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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

May 7, 2026cs.AI

Safety Certification is Classification

The goal of this paper is certifying safety of dynamical systems subject to uncertainty. Existing approaches use trajectory data to estimate transition probabilities, and compute safety probabilities recursively via dynamic programming (DP). This recursion may lead to compounding errors in the certified safety probability, thus collapsing to a vacuous lower bound for growing horizons

T

. We propose a kernel embedding framework that treats safety certification as a classification problem on trajectory data, directly estimating the

T

-step safety probability without recursion. We show that the framework subsumes well-established approaches from the literature (e.g., barrier certificates, robust Markov models) as special cases, and allows us to go beyond their limitations. As the main consequence, it bypasses compounding error across the horizon and enables certification for systems with non-Markovian dynamics. We demonstrate that direct estimators remain stable independent of the certification horizon and in the non-Markovian setting, whilst DP-based certificates silently go unsound -- confirmed in simulation on a neural-controlled quadrotor.

Oliver Schön, Licio Romao, Sadegh Soudjani

ETH Zürich · Zürich, Switzerland · Technical University of Denmark +3

Kernel Mean Embeddings Dynamical Systems Safety-Critical Control

May 7, 2026eess.SY

Maximal Controlled Invariant-MPC: Enhancing Feasibility and Reducing Conservatism through Terminal CBF Constraint in Safety-Critical Control

Optimal control for safety-critical systems is often dependent on the conservativeness of constraints. Control Barrier Functions (CBFs) serve as a medium to represent such constraints, but constructing a minimally conservative CBF is a computationally intractable problem. Therefore, approaches that can guarantee safety while reducing conservatism will help improve the optimality of the system under consideration. Here, we present a Model Predictive Control (MPC) formulation using CBF as a terminal constraint, which is proven to improve feasibility and reachable sets with increasing prediction horizon. The constructive nature of the proofs allows for warm-starting the nonlinear optimization problem, thereby reducing the computational time substantially. Simulations are set up for a simple nonholonomic system to numerically validate the results, and it is observed that the number of infeasible points decreased by a factor of 1.7 to 2.7. The increase in reachable state space was demonstrated by the ability of the system to track trajectories that are entirely inside the unsafe region of the control barrier function.

Tanmay Dokania, Yashwanth Kumar Nakka

Graduate Student, Daniel Guggenheim School of Aerospace Engineering. · Assistant Professor, Director, Aerospace Robotics Lab, Daniel Guggenheim School of Aerospace Engineering.

Dynamical Systems Safety-Critical Control Nonlinear MPC Control Barrier Functions Model Predictive Control Optimal Control

May 6, 2026cs.LG

Differentiable Parameter Optimization for DAEs with State-Dependent Events

Differential-algebraic equations (DAEs) with state-dependent events arise in systems whose continuous dynamics are constrained by algebraic equations and interrupted by mode changes, switching logic, impacts, or state reinitializations. Gradient-based parameter learning for such systems is challenging because algebraic variables are implicitly defined, event times depend on the parameters, and reset maps introduce discontinuities. This paper studies differentiable parameter optimization for semi-explicit DAEs with events. We formulate the learning problem as a constrained least-squares problem with DAE dynamics, algebraic constraints, guard equations, and reset maps. We then develop two complementary gradient-computation strategies. The first is an automatic-differentiation-through-simulation method that solves algebraic variables inside the vector field, differentiates the algebraic solve using the implicit function theorem, and handles events through segmented differentiable integration. The second is an explicit discrete-adjoint method that represents the forward simulation as an event-split residual system and computes gradients by solving for the Lagrange multipliers of smooth-segment and event residuals. The formulation clarifies that residual terms in the adjoint method are equality constraints, not heuristic penalties. We compare the two approaches in terms of gradient interpretation, event-time handling, implementation complexity, and local validity. Both methods provide gradients for the event path selected by the forward simulation and are valid under fixed event ordering and transversal guard crossings.

Ion Matei, Maksym Zhenirovskyy, Anthony Wong

Fujitsu Research of America

Dynamical Systems Automatic Differentiation Constrained Optimization Differentiable Physics Differentiable Optimization

May 6, 2026cs.LG

Graph Normalization: Fast Binarizing Dynamics for Differentiable MWIS

We introduce Graph Normalization (GN), a principled dynamical system on graphs that serves as a differentiable approximation engine for the NP-hard Maximum Weight Independent Set (MWIS) problem. MWIS encompasses many combinatorial challenges, including optimal assignment, scheduling, set packing, and MAP inference in discrete Markov Random Fields. Unlike Belief Propagation, we prove GN always converges to a binary indicator of a Maximum Independent Set. GN realizes a fast quasi-Newton descent through an exact Majorization-Minimization step, systematically improving the MWIS relaxed primal objective. We establish an equivalence between GN and the Replicator Dynamics of a nonlinear evolutionary game, where vertices compete for inclusion in an independent set. While a non-potential game, the GN game follows Fisher's Fundamental Theorem of Natural Selection, where the average fitness equals the MWIS primal objective and strictly increases. This connection leads to a weighted extension of the Motzkin-Straus theorem, showing MISes are in bijection with the local minima of a quadratic form over a tilted simplex. For the Assignment Problem, GN acts as a variant of the Sinkhorn algorithm that naturally converges to a hard assignment while generalizing to arbitrary constraint graphs. We demonstrate GN's performance as a fast binarization engine for the state-of-the-art Bregman-Sinkhorn relaxed MWIS solver. On real-world benchmarks with up to 1M edges, GN identifies solutions within 1% of the best known results in seconds on a CPU. GN opens new avenues for deep learning architectures requiring differentiable, "hard" decisions under constraints, with applications in structured sparse attention, dynamic network pruning, and Mixture-of-Experts. Beyond core AI, the GN framework enables end-to-end learning of constrained optimization in computer vision, computational biology, and resource allocation.

