Nonlinear Dynamics

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11 papers in the last four weeks, up 175% on the four weeks before. 0.1% of all new papers.

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Latest papers 88

May 12, 2026cs.LG

Identifying the nonlinear string dynamics with port-Hamiltonian neural networks

Hybrid machine learning combines physical knowledge with data-driven models to enhance interpretability and performance. In this context, Port-Hamiltonian Systems (PHS), which generalize Hamiltonian mechanics to describe open, non-autonomous dynamical systems, have been successfully integrated with neural networks under the name Port-Hamiltonian Neural Networks (PHNNs). While the ability of PHNNs to identify Hamiltonian ordinary differential equation (ODE) systems has already been demonstrated, their application to learning Hamiltonian partial differential equation (PDE) systems remains largely unexplored. This limitation restricts their use in musical acoustics, where instruments are typically modeled as distributed parameter systems governed by PDEs. In this work, we demonstrate how to learn the nonlinear string dynamics from data in a physically-consistent framework through a PHNN extension to PDEs. By constructing structured neural network architectures based on PHS, we can recover both the Hamiltonian governing the string and the dissipation affecting it. This approach outperforms baseline, non-physics-informed methods in terms of both accuracy and interpretability. Numerical experiments using synthetic data demonstrate the ability of the proposed PHNN model to identify and emulate the nonlinear dynamics of the system.

Maximino Linares, Guillaume Doras, Thomas Hélie

1IRCAM, 1, place Igor-Stravinsky, 75004, Paris, France · 2Sorbonne Université, 4, place Jussieu, 75005, Paris, France · 3CNRS, 3 rue Michel-Ange, 75016, Paris, France

Parametric Physics-Informed Neural Network Nonlinear Dynamics Neural Dynamics Hamiltonian Neural Network

May 12, 2026cs.LG

Parallel-in-Time Training of Recurrent Neural Networks for Dynamical Systems Reconstruction

Reconstructing nonlinear dynamical systems (DS) from data (DSR) is a fundamental challenge in science and engineering, but it inherently relies on sequential models. Recent breakthroughs for sequential models have produced algorithms that parallelize computation along sequence length

T

, achieving logarithmic time complexity,

\mathcal{O}(\log T)

. Since sequence lengths have been practically limited due to the linear runtime complexity

\mathcal{O}(T)

of classical backpropagation through time, this opens new avenues for DSR. This paper studies two prominent classes of parallel-in-time algorithms for this task, both of which leverage parallel associative scans as their core computational primitive. The first class comprises models with linear yet non-autonomous dynamics and a nonlinear readout, such as modern State Space Models (SSMs), while the second consists of general nonlinear models which can be parallelized using the DEER framework. We find that the linear training-time recurrence of the first class of models imposes limitations that often hinder learning of accurate nonlinear dynamics. To address this, we augment DEER with Generalized Teacher Forcing (GTF), a novel variant within the more general nonlinear framework that ensures stable and effective learning of nonlinear dynamics across arbitrary sequence lengths. Using GTF-DEER, we investigate the benefits of training on extremely long sequences (

T>10^4

) for DSR. Our results show that access to such long trajectories significantly improves DSR if the data features long time scales. This work establishes GTF-DEER as a robust tool for data-driven discovery and underscores the largely untapped potential of long-sequence learning in modeling complex DS.

Florian Hess, Florian Götz, Daniel Durstewitz

1Dept. of Theoretical Neuroscience, Central Institute of Mental Health,2026 Mannheim, Germany · Faculty of Mathematics and Computer Science, Heidelberg University, Germany · Faculty of Physics and Astronomy, Heidelberg University, Germany +1

Recurrent Neural Networks Recurrent Model Nonlinear Dynamics Sequence Modeling Backpropagation

May 12, 2026cs.LG

Approximation Theory of Laplacian-Based Neural Operators for Reaction-Diffusion System

