficient alternatives to computationally expensive numerical PDE solvers. In this work, we propose a physics-constrained deep neural network as a surrogate model to learn the microstructural evolution of a binary mixture, in which conservation of the order parameter is imposed directly on the network output as a hard constraint. We train the model to accurately predict the time-evolution of phase separation in binary mixtures governed by the Cahn-Hilliard equation. We show that predictions from our trained surrogate model remain stable and accurate over long-time rollouts for both critical and off-critical mixtures and preserve the mixture composition throughout evolution. In contrast, a variant in which conservation is enforced only via a penalty term in the loss function drifts away from the initial composition and significantly loses predictive accuracy over the same rollout. This establishes that the hard constraint is essential for long-time stability and order parameter conservation. We also show that our model accurately captures the growth of domain size and is consistent with the Lifshitz-Slyozov domain-growth law. These results demonstrate the effectiveness of the proposed framework for modeling systems with conserved kinetics, and the construction extends directly to other systems with conserved quantities.

Vijay Yadav, Pallvi Pandey, Madhu Priya +2

Department of Physics, Indian Institute of Technology Jodhpur, Jodhpur, Rajasthan 342 030, India · Department of Physics, Birla Institute of Technology Mesra, Ranchi, Jharkhand 835 215, India · Department of Physics, Central University of Rajasthan, Ajmer, Rajasthan 305 817, India

Dynamical Systems PDE Surrogate Modeling Physics-Informed ML

Jun 9, 2026cs.LG

Machine Learning Methods for Studying Latent Neural Activity Dynamics

Recent developments in brain recording are driving a demand for machine learning tools capable of decoding the latent structure of large populations of neurons. In this paper, we provide a comprehensive survey that outlines the trajectory of Latent Variable Models (LVMs) from early state-space models to more recent deep generative models. We organize the literature into three closely related domains: (1) Single-Region Latent Dynamics, which includes models such as linear dynamical systems to more complex dynamics represented by Recurrent Neural Networks (RNNs) and Neural Ordinary Differential Equations (ODEs); (2) Multi-Region Communication, which employs probabilistic as well as subspace methods to study how information is transferred across different brain areas considering synaptic propagation delays and network connectivity; and (3) Behavior-Aligned Modeling, which seeks to disentangle neural activity related to task performance from other internal states via supervised or contrastive learning. This survey also includes large-scale neural foundation models, such as Transformers and diffusion models, that rely on large-scale pre-training for optimal performance across subjects. Finally, we conclude and discuss benchmarks, evaluation criteria, and open challenges, such as the ability to identify causal links or directionality of communication, to facilitate future research for bridging interpretable brain dynamics with reliable neural decoding.

Shufeng Kong, Fumei Deng, Xinyi Dong +7

School of Software Engineering, Sun Yat-sen University, Zhuhai, China · School of Computer Science and Artificial Intelligence, Foshan University, Foshan, China · Department of Ecology and Evolutionary Biology, Cornell University, Ithaca, NY, USA +2

Dynamical Systems Latent Variable Models Latent Dynamics Modeling Neural Processes Neural Decoding

Jun 9, 2026cs.LG

Loss Landscape Diagnosis for Gradient-Based Gray-Scott System Inversion: Disentangling the Roles of PINN Components

Gradient-based inversion of reaction-diffusion systems is typically approached via surrogate models or physics-informed neural networks (PINNs), while the most direct route, backpropagation through the PDE's structure itself, has largely been avoided. We pursue this direct route as a diagnostic probe, backpropagating a steady-state loss through unrolled Gray-Scott simulation to recover its parameters, with no surrogate or neural-network augmentation. Optimization fails to converge, and plotting the landscape directly locates the failure in its geometry -- flat plateaus with no gradient signal, bounded by sharp cliffs that align with bifurcation boundaries -- a structure that recurs across loss functions and is inherited however the gradients are routed to parameters. Reading this minimal setup as an ablation of PINN, we disentangle each component's role: with the neural network fixed, the residual loss is quadratic in the PDE parameters and yields a smooth landscape, so it alone already avoids the pathology, by implicitly encoding the full PDE dynamics across all initial conditions. The neural network, for its part, cannot repair an ill-posed parameter subspace, and so serves only to complete the observed data -- a division of labor not previously made explicit. These findings carry concrete design implications for PINN-type methods and a broader heuristic on when added dimensions actually help.