Laurent Guigues

Dynamical Systems Replicator Dynamics Combinatorial Optimization Nonconvex Optimization Differentiable Optimization

May 6, 2026cs.LG

Bilinear Mamba-Koopman Neural MPC for Varying Dynamics

Koopman-based neural MPC models generate time-varying dynamics from historical data, but preserve convexity by enforcing that the system operator is independent of the current control input. This conditional independence constraint limits adaptation to changing dynamics within a single MPC horizon, particularly under time-varying conditions and under stale-plan execution. We propose Bilinear Mamba-Koopman Neural MPC, a minimal extension that introduces control-dependent coupling in the latent dynamics, allowing the effective operator to adapt to the current input. The resulting model is a strict generalization of the standard linear, conditional-independence formulation, adds less than 1% parameters through a low-rank structure, and admits exact model Jacobians that enable efficient Sequential Convex Programming (SCP) with monotone-descent and KKT convergence results under standard trust-region assumptions. Across CartPole and RSCP benchmarks in time-invariant and time-varying regimes, the proposed model matches or improves forecasting accuracy on every cell when training noise is averaged out, with strict gains where control-state coupling is structurally present. Its main closed-loop gains appear in the RSCP TV task, where iterative SCP improves adaptation within the horizon and substantially stabilizes training; in CartPole TV, the gains are modest but consistent. In delayed re-planning experiments on the time-varying variants, the bilinear model degrades more gracefully under stale-plan execution, maintaining a consistent advantage on CartPole TV and a substantially larger robustness margin on RSCP TV. These results show that control-dependent latent dynamics provide a simple and effective mechanism for robust MPC under varying conditions.

Matan Pagi, Zohar Sorek

Psistar AI

Dynamical Systems Koopman Operator Learning Nonlinear MPC Robust Control Model Predictive Control Optimal Control

May 6, 2026cs.LG

Differentiable Chemistry in PINNs for Solving Parameterized and Stiff Reaction Systems

From neural ODEs to continuous-time machine learning, differentiable solvers allow physics, optimization, and simulation to become trainable components within deep learning systems. This has opened the path to a new generation of deep learning frameworks for scientific computing, with many promising applications still emerging. In this paper, we integrate a differentiable chemistry solver into a modified physics-informed neural network to solve parameterized reaction systems that are inherently stiff. The proposed framework introduces several key components required to overcome limitations of standard physics-informed neural networks. These include a differentiable chemistry solver, a network architecture for parameterized solutions, and residual weighting tailored to stiff reactions. We evaluate the framework on a set of differential equations related to hydrogen combustion, which include initial/boundary value problems, inverse parameter identification, and a parameterized partial differential equation. Our results highlight the ability of the proposed approach to extend physics-informed neural networks to stiff chemical systems that were previously inaccessible.

Miloš Babić, Franz M. Rohrhofer, Stefan Posch

CD Laboratory for Physics-driven Machine Learning in Industrial Applications, Graz, Austria · The Institute of Thermodynamics and Sustainable Propulsion Systems, Graz University of Technology, Graz, Austria · Know Center Research GmbH, Graz, Austria

Dynamical Systems Differentiable Physics Physics-Informed ML

May 6, 2026cs.LG

Gradient Scaling Effects in Adaptive Spectral PINNs for Stiff Nonlinear ODEs

Physics-Informed Neural Networks (PINNs) often struggle to train reliably on stiff and oscillatory dynamical systems due to poor optimization conditioning. While prior work has emphasized representational remedies such as spectral parameterizations, the optimization implications of initial-condition (IC) embeddings in adaptive spectral PINNs have not been well characterized. In this work, we show that the choice of IC gating function induces explicit time-dependent gradient scaling, which interacts with spectral representations during training. Using a nonlinear stiff spring-pendulum ODE as a controlled benchmark, we compare exponential and linear IC gates in combination with fixed and adaptive Fourier spectral trunks. We observe stiffness-dependent changes in relative dominance for adaptive PINNs: at moderate stiffness (

k=20

), exponential gating often yields lower error but exhibits heterogeneous behavior across random seeds, whereas at higher stiffness (

k=60

), linear gating becomes preferable, with additional reversals observed at larger

k

. These trends hold for both relative

L^2

error and maximum pointwise error and are confirmed by paired Wilcoxon signed-rank tests with Holm correction. Overall, our results demonstrate that IC embeddings are not a neutral design choice in PINNs: the induced gradient scaling materially shapes optimization conditioning in stiff regimes, with distinct sensitivity patterns in baseline and adaptive spectral models.