Neural operators provide a framework for learning solution operators of partial differential equations (PDEs), enabling efficient surrogate modeling for complex systems. While universal approximation results are now well understood, approximation analysis specific to nonlinear reaction-diffusion systems remains limited. In this paper, we study neural operators applied to the solution mapping from initial conditions to time-dependent solutions of a generalized Gierer-Meinhardt reaction-diffusion system, a prototypical model of nonlinear pattern formation. Our main results establish explicit approximation error bounds in terms of network depth, width, and spectral rank by exploiting the Laplacian spectral representation of the Green's function underlying the PDE. We show that the required parameter complexity grows at most polynomially with respect to the target accuracy, demonstrating that Laplacian eigenfunction-based neural operator architectures alleviate the curse of parametric complexity encountered in generic operator learning. Numerical experiments on the Gierer-Meinhardt system support the theoretical findings.

Takashi Furuya, Ryo Ozawa, Jenn-Nan Wang

Doshisha University · RIKEN AIP · Tohoku University +1

Fourier Neural Operators Neural Operators Neural Approximations Nonlinear Dynamics Structural Complexity

May 10, 2026cs.LG

Discovery of Nonlinear Dynamics with Automated Basis Function Generation

Discovering governing equations from observational data remains a fundamental challenge in scientific modeling, particularly when the underlying mathematical structure is unknown. Traditional sparse identification methods like SINDy excel at discovering parsimonious models but require researchers to specify candidate basis functions a priori, a limitation that often leads to model failure when critical terms are omitted or when systems exhibit unconventional dynamics. Purely symbolic regression approaches offer unlimited flexibility but struggle with noise sensitivity and frequently produce overly complex, unstable equations. We present AutoSINDy, a hybrid Discovery-then-Solve framework that combines the exploratory power of symbolic regression with the robust sparsity-promoting capabilities of SINDy. Our method operates in three stages: (1) PySR-based symbolic regression discovers candidate functional forms from bootstrapped data chunks; (2) a curation pipeline decomposes, expands, and filters these expressions using collinearity analysis to construct a minimal yet comprehensive library; and (3) SINDy identifies sparse governing equations from this custom-tailored library. Extensive experiments across canonical nonlinear systems demonstrate that AutoSINDy consistently recovers ground-truth equations even under high observational noise, achieving a ground-truth recovery rate of 92.8% across all trials. Compared with standard SINDy using enriched libraries and standalone symbolic regression, AutoSINDy achieves higher predictive accuracy, superior generalization to unseen trajectories, and substantially lower symbolic complexity.

Mohammad Amin Basiri, Charles Nicholson

Data Science and Analytics Institute, University of Oklahoma, Norman, OK, USA · School of Industrial and Systems Engineering, University of Oklahoma, Norman, OK, USA

Symbolic Regression System Identification Nonlinear Dynamics Model Discovery Basis Functions Discovery

May 9, 2026physics.comp-ph

Nonlinear GENERIC Informed Neural Networks (N-GINNs): learning GENERIC dynamics with non-quadratic dissipation potentials

We introduce Nonlinear GENERIC Informed Neural Networks (N-GINNs), a deep learning framework for discovering evolution equations of systems governed by the nonlinear GENERIC formalism (General Equation for Non-Equilibrium Reversible-Irreversible Coupling). Such systems exhibit coupled conservative and dissipative dynamics, and can be described via the superposition of a Hamiltonian flow and a generalized gradient flow. In contrast to existing approaches, our formulation incorporates generalized gradient flows via convex dissipation potentials, enabling the identification of a broader class of thermodynamically consistent dynamics, including systems with non-quadratic dissipation potentials. Thermodynamic structure is strongly enforced by construction through suitable reparameterizations of both the bivector operator and the dissipation potential, ensuring exact compliance with the first and second laws of thermodynamics. We validate the proposed approach on three representative examples: a harmonic oscillator coupled to a heat bath, an idealized chemical motor, and a one-dimensional viscoplastic model of Perzyna type. These results demonstrate the method's ability to accurately infer thermodynamically consistent models from data for systems incorporating both conservative and nonlinear dissipative dynamics.