Yan Yang

Independent Researcher.

Dynamical Systems Parameter Estimation Physics-Informed ML Loss Landscape Geometry PDE Inverse Problems

Jun 8, 2026cs.AI

Predictive Assistance and the Temporal Dynamics of Exploratory Compression

Classical theories of cognition describe problem solving as exploratory search through structured problem spaces in which repeated interaction gradually compresses search into efficient representational structures. Predictive artificial intelligence systems introduce a distinct regime in which stabilization may occur before exploratory diversification unfolds, supplying solutions and decision trajectories prior to internally generated search. This paper develops a geometric dynamical framework in which attention evolves over a landscape of strategies shaped by stabilizing drift, endogenous exploratory perturbation, and responsiveness-gated learning. Predictive assistance is modeled as a process of exogenous exploratory compression that stabilizes trajectories before self-generated exploration broadens the accessible regions of strategy space. The framework yields three main results. First, sustained predictive stabilization reduces exploratory responsiveness by attenuating the effective influence of intrinsic perturbations even when exploratory variability remains present. Second, curvature accumulates and relaxes asymmetrically, producing hysteresis and delayed recovery of exploratory mobility after assistance withdrawal. Third, developmental outcomes depend critically on the timing of stabilization, with early intervention narrowing future exploratory traversal before broad representational diversification has occurred. The framework generates empirically testable predictions concerning exploratory entropy, premature convergence, and delayed recovery following predictive stabilization. More broadly, the results suggest that predictive systems may reshape the geometry of exploratory cognition itself.

Balaraju Battu

European University Institute, Florence, Italy · New York University Abu Dhabi, UAE

Dynamical Systems Computational Cognitive Modeling AI-Assisted Decision Making

Jun 8, 2026cs.LG

Mechanical Field Networks: Structured Neural Dynamics for Multivariate Systems

Many multivariate dynamical systems are observed only through trajectories, leaving the mechanisms governing their joint dynamics hidden. Existing approaches can impose interpretable dynamics or learn flexible state transitions, yet the resulting interaction structure is typically either specified in advance or left implicit within the learned dynamics. We introduce MF-Net, a recurrent dynamical model that represents all variables in a shared field state and updates this state through a learned relation law. Each variable carries a field component, and these components evolve jointly through a learnable mechanical transition. Here, mechanical refers to the relation-to-motion organization of the transition, where learned relations shape state-dependent flows, field responses, and motion tendencies that move the field state forward. The resulting structure is part of the rollout itself: learned relations influence how the field moves, and the same internal quantities support both forecasting and structural readout. Across known-law interaction systems, chaotic benchmarks, real neural recordings, and ecological time series, MF-Net achieves competitive short- and medium-horizon forecasting while retaining inspectable structural readout. On the 40-dimensional Lorenz--96 testbed, MF-Net achieves an eight-step

R^2

of

0.798\pm0.018

; across five seeds, its learned relation matrix recovers the local coupling support with a local/nonlocal strength ratio of

19.80\pm1.00

and Precision@

K

of

1.000\pm0.000

. MF-Net provides a structure-readable dynamical modeling framework in which learned relations are trained through forward evolution and, on real data, interpreted as functional predictive couplings under appropriate observational limits.