Isabela M. Yepes, Pavlos Protopapas

Harvard University

Dynamical Systems Neural Network Optimization Physics-Informed ML Neural Network Training Dynamics

May 5, 2026math.DS

Reentrant value fields as delayed coupled reaction-diffusion systems on finite graphs

We describe a dynamical system in which a symbolic field is coupled to a geometric field via a bipartite Hilbert-Schmidt kernel. The system is fully described by a retarded functional differential equation (RFDE) on the history space, subject to Lipschitz and small gain conditions. We show that the RFDE is well-posed under constant input and that it admits a compact global attractor. The principal subsystem

(H_L, X_R, P)

, which is comprised of the two primary fields as well as an executive field, is shown to be globally stable independent of delay, provided that the interfield coupling satisfies

C_{\mathcal{K}}^2<μ_Lμ_R

. In addition, we describe design specifications that fulfill the hypotheses of the main Theorem.

Karsten Bohlen

arXiv:2605.03940v4

Dynamical Systems

May 5, 2026cs.RO

Neural Control: Adjoint Learning Through Equilibrium Constraints

Many physical AI tasks require sequential implicit computation: at each step, boundary controls are applied, and the resulting configuration is obtained by solving an equilibrium problem. This setting arises naturally in deformable object manipulation, where even bending a deformable linear object (DLO) to a target shape can be nonlinear and multistable: identical boundary conditions may produce different configurations depending on actuation history. Unlike explicit transition models, the control-to-configuration relation is implicit and history-dependent, making long-horizon learning and control brittle; backpropagating through iterative solves is also memory- and compute-intensive. We propose Neural Control, a boundary-control framework that propagates gradients through branch-dependent sequences of equilibrium solves rather than a single fixed point. Neural Control computes trajectory-dependent proxy gradients by differentiating equilibrium conditions with an adjoint formulation, avoiding solver unrolling while keeping forward rollouts on converged equilibria. Combined with receding-horizon continuation, Neural Control re-anchors optimization to realized equilibria and mitigates basin switching. We validate Neural Control on simulated and real DLO manipulation, compare against SPSA and iCEM, and demonstrate applicability to a learned DEQ-style implicit equilibrium model.

Dezhong Tong, Jiawen Wang, Hengyi Zhou +3

The Robotics Department, University of Michigan, Ann Arbor, US · The Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, US

Robotic Control Dynamical Systems Deformable Object Manipulation

May 3, 2026eess.SY

Observability Conditions and Filter Design for Visual Pose Estimation via Dual Quaternions

This paper presents a dual quaternion framework for 6-DOF visual target tracking that addresses key limitations of perspective-n-point (P

n

P) solvers: sensitivity to noise and outliers, and inability to propagate estimates through measurement dropouts. A nonlinear observability analysis is performed using a Lie algebraic approach, deriving sufficient conditions for local observability under two sensing modalities: relative position vector and unit vector measurements. For the unit vector case, the classical collinear feature point degeneracy of the perspective-three-point problem is recovered through rank analysis of the observability codistribution matrix, providing a control-theoretic interpretation of a previously geometric result. A dual quaternion Lie group unscented Kalman filter is then developed, directly modeling relative dynamics without assumptions about cooperative measurements or slowly-varying motion. Simulations demonstrate improved pose estimation accuracy and robustness to occlusions compared to an off-the-shelf P

n

P solver. Results are broadly applicable to visual-inertial navigation, simultaneous localization and mapping, and P

n

P solver development.

Nicholas B. Andrews, Kristi A. Morgansen

Department of Aeronautics and Astronautics, University of Washington, Seattle, WA 98195 USA

Dynamical Systems Kalman Filtering Relative Pose Estimation Object Pose Estimation Unscented Kalman Filtering Robot State Estimation

May 3, 2026cs.LG

Learning Koopman operators for coupled systems via information on governing equations of subsystems

Nonlinear coupled systems are ubiquitous in science and engineering. The analysis and modeling of such systems is challenging due to their high dimensionality and complex interactions among subsystems. In recent years, operator-theoretic methods based on the Koopman operator have attracted attention as a powerful tool for analyzing and modeling nonlinear dynamical systems. Extended dynamic mode decomposition (EDMD) is one of the most popular methods to approximate the Koopman operator. However, EDMD is a purely data-driven method, and it could be unstable and inaccurate for coupled systems under limited data availability. In this paper, we propose a method to learn the Koopman operator for coupled systems using the differential equations governing each subsystem. We also demonstrate its effectiveness through numerical experiments on coupled oscillator systems.

Tatsuya Naoi, Jun Ohkubo

Graduate School of Science and Engineering, Saitama University, Sakura, Saitama, 338-8570, Japan

Dynamical Systems Koopman Operator Learning Nonlinear System Identification

May 1, 2026cs.LG

Mesh Field Theory: Port-Hamiltonian Formulation of Mesh-Based Physics

We present Mesh Field Theory (MeshFT) and its neural realization, MeshFT-Net: a structure-preserving framework for mesh-based continuum physics that cleanly separates the physics' topological structure from its metric structure. Imposing minimal physical principles (locality, permutation equivariance, orientation covariance, and energy balance/dissipation inequality), we prove a reduction theorem for mesh-based physics. Under these conditions, the physical dynamics admit a local factorization into a port-Hamiltonian form: the conservative interconnection is fixed uniquely by mesh topology, whereas metric effects enter only through constitutive relations and dissipation. This reduction clarifies what must be fixed and what should be learned, directly informing MeshFT-Net's design. Across evaluations on analytic and realistic datasets, physics-consistency tests, and out-of-distribution validation, MeshFT-Net achieves near-zero energy drift and strong physical fidelity (correct dispersion and momentum conservation) along with robust extrapolation and high data efficiency. By eliminating non-physical degrees of freedom and learning only metric-dependent structure, MeshFT provides a principled inductive bias for stable, faithful, and data-efficient learning-based physical simulation.