Vojtěch Votruba, Zequn He, Weilun Qiu +2

Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Prague, 18675, Czech Republic · Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, 19104, PA, United States of America

Thermodynamics Nonlinear Dynamics Parametric Physics-Informed Neural Network

May 8, 2026cs.LG

StreamPhy: Streaming Inference of High-Dimensional Physical Dynamics via State Space Models

Inferring the evolution of high-dimensional and multi-modal (e.g., spatio-temporal) physical fields from irregular sparse measurements in real time is a fundamental challenge in science and engineering. Existing approaches, including diffusion-based generative models and functional tensor methods, typically operate in offline settings, depend on full temporal observations, or incur substantial inference cost. We propose StreamPhy, an end-to-end framework that enables efficient and accurate streaming inference of full-field physical dynamics from incoming irregular sparse measurements. The framework integrates a data-adaptive observation encoder that is robust to arbitrary observation patterns, a structured state-space model that supports memory-efficient online updates across irregular time intervals, and an expressive Functional Tensor Feature-wise Linear Modulation (FT-FiLM) decoder for continuous-field generation. We prove that FT-FiLM is more expressive than the functional Tucker model, admitting a richer function class for handling complex dynamics. Experiments on three representative physical systems under challenging sampling patterns show that StreamPhy consistently outperforms state-of-the-art baselines, with at least 48% improvement in accuracy and up to 20--100X faster inference than diffusion-based methods.

Panqi Chen, Yifan Sun, Shikai Fang +2

College of Information Science and Electronic Engineering, Zhejiang University · School of EECS,Oregon State University

Streaming Inference Spatiotemporal Fields Inference Dynamics State Space Models Streaming Nonlinear Dynamics

May 8, 2026eess.SY

A Behavioral Framework for Data-Driven Modeling of Nonlinear Systems in Vector-Valued Reproducing Kernel Hilbert Spaces

We generalize Jan Willems' behavioral approach to a class of discrete-time nonlinear systems in a vector-valued reproducing kernel Hilbert space (RKHS). Apart from linear time-invariant systems, this class covers nonlinear systems modeled by Volterra series and their autoregressive variants, as well as systems admitting Hammerstein-type state-space realizations. We apply the proposed framework to the problem of data-driven modeling of such systems, i.e., when simulation or control objectives for an unknown system are carried out without an explicit system identification step. To that end, we link the behavioral approach to two data-driven modeling methods in a vector-valued RKHS: (1) minimum-norm interpolation and (2) subspace identification.

Boya Hou, Maxim Raginsky

Carl R. Woese Institute for Genomic Biology and the Coordinated Science Laboratory, University of Illinois Urbana-Champaign, Urbana, IL, 61801 USA · Department of Electrical and Computer Engineering and the Coordinated Science Laboratory, University of Illinois Urbana-Champaign, Urbana, IL, 61801 USA

System Identification Reproducing Kernel Hilbert Spaces Nonlinear Dynamics Kernel Hilbert Spaces State Space Models Data-Driven

May 7, 2026cs.LG

Towards Scalable One-Step Generative Modeling for Autoregressive Dynamical System Forecasting

Fast surrogate modeling for high-dimensional physical dynamics requires more than low short-term error: useful models must roll out efficiently while preserving the statistical structure of long trajectories. Neural operators provide inexpensive autoregressive forecasts but can drift in turbulent regimes, whereas rolling diffusion and latent generative surrogates can represent stochastic transitions at the cost of multi-step denoising, noise-schedule design, or auxiliary compression models. We propose MeanFlow Long-term Invariant Spatiotemporal Consistency Autoregressive Models (MeLISA), a latent-free autoregressive generative surrogate built on pixel-space MeanFlow. MeLISA defines a blockwise stochastic transition kernel that generates each forecast block with a single model evaluation, avoiding latent encoders and iterative diffusion solvers at inference time. To stabilize long-horizon rollouts, MeLISA combines a Window-Consistency MeanFlow objective that learns conditional spatiotemporal generation from partially observed temporal windows with a Time Increment Consistency loss that constrains multi-lag finite increments and targets temporal-correlation structure. We evaluate MeLISA with compact UNet and scalable DiT backbones on two high-resolution benchmarks, extended 2D Kolmogorov flow at