Xingji Cui

Xi’an Jiaotong University, China

Multivariate Time Series Forecasting Dynamical Systems Latent Dynamics Modeling Nonlinear System Identification Recurrent Neural Networks

Jun 8, 2026cs.CV

An Opticalmechanics Framework for Dynamic Estimation of Multibody Systems

Conventional dynamics analysis of the human body is often constrained by the need for contact force and torque sensors and controlled laboratory environments. To address this issue, this study proposes an opticalmechanics kinematic-dynamic integrated estimation framework for multibody systems. Specifically, a constrained multibody model is established to describe the system dynamics, while image-measured kinematic quantities are used as non contact inputs for dynamic estimation. The unknown joint torque is then identified through a genetic-algorithm based optimization by minimizing the discrepancy between model-predicted and image-measured kinematic quan tities. Experimental validation on an air-bearing platform showed that the wrist joint torque estimated from image data achieved a mean absolute error of 0.46 Nm compared with sensor measurements. In the forward prediction test, the model-predicted angular velocity achieved a mean absolute error of 0.006 rad/s relative to the image-measured results. This study demonstrates the potential of combining image measurement and mechanical modeling for non-contact dynamic estimation in scenarios where direct force and torque measurement is difficult.

Banglei Guan, Xuanyu Bai, Qingquan Chen +5

National University of Defense Technology

Dynamical Systems Multibody Dynamics Inverse Dynamics

Jun 8, 2026cs.RO

Stabilizing Fast Dynamical Systems Using Time-of-Flight Cameras

Time-of-flight cameras are popular in robotics because they provide direct depth information and are compact, inexpensive, and less dependent on ambient visible light. However, their low spatial resolution and depth noise raise questions about their suitability for stabilizing fast, unstable dynamics. In this paper, we show that an inexpensive, low-resolution time-of-flight camera can provide angle feedback for reliable and precise balancing of an inverted pendulum on a cart, a canonical control benchmark. These results open the door to using compact, inexpensive, low-resolution time-of-flight cameras for precise feedback control of fast, unstable systems.

Anthony Czubarow, Antonio Terpin, Raffaello D'Andrea

Institute for Dynamic Systems and Control, ETH Zürich, Zürich, Switzerland

Dynamical Systems Feedback Control

Jun 7, 2026cs.LG

Between Amnesia and Chaos: A Memory Stability Expressivity Trilemma for Trainable Dissipative Oscillator Networks

Physical reservoir computing harnesses nonlinear mechanical dynamics but, by convention, freezes the substrate and trains only a linear readout, presuming the substrate is not usefully trainable. We revisit that premise for networks of nonlinear oscillators whose mass, damping, and stiffness are learned end-to-end through a symplectic integrator. Our central result is a trilemma: memory horizon, gradient stability, and dynamical expressivity cannot be simultaneously maximized, because all three are governed by the damping. The backward gradient decays at a rate set by the damping, capping how far back credit can propagate, while forward sensitivities grow exponentially in the largest Lyapunov exponent, so usable gradients require damping above a stability floor. Since the Lyapunov exponent falls as damping rises while the memory ceiling falls as the horizon grows, stable training is confined to a band that contracts with horizon and closes at a critical point. We test every step on a twenty-oscillator network. A damping sweep finds the largest Lyapunov exponent monotone and crossing zero at a well-defined stability floor, confirming the theorem's key assumption. A compute-matched comparison of learned versus frozen substrate on delayed recall across nine horizons shows the learned substrate dominating at short horizons and the advantage closing and reversing near a horizon of eleven steps, the predicted signature of band closure; trained models settle near the stability floor, seeking the edge of chaos unprompted. The analytic ceiling overestimates the empirical crossover roughly fivefold, a gap between detectable and learnable gradient that we report rather than tune away. The contribution is a confirmed account of when training a physical substrate beats freezing it.