Satoshi Noguchi, Yoshinobu Kawahara

Center for Mathematical Science and Advanced Technology, JAMSTEC · Center for Advanced Intelligence Project, RIKEN · Graduate School of Information Science and Technology, The University of Osaka.

Dynamical Systems Scientific ML Physics-Informed ML Port-Hamiltonian Systems Physics-Based Simulation

May 1, 2026cs.NE

Geometric and dynamical analysis of attractor boundaries and storage limits in kernel Hopfield networks

High-capacity associative memories based on Kernel Logistic Regression (KLR) exhibit strong storage capabilities, but the dynamical and geometric mechanisms underlying their stability remain poorly understood. This paper investigates the global geometry of attractor basins and the mechanisms governing the storage limit in KLR-trained Hopfield networks. We combine empirical evaluations using random sequences and real-world image embeddings (CIFAR-10) with morphing experiments and statistical Signal-to-Noise Ratio (SNR) analysis. Our experiments show that the network achieves a storage capacity for random sequences up to

P/N \approx 16

, while maintaining stable retrieval for structured data at effective loads near

P/N \approx 20

. Morphing analysis indicates that attractors on the "Ridge of Optimization" are separated by sharp, phase-transition-like boundaries, characterized by steep effective potential barriers and critical slowing down. Furthermore, by comparing an SNR analysis with a geometric reference point inspired by Cover's theorem, we show that the practical storage limit is governed primarily not by a lack of geometric separability in the feature space, but by the loss of dynamical stability against crosstalk noise. These findings suggest that KLR networks function as highly localized exemplar-based memories that operate near the onset of dynamical collapse, providing a useful perspective on the design of robust, large-scale retrieval systems.

Akira Tamamori

Faculty of Information Science, Aichi Institute of Technology 1247 Yachigusa, Yakusa-cho, Toyota-shi, Aichi 470-0392, Japan

Dynamical Systems Neural Representation Geometry Hopfield Networks Associative Memory

Apr 30, 2026cond-mat.soft

Active-learning mapping of the Vicsek model phase diagram

The Vicsek model is a minimal model of collective motion, capturing how local alignment interactions can generate macroscopic nonequilibrium order in systems such as bird flocks. In this work, we use active learning to map the Vicsek phase diagram as a function of noise strength, density, and particle speed. A neural-network classifier is trained on global polar-order labels, and classifier entropy is used to select new simulations near uncertain crossover regions. The resulting phase map resolves a high-noise disordered gas, a low-noise polar ordered regime, and an intermediate coexistence-candidate regime whose noise window shifts upward and broadens with increasing density. Independent density and local-order diagnostics indicate that the intermediate regime contains dense, locally ordered bands coexisting with a dilute, weakly ordered background. Comparison with the ordered regime shows that banded coexistence is identified by the joint enhancement of band contrast, local-order heterogeneity, and positive density-order correlation. Overall, these results establish a machine-learning-guided workflow for active matter, in which active learning constructs an operational phase map and independent spatial diagnostics convert classifier-defined regimes into physically interpretable nonequilibrium morphologies.

Grace T. Bai, Brandon B. Le

Department of Computer Science, University of Virginia, Charlottesville, Virginia 22904, USA · Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA

Dynamical Systems Phase Transitions

Apr 29, 2026math.PR

Stochastic Scaling Limits and Synchronization by Noise in Deep Transformer Models

We prove pathwise convergence of the layerwise evolution of tokens in a finite-depth, finite-width transformer model with MultiLayer Perceptron (MLP) blocks to a continuous-time stochastic interacting particle system. We also identify the stochastic partial differential equation describing the evolution of the tokens' distribution in this limit and prove propagation of chaos when the number of such tokens is large. The bounds we establish are quantitative and the limits we consider commute. We further prove that the limiting stochastic model displays synchronization by noise and establish exponential dissipation of the interaction energy on average, provided that the common noise is sufficiently coercive relative to the deterministic self-attention drift. We finally characterize the activation functions satisfying the former condition.

Andrea Agazzi, Giuseppe Bruno, Eloy Mosig García +2

Department of Mathematics and Statistics, University of Bern (CH) · Department of Mathematics, University of Pisa (IT)

Dynamical Systems Transformer Transformer Attention

Apr 29, 2026cs.AI

A Dynamical Framework for Cognitive Processes Based on Transformations and Semantic Equivalence

This paper proposes a structural and dynamical framework for modeling cognitive processes within a cybernetic perspective. Cognitive states are represented as elements of a state space evolving through an iterative update rule of the form

X_{t+1} = π\big(F(f(X_t))\big),

where

f

describes internal transformations,

F

represents interpretative mappings, and

π

enforces semantic equivalence. The model is interpreted as a feedback system integrating transformation, observation, and stabilization. A categorical formulation is introduced to capture compositional structure, while the associated dynamics are analyzed through fixed-point arguments and contraction conditions ensuring stability. To demonstrate the operational character of the framework, a computational illustration is provided, together with a qualitative analysis of the induced dynamics. A concrete linguistic application shows how context-dependent interpretation can be modeled as a trajectory toward a stable semantic class. The proposed approach connects dynamical systems, category theory, and cognitive modeling, and provides a unified representation of cognition as a feedback-driven process evolving toward invariant interpretations.