256 \times 256

and turbulent channel-flow slice at

192 \times 192

. MeLISA outperforms neural-operator baselines on short-term forecasting accuracy and long-horizon statistical metrics, including energy spectra, turbulent kinetic energy, and mixing-rate-related dynamics, while achieving inference speeds comparable to, and in some cases faster than, neural operators. Compact 3.7-5.7M-parameter variants already deliver strong parameter efficiency, and DiT variants provide a scalable path up to 150M parameters. Overall, MeLISA benefits both rollout efficiency and long-horizon statistical accuracy.

Tianyue Yang, Xiao Xue

The Center for Computational Science University College London

Long-Horizon Forecasting Autoregressive Model Generative Models Spatiotemporal Representations Nonlinear Dynamics

May 6, 2026cs.LG

Koopman Identification of Nonlinear Systems via Reservoir Liftings

Learning tractable linear representations of nonlinear dynamical systems via Koopman operator theory is often hindered by dictionary selection, temporal memory encoding, and numerical ill-conditioning. Inspired by Reservoir Computing (RC) paradigm, this paper introduces the RC-Koopman framework, which interprets reservoir as a stateful, finite-dimensional Koopman dictionary whose temporal depth is explicitly controlled by its spectral radius. We show that the Echo State Property (ESP) guarantees well-posedness and favorable numerical conditioning of the lifted Koopman approximation. A correlation-based spectral radius selection algorithm aligns reservoir memory with dominant system timescales. Analysis reveals how the finite memory of the reservoir determines which Koopman eigenfunctions remain observable from the lifted features. Evaluation on synthetic benchmarks demonstrates that RC-Koopman achieves a favorable balance between reconstruction accuracy of the underlying nonlinear dynamics and dynamical stability, compared to Extended Dynamic Mode Decomposition (EDMD) and Hankel-based lifting approaches. Code available at: https://github.com/NEAR-the-future/RC-Koopman.git

Weibin Gu, Chen Yang, Lu Shi

Institute for AI Industry Research (AIR), Tsinghua University, Beijing, PR China · School of Engineering, China University of Petroleum-Beijing at Karamay, Xinjiang Uygur Autonomous Region, PR China

Koopman Operator Reservoir Computing Nonlinear Dynamics

May 5, 2026cs.RO

Robust Path Tracking for Vehicles via Continuous-Time Residual Learning: An ICODE-MPPI Approach

Model Predictive Path Integral (MPPI) control is a powerful sampling-based strategy for nonlinear autonomous systems. However, its performance is often bottlenecked by the fidelity of nominal dynamics. We propose ICODE-MPPI, a robust framework that leverages Input Concomitant Neural Ordinary Differential Equations (ICODEs) to learn and compensate for unmodeled residual dynamics. Unlike discrete-time learners, ICODEs maintain physical consistency and temporal continuity during the MPPI prediction horizon. High-fidelity simulations on complex trajectories demonstrate that ICODE-MPPI achieves up to a 69% reduction in cross-tracking error under persistent disturbances compared to standard MPPI control. Furthermore, our analysis confirms that ICODE-MPPI significantly suppresses control chattering, yielding smoother steering commands and superior robust performance.