Caleb Munigety

Independent Researcher

Dynamical Systems Lyapunov Stability Reservoir Computing Neural Network Training Dynamics

Jun 7, 2026cs.LG

Inferring hidden forcing in a biological oscillator using Kolmogorov-Arnold networks

Inferring the forces that drive a dynamical system from partial observations is a fundamental challenge across physics, particularly when distinct underlying mechanisms produce similar observable dynamics. Here we show that the effective muscular forcing underlying avian respiratory dynamics can be reconstructed from measurements of air-sac pressure alone. Using an interpretable learning framework based on Kolmogorov-Arnold networks, we infer the governing equations of the system directly from data and uncover a nontrivial structure in the underlying forcing that is not apparent from the pressure signal, which instead suggests a relaxation-like oscillation. The reconstructed dynamics predict a two-phase activation pattern within each respiratory cycle, which we independently validate through electromyographic recordings of expiratory muscles. These results demonstrate that data-driven reconstruction of dynamical laws can reveal hidden physical structure and provide access to unobserved driving variables, establishing a general route to infer latent forces in partially observed dynamical systems.

Julian Szereszewski, Facundo Fainstein, Leandro E. Fernandez +1

Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Física, Ciudad Universitaria, 1428 Buenos Aires, Argentina. · CONICET - Universidad de Buenos Aires, Instituto de Física Interdisciplinaria y Aplicada (INFINA), Ciudad Universitaria, 1428 Buenos Aires, Argentina.

Dynamical Systems Nonlinear System Identification

Jun 6, 2026cs.LG

Towards Graph Foundation Models for Dynamics in Complex Networked Systems: Lessons from Super-Spreader Identification in Multilayer Networks

Network dynamics - including spreading, influence maximisation, and epidemic modelling - remain largely confined to the transductive paradigm, where models are trained on a single network and cannot be reused on unseen graphs without retraining. We argue that inductive cross-network generalisation is a necessary prerequisite for Graph Foundation Models (GFMs) in this domain and propose four design properties towards this goal. As a proof of concept, ts-net (TopSpreadersNetwork), trained solely on synthetic multilayer networks (MLNs), demonstrates zero-shot generalisation to real-world MLNs of varying size and layer count, outperforming classical heuristics and transductive baselines on three of four metrics. Based on ts-net's performance, we further outline five open challenges towards building GFMs for network dynamics: scale, many-layer generalisation, self-supervised pretraining, cross-task transfer, and node-attribute integration.

Michał Czuba, Mateusz Stolarski, Adam Piróg +2

Department of Artificial Intelligence, Wrocław University of Science and Technology, Wrocław, Poland · Semantics, Warsaw, Poland

Dynamical Systems Graph Foundation Models Cross-Dataset Generalization

Jun 6, 2026math.NA

Stable and Scalable Probabilistic Numerical Solvers for Stiff and High-Dimensional ODEs

Filtering-based probabilistic numerical solvers for ordinary differential equations (ODEs) have been established as a flexible and efficient simulation framework with built-in numerical uncertainty quantification. However, problems that are both stiff and high-dimensional remain a challenge, as current methods are either stable and have cubic cost in the ODE dimension, or scale linearly at the expense of stability. In this paper, we close this gap and develop probabilistic ODE solvers that are both stable and scalable. We propose two complementary strategies. First, we develop a matrix-free update step that uses Jacobian-vector products, iterative linear solvers, and stochastic covariance estimation to enable linear scaling, all while retaining stability. Second, we propose iterative re-linearization to further improve stability without sacrificing scalability, turning probabilistic ODE solvers into fully implicit methods. We evaluate the proposed approaches on a range of stiff and high-dimensional problems and demonstrate improved stability and scalability over established probabilistic solvers.

Nathanael Bosch

EPFL, Lausanne, Switzerland

Dynamical Systems Uncertainty Quantification Algorithmic Stability

Jun 5, 2026eess.SY

On orbital stabilization of a circular motion primitive for a dynamic extension of the Dubins car model

This paper addresses orbital stabilization of a circular motion primitive for a dynamic extension of the Dubins car model within a transverse-linearization framework. We show that the corresponding transverse linearization is unstable and not stabilizable by linear state feedback. Therefore, the standard linearization-based approach to orbital stabilization cannot be applied directly. The main contribution is a set of explicit and verifiable conditions that characterize when a controller design based on transverse linearization remains applicable. These conditions rely on the specific structure of the dynamics in a neighborhood of the motion and on the use of non-standard transverse coordinates for controller design and analysis. Numerical simulations illustrate the proposed design procedure.