Carlo Cattani, Dioneia Motta Monte-Serrat

Engineering School, DEIM, University of Tuscia, (VT), 01100, Italy · Department of Physics, University of Sao Paulo, USP; Department of Law, University of Ribeirao Preto, Unaerp, Brazil.

Dynamical Systems Cognitive Modeling

Apr 29, 2026cs.NE

Population Dynamics in ARIEL Robotics Systems Featuring Embodied Evolution via Spatial Mating Mechanisms

We present a Spatially Embedded Evolutionary Algorithm where robot individuals exist in a physically simulated 2D environment, must navigate to encounter potential mates, and compete for survival under various spatially-aware selection pressures. Using HyperNEAT evolved neural controllers for ARIEL gecko-inspired quadrupeds in MuJoCo, we investigate how spatial structure fundamentally alters evolutionary dynamics. Our experiments show a modest 4.9% difference in peak fitness between proximity-based and random pairing possibly within stochastic variation while combining spatial parent selection with stochastic death selection produces unstable population dynamics. We discover a continuous phase transition in energy-based selection experiments, with critical zone count separating extinction-dominated and explosion-dominated regimes. Our density-dependent death selection mechanism achieves 97% completion rates but causes fitness decline, revealing a fundamental dilemma where decoupled mechanisms produce bistable dynamics, positively coupled mechanisms create counter-selection pressures, and only deterministic fitness-based selection maintains stability. These findings provide important constraints for future spatial EA design.

Victoria Peterson, Akshat Srivastava, Raghav Prabhakar

Vrije Universiteit Amsterdam, Netherlands

Dynamical Systems Evolutionary Optimization Evolutionary Robotics

Apr 29, 2026nlin.CD

Inferring bifurcation diagrams of two distinct chaotic systems by a single machine

We propose a dual-channel reservoir-computing scheme for inferring the dynamics of two distinct chaotic systems with a single machine. By augmenting a standard reservoir with a system-label channel and a parameter-control channel, the machine can be trained from time series collected from a few sampled states of the two systems. We show that the trained machine not only predicts the short-time evolution of the sampled states, but also reproduces the long-term statistical properties of unseen states, thereby enabling reconstruction of the bifurcation diagrams of both systems from partial observations. The effectiveness of the scheme is demonstrated for the Lorenz and Rössler systems in numerical simulations and for the Chua and Rossler circuits in experiments. Functional-network analysis further shows that the two target systems are encoded by distinct dynamical patterns in the reservoir. These results extend multifunctional and parameter-aware reservoir computing, and provide a route to data-driven inference of multiple nonlinear systems using a single machine.

Jianmin Guo, Yao Du, Yizhen Yu +2

School of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, China · College of Physics and Electronic Information Engineering, Qinghai Normal University, Xining 810008, China · School of Physics and Electronic Science, East China Normal University, Shanghai 200062, China

Dynamical Systems Nonlinear System Identification Reservoir Computing Chaotic Dynamical Systems

Apr 28, 2026eess.SY

Co-Learning Port-Hamiltonian Systems and Optimal Energy-Shaping Control

We develop a physics-informed learning framework for energy-shaping control of port-Hamiltonian (pH) systems from trajectory data. The proposed approach co-learns a pH system model and an optimal energy-balancing passivity-based controller (EB-PBC) through alternating optimization with policy-aware data collection. At each iteration, the system model is refined using trajectory data collected under the current control policy, and the controller is re-optimized on the updated model. Both components are parameterized by neural networks that embed the pH dynamics and EB-PBC structure, ensuring interpretability in terms of energy interactions. The learned controller renders the closed-loop system inherently passive and provably stable, and exploits passive plant dynamics without canceling the natural potential. A dissipation regularization enforces strict energy decay during training, thereby enhancing robustness to sim-to-real gaps. The proposed framework is validated on state-regulation and swing-up tasks for planar and torsional pendulum systems.

Ankur Kamboj, Biswadip Dey, Vaibhav Srivastava

Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824 · Meta Reality Labs, Redmond, WA 98052

Dynamical Systems Feedback Control System Identification Passivity-Based Control Physics-Informed ML Port-Hamiltonian Systems

Apr 28, 2026cs.LG

Teacher Forcing as Generalized Bayes: Optimization Geometry Mismatch in Switching Surrogates for Chaotic Dynamics

Identity teacher forcing (ITF) enables stable training of deterministic recurrent surrogates for chaotic dynamical systems and has been highly effective for dynamical systems reconstruction (DSR) with recurrent neural networks (RNNs), including interpretable almost-linear RNNs (AL-RNNs). However, as an intervention-based prediction loss (and thus a generalized Bayes update), teacher forcing need not match the free-running model's marginal likelihood geometry. We compare the objective-induced curvatures of ITF and marginal likelihood in a probabilistic switching augmentation of AL-RNNs, estimating ambiguity-aware observed information via Louis' identity. In the switching setting studied here, conditioning on a single forced regime path (as ITF does) inflates curvature, while marginal likelihood curvature is reduced by a missing-information correction when multiple switching explanations remain plausible. In Lorenz-63 experiments, windowed evidence fine-tuning improves held-out evidence but can degrade dynamical quantities of interest (QoIs) relative to ITF-pretrained models.