Artem Angelchev-Shiryaev, Pavel E. Aleshin, Anton S. Shiriaev +2

Department of Industrial and Mechanical Sciences, Lund University, Lund, Sweden · Department of Information Technologies, Sirius University, Sirius, Russia · Department of Engineering Cybernetics, NTNU, Trondheim, Norway +1

Dynamical Systems Feedback Control

Jun 5, 2026nlin.CD

Unified Geometry-Guided ML-FTLE for Tracking Transient Chaos from Scalar Time Series

Detecting transient chaos from scalar observations without governing equations represents a fundamental challenge in nonlinear dynamics. We propose a geometry-guided machine learning framework that unifies predictive trajectory divergence with macroscopic attractor morphology to track abrupt regime shifts. The methodology extracts a local instability scale via out-of-sample k-nearest neighbor forecast errors to establish the ML-FTLE estimator, subsequently mapping this temporal divergence onto a structural closeness matrix derived from a minimal dictionary of Poincare occupancy grids. By employing partial least squares regression, we extract a latent geometric component calibrated directly to the empirical finite-time Lyapunov spectrum, yielding the Poincare-based geometric-guided FTLE. Validation against analytical QR-FTLE baselines confirms that fusing topological state spaces with predictive divergence systematically improves continuous transition tracking. The Structural Similarity Index optimally resolves gradual damping, while Hausdorff Distance exhibits extreme resilience during abrupt phase-space collapses. Furthermore, macroscopic spatial discretization acts as a robust topological regularizer against additive Gaussian noise, preserving deterministic signatures even at moderate signal thresholds. This equation-free framework provides a highly accurate, noise-resilient diagnostic for monitoring structural transitions in complex non-stationary systems.

S. V. Manivelan, Andrei Velichko, I. Manimehan

Department of Physics, M. R. Government Arts College (Affiliated to Bharathidasan University, Tiruchirappalli), Mannargudi, 614 001, Tamilnadu, India. · Institute of Physics and Technology, Petrozavodsk State University, Petrozavodsk, 185910, Russia.

Dynamical Systems Chaotic Dynamical Systems Change-Point Detection

Jun 5, 2026stat.ML

Empirical Transfer Operators and Finite-Sample Change Detection for Noisy Expanding Interval Maps

We study finite-sample change detection for one-dimensional noisy dynamical systems using partition-based empirical approximations of stationary behaviour. Given observations from an interval-valued process, we partition the state space, estimate a finite transition matrix from observed transitions between partition elements, and apply a small Doeblin-type regularisation to ensure a unique stationary distribution. From an initial reference segment, we compute a baseline empirical stationary distribution

\widehatπ_{0,ρ}

. For each later sliding window, we compute

\widehatπ_{t,ρ}

and define the score

S_t=\|\widehatπ_{t,ρ}-\widehatπ_{0,ρ}\|_1.

Large values of

S_t

indicate a change in stationary behaviour relative to the baseline. The statistic detects changes in invariant density or stationary law, but not all possible changes in transition dynamics. Under explicit assumptions on empirical transition concentration, finite-state stationary distribution stability, partition approximation, regularisation bias, and noise stability, we derive a finite-sample bound for the empirical stationary density. The bound separates sampling error, regularisation bias, partition approximation error, and noise bias. We then obtain a single-window false-alarm guarantee and a sufficient detection condition when the invariant density changes by more than the estimation error. We illustrate the method on synthetic noisy beta-map change-point experiments.

Aparna Rajput

Dynamical Systems Distribution Shift Detection Change-Point Detection Temporal Distribution Shift

Jun 4, 2026cs.LG

Capturing non-Markovian dynamics in non-equilibrium stochastic systems using flow matching

Hydrodynamic models of stochastic particle systems represented by coarse-grained stochastic partial differential equations (SPDE), such as the regularized Dean-Kawasaki (DK) equation, do not accurately capture the short-time system dynamics that is dominated by non-Markovian effects, and low particle density regimes where the distributions are highly non-Gaussian. We develop a generative flow matching method that directly models the probability distribution of fluxes from particle simulations that explicitly incorporates non-Markovian and non-Gaussian effects. As a demonstration, we use this method to simulate the Kramers first passage time problem for a system of non-interacting Brownian particles. We show the model accurately captures the short-time behavior and provides better predictions of the statistical moments of the number density when compared against the solution of the Markovian baseline, regularized DK equation.

Bhargav Sriram Siddani, John B. Bell, Alejandro L. Garcia +1

Lawrence Berkeley National Laboratory · San Jose State University

Dynamical Systems Flow Matching

Jun 3, 2026cs.LG

Learning Manifold and Itô Dynamics with Branched Neural Rough Differential Equations

Neural rough differential equations (NRDEs) stay accurate under irregular sampling while taking far fewer integration steps than standard neural differential equations, summarising a finely sampled driver by its log-signature and advancing the hidden state over coarse intervals using the log-ODE method. This efficiency rests on the shuffle algebra, the algebraic counterpart of Stratonovich calculus. This reliance means NRDEs cannot expose the quadratic-variation terms Itô dynamics require, nor the ordered covariant derivatives that govern Itô flows on connection-equipped manifolds. Ameliorating this, we introduce Branched Neural Rough Differential Equations (B-NRDEs), a Hopf-algebraic framework that recasts the NRDE log-ODE step as geometric numerical integration on the state-space manifold, matching the driving algebra to the governing calculus: Grossman--Larson rooted trees for Euclidean Itô dynamics, Munthe-Kaas--Wright planar rooted trees for ordered covariant derivatives on manifolds, and the shuffle algebra in the classical Stratonovich case. This yields intrinsic coarse-step dynamics that exactly preserve manifold constraints. Finally, we introduce a branched signature-kernel objective to enable Itô-consistent law matching by making quadratic-variation terms visible during training. On rough Bergomi volatility, sim-to-real

\mathrm{SO}(3)

dynamics forecasting, and SPD covariance dynamics, B-NRDEs offer a unified, effective approach to stochastic and manifold-valued dynamics beyond the Euclidean--Stratonovich setting.

Luke Thompson, Dai Shi, Lequan Lin +2

University of Sydney, Australia · Cambridge University, United Kingdom.

Dynamical Systems Stochastic Differential Equations Neural ODEs

Jun 3, 2026cs.LG

Reconciling Causality and Non-Equilibrium Thermodynamics with Hamiltonian Causal Models

Causal modeling of physical temporal phenomena must handle interventions that act along trajectories, nonstationary induced laws, path-dependent effects, and feedback mediated by dynamics, all challenging in standard causal models. We introduce Hamiltonian Causal Models (HCMs), a trajectory-level framework in which observed variables interact with local environments and interventions act as controls of Hamiltonian mechanisms. HCMs separate immutable equations of motion from intervenable mechanisms and define causal effects as discrepancies between interventional path laws. A key motivation for HCMs is their natural interface with non-equilibrium thermodynamics. Entropy production quantifies the irreversibility of a process and is a central causal observable: it is estimable from data and witnesses causal effects along the system's evolution that are invisible to endpoint and cumulative versions of the standard average treatment effect. As in physics, cause and effect are not primitives of the relation between two random variables but arise from the non-invertibility of the thermodynamic arrow. With this, our paper reconciles the language of statistical causal models and non-stationary thermodynamics, offering new tools to describe causality in a wide range of physical systems.