Andre Herz, Daniel Durstewitz, Georgia Koppe

Interdisciplinary Center for Scientific Computing, Heidelberg University, Germany · Dept. of Theoretical Neuroscience, Central Institute of Mental Health (CIMH), Mannheim, Germany · Hector Institute for AI in Psychiatry and Dept. of Psychiatry and Psychotherapy, CIMH, Mannheim, Germany +1

Neural Surrogate Modeling Dynamical Systems Bayesian Inference Recurrent Neural Networks Chaotic Time-Series Prediction

Apr 28, 2026stat.ML

Residual-loss Anomaly Analysis of Physics-Informed Neural Networks: An Inverse Method for Change-point Detection in Nonlinear Dynamical Systems with Regime Switching

Nonlinear dynamical systems with regime transitions are typically described by ordinary differential equations with jumping parameters parameters. Traditional methods often treat change-point detection and parameter estimation as separate tasks, ignoring the inherent coupling between them. To address this, we propose residual-loss anomaly analysis of physics-informed neural networks, a unified framework that leverages dynamical consistency within the physics-informed learning paradigm. This approach jointly infers piecewise parameters and transition points under a single set of constraints. The method follows a two-stage strategy: First, local physical residuals are analyzed through overlapping subinterval decomposition. When a subinterval spans a true transition point, the residual exhibits a distinct structural elevation in noise-free conditions, which has a non-zero lower bound, enabling effective localization of potential transition intervals. Second, within our framework, change-point locations and piecewise parameters are integrated into a unified physical loss function for joint optimization, enabling simultaneous identification. Experiments on benchmark nonlinear dynamical systems, including Malthusian and logistic growth models, Van der Pol oscillator, Lotka-Volterra model and Lorenz system, demonstrate that the proposed method outperforms traditional decoupled approaches in both change-point localization and parameter estimation accuracy. This study provides an efficient, unified solution for structurally coupled inverse problems in nonlinear dynamical systems with regime switching.

Yuhe Bai, Chengli Tan, Jiaqi Li +2

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China · School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710072, China

Dynamical Systems Regime-Switching Dynamical Systems Nonlinear System Identification Change-Point Detection Physics-Informed ML Anomaly Detection

Apr 28, 2026physics.data-an

Emergent Self-Attention from Astrocyte-Gated Associative Memory Dynamics

We introduce a Hopfield-type associative memory in which effective connectivity is multiplicatively modulated by astrocytic gains evolving under an entropy-regularized replicator equation. The coupled neuron-astrocyte dynamics admit a Lyapunov function, ensuring global convergence. At fixed points, astrocytic gains implement a softmax-normalized allocation over pattern similarity scores, yielding a mechanistic realization of self-attention as emergent routing on the gain simplex. In regimes of high memory load and interference, the model significantly improves retrieval accuracy relative to classical Hopfield dynamics and recent neuron-astrocyte baselines. These results establish a dynamical systems framework linking glial modulation, competitive resource allocation, and attention-like computation.

Arnau Vivet, Alex Arenas

1Departament d’Enginyeria Inform`atica i Matem`atiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain · 2Complexity Science Hub Vienna, Metternichgasse 8, 1030 Vienna, Austria

Softmax Attention Dynamical Systems Self-Attention Replicator Dynamics Neural Processes Hopfield Networks+2

Apr 27, 2026physics.data-an

Information bottleneck for learning the phase space of dynamics from high-dimensional experimental data

Identifying the dynamical state variables of a system from high-dimensional observations is a central problem across physical sciences. The challenge is that the state variables are not directly observable and must be inferred from raw high-dimensional data without supervision. Here we introduce DySIB (Dynamical Symmetric Information Bottleneck) as a method to learn low-dimensional representations of time-series data by maximizing predictive mutual information between past and future observation windows while penalizing representation complexity. This objective operates entirely in latent space and avoids reconstruction of the observations. We apply DySIB to an experimental video dataset of a physical pendulum, where the underlying state space is known. The method, with hyperparameters of the learning architecture set self-consistently by the data, recovers a two-dimensional representation that matches the dimensionality, topology, and geometry of the pendulum phase space, with the learned coordinates aligning smoothly with the canonical angle and angular velocity. These results demonstrate, on a well-characterized experimental system, that predictive information in latent space can be used to recover interpretable dynamical coordinates directly from high-dimensional data.

K. Michael Martini, Eslam Abdelaleem, Paarth Gulati +1

Department of Physics, Emory University, Atlanta, GA, USA · Schools of Physics and Psychology, Georgia Institute of Technology, Atlanta, GA, USA · Initiative in Theory and Modeling of Living Systems, Emory University, Atlanta, GA, USA +1

Dynamical Systems Information Bottleneck Predictive Representation Learning State Representation Learning

Apr 27, 2026cs.CC

Primitive Recursion without Composition: Dynamical Characterizations, from Neural Networks to Polynomial ODEs