Dario Rancati, Max Welling, Francesco Locatello

Institute of Science and Technology Austria · University of Amsterdam

Dynamical Systems Causal Inference Causal Intervention

Jun 3, 2026cs.LG

Learning symplectic model reduction based on a approximation theorem of symplectic embeddings

High-dimensional Hamiltonian systems play a central role in many scientific and engineering disciplines, with dynamics evolving on symplectic manifolds. Although deep learning provides powerful tools for constructing low-dimensional surrogates from data, the intrinsic symplectic structure is easily destroyed during model reduction. As a result, a standard autoencoder may produce latent coordinates that do not support a Hamiltonian flow, leading to unstable long-time prediction. In this paper, we first establish a universal approximation theorem for symplectic embeddings. Based on this theory, we propose symplecticity-preserving autoencoders (SpAE), in which the decoder is parameterized as a symplectic embedding and the encoder is constructed as the corresponding symplectic projection. This architecture is expressive enough to approximate nonlinear symplectic embeddings and the associated symplectic projections, preserves the symplectic structure exactly by construction, and can be trained by standard unconstrained optimization, thereby improving both reconstruction and prediction accuracy. Extensive experiments on high-dimensional lattice and particle systems demonstrate the effectiveness of the proposed method.

Liyi Feng, Yifa Tang, Yulin Xie +2

School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, China · State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China · Department of Mathematics, National University of Singapore, 119076, Singapore

Dynamical Systems Autoencoders Reduced-Order Modeling Hamiltonian Dynamics

Jun 1, 2026cs.RO

Learning Action-Conditional and Object-Centric Gaussian Splatting World Models for Rigid Objects

World models enable intelligent agents to predict the consequences of their actions on the environment. In this paper, we propose Multi Rigid Object Gaussian World Model (MRO-GWM), a novel model that learns action-conditional dynamics of rigid objects in 3D. By representing the scene by object-centric Gaussians, we can represent arbitrary object shapes and multi-object scenes. We develop a novel spatio-temporal transformer architecture that predicts future rigid body motion from a history of object Gaussians and future actions. Objects are represented by their Gaussians in a canonical frame, which allows for describing object motion as rigid body transformation. Our model is trained on reconstructions from multiple viewpoints, which requires the model to handle partial observations of objects due to occlusions. We analyze prediction performance of our approach on synthetic datasets composed of typical household objects with multi-object dynamics and interactions by a robot end effector. We also evaluate our model in model-predictive control for non-prehensile manipulation in simulation.

Jens U. Kreber, Lukas Mack, Joerg Stueckler

Intelligent Perception in Technical Systems Group University of Augsburg Augsburg, Germany

3D Gaussian Splatting Dynamical Systems World Model Learning World Models for Robotics Action-Conditioned World Models Object-Centric World Models+1

May 31, 2026stat.ML

Computation-Aware Kalman Filtering with Model Selection for Neural Dynamics

Due to their explicit priors and ability to model uncertainty, Bayesian methods have played a major role in dynamical latent variable modeling of single-cell neural recordings. However, modern-sized datasets have made overparameterized deep networks the preferred methods of choice due to their predictive power and favorable computational scaling. While many posterior approximations exist, all incur approximation errors. Recent work accounts for this error in the form of computational uncertainty but comes at the cost of quadratic complexity and assumes fixed model hyperparameters. Here we extend this development to model selection, including a novel training loss and optimization scheme, which yields tractable inference in large state-spaces. We introduce a framework, the Computation-Aware State-Space Model (CASSM), specifically designed for the scale-imbalanced regime, where the number of trials is significantly lower than the number of recorded neurons. In this regime, for both synthetic and real data, we show that our method is competitive with data-hungry deep networks, with significantly improved uncertainty calibration over previous attempts to scale Bayesian methods. Our experiments provide a roadmap to neuroscience researchers in choosing from a host of potential dynamical latent variable models given key dataset properties and constraints.

JR Huml, Jonathan Wenger, John P. Cunningham

Department of Statistics, Columbia University; Zyphra Technologies · Department of Statistics, Columbia University; Zuckerman Institute

Dynamical Systems Bayesian Filtering Uncertainty Quantification Latent Dynamics Modeling State Space Models