What do recurrent neural networks, polynomial ODEs, and discrete polynomial maps each bring to computation, and what do they lack? All three operate over the continuum--real-valued states evolved by real-valued dynamics--even when the target functions are discrete. We study them through primitive recursion. We prove that primitive recursion admits equivalent characterizations in all three frameworks: bounded iteration of a fixed recurrent ReLU network, robust computation by a fixed polynomial ODE, and iteration of a fixed polynomial map with an externally supplied step-size parameter. In each, the time bound is itself primitive recursive, composition emerges from the dynamics rather than as a closure rule, and inputs are raw integer vectors. Every primitive recursive function is first compiled into bounded iteration of a single threshold-affine normal form, then interpreted as a ReLU computation and as a polynomial ODE. The equivalences expose a structural asymmetry: no fixed polynomial map can round uniformly to the nearest integer or realize exact phase selection--operations polynomial ODEs perform robustly via continuous-time flow. Each formalism compensates for a limitation the others lack: the ReLU gate provides exact branching, continuous time provides autonomous rounding and control, and the step-size parameter recovers both at the cost of discretization precision. This opens dynamical characterizations of subrecursive hierarchies and complexity classes by restricting time bounds, polynomial degrees, or discretization resources within one framework. More broadly, these models do not compute by composing subroutines: they shape the trajectory of a dynamical system through clocks, phase selectors, and error correction built into the dynamics. This differs structurally from symbolic programming, and our theorem gives a precise framework to study the difference.

Olivier Bournez

Institut Polytechnique de Paris, Ecole Polytechnique, LIX, Palaiseau, France

Dynamical Systems ReLU Neural Networks Recurrent Neural Networks

Apr 27, 2026cs.LG

Latent-Hysteresis Graph ODEs: Modeling Coupled Topology-Feature Evolution via Continuous Phase Transitions

Graph neural ordinary differential equations (Graph ODEs) extend graph learning from discrete message-passing layers to continuous-time representation flows. While it supports adaptive long-range propagation, we show that Graph ODEs with strictly positive irreducible mixing operators face an inherent \emph{monostability trap}: in the long-time regime, information leakage is unavoidable and the dynamics converge to a single global consensus attractor. We propose the \textbf{Hysteresis Graph ODE (HGODE)}, which couples feature evolution with a latent topological potential driven by a learned pairwise force. A double-well edge potential and bipolarized gate allow edge states to polarize into connected or insulated phases while preserving differentiability. We provide asymptotic analysis of the collapse mechanism and the proposed hysteretic topology dynamics, and validate HGODE on theory-driven synthetic diagnostics and real-world graph benchmarks.

Qinhan Hou, Jing Tang

Doctoral Program of Computer Science, University of Helsinki, Helsinki, Finland · Research Program in Systems Oncology, Faculty of Medicine, University of Helsinki, Helsinki, Finland

Dynamical Systems Graph Structure Learning Graph Neural Networks Graph Representation Learning Phase Transitions Neural ODEs

Apr 26, 2026cs.LG

Cardiac Stability Theory: An Axiomatically Grounded Framework for Continuous Cardiac Health Monitoring via Smartphone Photoplethysmography

We present Cardiac Stability Theory (CST), an axiomatically grounded framework formally defining cardiovascular health as a stability margin around a cardiac dynamical attractor. From four axioms we derive the Cardiac Stability Index (CSI), a composite scalar in [0,1] integrating the largest Lyapunov exponent, recurrence determinism, and signal entropy via time-delay embedding. The ECG-based model (CSISurrogateV2, CNN-Transformer) achieves

R^2=0.8788

, MAE

=0.0234

on PTB-XL (21,799 recordings). We extend CSI to smartphone PPG via Complementary Domain Transfer (CDT): CSISurrogateV2 generates pseudo-labels for the BUT PPG dataset (48 recordings, 12 subjects), training TinyCSINet (122,849 parameters), achieving MAE

=0.0557

,

ρ=0.660

on the held-out test set (

n=1065

windows) at

{<}30

ms mobile latency. CDT is validated on BIDMC, Welltory, and RWS-PPG. Paired validation on 5,035 BIDMC windows yields

r=0.454

(

ρ=0.485

,

p<10^{-295}

), confirming correlated cardiac stability across modalities. CSI is negatively correlated with age (slope

= -0.000225

CSI/year, PTB-XL), discriminates atrial fibrillation from normal sinus rhythm (AUROC

=0.89

), and is robust under Perturbation Invariance Training (max AUC drop 1.65%). We derive HeartSpan, a longitudinal stability metric relative to population age norms, enabling continuous non-invasive cardiac monitoring from commodity smartphones for longevity tracking and cardiac risk stratification.

Timothy Oladunni, Farouk Ganiyu Adewumi

Department of Computer Science, Morgan State University, Baltimore, MD 21251, USA

Dynamical Systems Photoplethysmography Wearable Health Monitoring

Apr 26, 2026cs.LG

Hamiltonian Graph Inference Networks: Joint structure discovery and dynamics prediction for lattice Hamiltonian systems from trajectory data

Lattice Hamiltonian systems underpin models across condensed matter, nonlinear optics, and biophysics, yet learning their dynamics from data is obstructed by two unknowns: the interaction topology and whether node dynamics are homogeneous. Existing graph-based approaches either assume the graph is given or, as in

α

-separable graph Hamiltonian network, infer it only for separable Hamiltonians with homogeneous node dynamics. We introduce the Hamiltonian Graph Inference Network (HGIN), which jointly recovers the interaction graph and predicts long-time trajectories from state data alone, for both separable and non-separable Hamiltonians and under heterogeneous node dynamics. HGIN couples a structure-learning module -- a learnable weighted adjacency matrix trained under a Hamilton's-equations loss -- with a trajectory-prediction module that partitions edges into physically distinct subgraphs via

k

-means clustering, assigning each subgraph its own encoder and thereby breaking the parameter-sharing bottleneck of conventional GNNs. On three benchmarks -- a Klein--Gordon lattice with long-range interactions and two discrete nonlinear Schrödinger lattices (homogeneous and heterogeneous) -- HGIN reduces long-time energy prediction error and trajectory prediction error by six to thirteen orders of magnitude relative to baselines. A symmetry argument on the Hamiltonian loss further shows that the learned weights encode the parity of the underlying pair potential, yielding an interpretable readout of the system's interaction structure.

Ru Geng, Panayotis Kevrekidis, Yixian Gao +2

School of Mathematics and Statistics, Changchun University, Changchun, 130022, China · Department of Mathematics and Statistics, University of Massachusetts, Amherst, 01003, MA, USA · Department of Physics, University of Massachusetts, Amherst, 01003, MA, USA +2

Dynamical Systems Graph Structure Learning Graph Neural Networks Graph Representation Learning System Identification Physics-Informed ML+2

Apr 24, 2026cs.RO

Learning Control Policies to Provably Satisfy Hard Affine Constraints for Black-Box Hybrid Dynamical Systems

Ensuring safety for black-box hybrid dynamical systems presents significant challenges due to their instantaneous state jumps and unknown explicit nonlinear dynamics. Existing solutions for strict safety constraint satisfaction, like control barrier functions (CBFs) and reachability analysis, rely on direct knowledge of the dynamics. Similarly, safe reinforcement learning (RL) approaches often rely on known system dynamics or merely discourage safety violations through reward shaping. In this work, we want to learn RL policies which provably satisfy affine state constraints in closed loop for black-box hybrid dynamical systems with affine reset maps. Our key insight is forcing the RL policy to be affine and repulsive near the constraint boundaries for the unknown nonlinear dynamics of the system, providing guarantees that the trajectories will not violate the constraint. We further account for constraint violation due to instantaneous state jumps that occur due to impacts or reset maps in the hybrid system by introducing a second repulsive affine region before the reset that prevents post-reset states from violating the constraint. We derive sufficient conditions under which these policies satisfy safety constraints in closed loop. We also compare our approach with state-of-the-art reward shaping and learned-CBF methods on hybrid dynamical systems like the constrained pendulum and paddle juggler environments. In both scenarios, we show that our methodology learns higher quality policies while always satisfying the safety constraints.

Aayushi Shrivastava, Kartik Nagpal, Sairam Jinkala +2

Department of Mechanical Engineering, University of California Berkeley, Berkeley, CA 94709, USA

Dynamical Systems Safety-Critical Control Constrained RL Safe RL

Apr 23, 2026math.OC

Model Predictive Control of Hybrid Dynamical Systems

The problem of controlling hybrid dynamical systems using model predictive control (MPC) is formulated and sufficient conditions for asymptotic stability of a set are provided. Hybrid dynamical systems are modeled in terms of hybrid equations, involving a differential equation and a difference equation with inputs and constraints. The proposed hybrid MPC algorithm uses a suitable prediction and control horizon construction inspired by hybrid time domains. Structural properties of the hybrid optimization problem, its feasible set, and its value function are provided. Checkable conditions to guarantee asymptotic stability of a set are provided. These conditions are given in terms of properties on the stage cost, terminal cost, and the existence of static state-feedback laws, related through a control Lyapunov function condition. Examples illustrate the results throughout the paper.

Ricardo G. Sanfelice, Berk Altin

Department of Electrical and Computer Engineering, University of California, Santa Cruz, CA, 94107 USA

Dynamical Systems Model Predictive Control Optimal Control

Apr 23, 2026cs.NE

Neuromorphic Computing Based on Parametrically-Driven Oscillators and Frequency Combs

Parametrically driven oscillators provide a natural platform for neuromorphic computation, where nonlinear mode coupling and intrinsic dynamics enable both memory and high-dimensional transformation. Here, we investigate a two-mode system exhibiting 2:1 parametric resonance and demonstrate its operation as a reservoir computer across distinct dynamical regimes, including sub-threshold, parametric resonance, and frequency-comb states. By encoding input signals into the drive amplitude and sampling the resulting temporal and spectral responses, we perform one step-ahead prediction of benchmark chaotic systems, including Mackey-Glass, Rossler, and Lorenz dynamics. We find that optimal computational performance is achieved within the parametric resonance regime, where nonlinear interactions are activated while temporal coherence is preserved. In contrast, although frequency-comb states introduce increased spectral dimensionality, their performance is not consistently good across their existence band and also degrades in the chaotic comb regime due to loss of phase coherence. Mapping prediction error over parameter space reveals a direct correspondence between computational capability and the underlying bifurcation structure, with low-error regions aligned with the parametric resonance boundary. We further show that the input modulation, the detuning from the frequency matching condition, damping ratio, and input data rate systematically control the accessible dynamical regimes and thereby the computational performance. These results establish parametric resonance as a robust operating regime for oscillator-based reservoir computing and provide design principles for tuning physical systems toward optimal neuromorphic functionality.

Mahadev Sunil Kumar, Adarsh Ganesan

Amrita School of Computing, Amrita Vishwa Vidyapeetham, Kollam, India 690525 · Department of Electrical and Electronics Engineering, Birla Institute of Technology and Science, Pilani – Dubai Campus, Dubai, United Arab Emirates 345055

Dynamical Systems Reservoir Computing Neuromorphic Computing Chaotic Time-Series Prediction