Shugen Song, Wenjie Mei, Chengyan Zhao

School of Automation, Southeast University, Nanjing 210096, China · School of Automation and the Key Laboratory of MCCSE of the Ministry of Education, Southeast University, Nanjing 210096, China · Graduate School of Computer Science and Systems Engineering, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka-shi, Fukuoka, Japan

Model Predictive Path Integral Residual Learning Nonlinear Dynamics Neural Ordinary Differential Equations

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

Koopman Operator Dynamic Mode Decomposition Nonlinear Dynamics

Apr 30, 2026cs.RO

RopeDreamer: A Kinematic Recurrent State Space Model for Dynamics of Flexible Deformable Linear Objects

The robotic manipulation of Deformable Linear Objects (DLOs) is a fundamental challenge due to the high-dimensional, non-linear dynamics of flexible structures and the complexity of maintaining topological integrity during contact-rich tasks. While recent data-driven methods have utilized Recurrent and Graph Neural Networks for dynamics modeling, they often struggle with self-intersections and non-physical deformations, such as tangling and link stretching. In this paper, we propose a latent dynamics framework that combines a Recurrent State Space Model with a Quaternionic Kinematic Chain representation to enable robust, long-term forecasting of DLO states. By encoding the DLO as a sequence of relative rotations (quaternions) rather than independent Cartesian positions, we inherently constrain the model to a physically valid manifold that preserves link-length constancy. Furthermore, we introduce a dual-decoder architecture that decouples state reconstruction from future-state prediction, forcing the latent space to capture the underlying physics of deformation. We evaluate our approach on a large-scale simulated dataset of complex pick-and-place trajectories involving self-intersections. Our results demonstrate that the proposed model achieves a 40.52% reduction in open-loop prediction error over 50-step horizons compared to the state-of-the-art baseline, while reducing inference time by 31.17%. Our model further maintains superior topological consistency in scenarios with multiple crossings, proving its efficacy as a compositional primitive for long-horizon manipulation planning.

Tim Missal, Lucas Domingues, Berk Guler +3

Technical University of Darmstadt · School of Electrical and Computer Engineering, Universidade Estadual de Campinas (UNICAMP), Brazil · Instituto de Pesquisas Eldorado, Brazil +5

Deformable Linear Objects Latent Dynamics Nonlinear Dynamics Recurrence

Apr 30, 2026cs.LG

Toward Scalable SDN for LEO Mega-Constellations: A Graph Learning Approach

Terrestrial network limitations drive the integration of non-terrestrial networks (NTNs), notably mega-constellations comprising thousands of low Earth orbit (LEO) satellites. While these satellites act as interconnected network switches via inter-satellite links (ISLs), their massive scale creates severe bottlenecks for network management. To address this, we propose a scalable, hierarchical software-defined networking (SDN) framework. Our architecture leverages graph neural networks (GNNs) to compactly represent the constellation topology, and Koopman theory to linearize nonlinear dynamics. Specifically, a Graph Koopman Autoencoder (GKAE) forecasts spatio-temporal behavior within a linear subspace for each orbital shell. A central SDN controller then aggregates these shell-level predictions for globally coordinated control. Simulations on the Starlink constellation demonstrate that our approach achieves at least a 42.8% improvement in spatial compression and a 10.81% improvement in temporal forecasting compared to established baselines, all while utilizing a significantly smaller model footprint.

Sivaram Krishnan, Bassel Al Homssi, Zhouyou Gu +3

School of Electrical and Mechanical Engineering, University of Adelaide, Adelaide 5000, Australia

Low Earth Orbit Satellite Networks Optical Networks Graph Representation Learning Nonlinear Dynamics Autoencoder Architectures

Apr 29, 2026cs.LG

PiGGO: Physics-Guided Learnable Graph Kalman Filters for Virtual Sensing of Nonlinear Dynamic Structures under Uncertainty

Digital twins provide a powerful paradigm for diagnostic and prognostic tasks in the monitoring and control of engineered systems; however, their deployment for complex structures remains challenged by model-form uncertainty, arising from unknown nonlinear dynamics, and by sparse sensing. These limitations hinder reliable online state estimation using either purely physics-based or purely data-driven approaches. This work introduces the Physics-Guided Graph Neural ODE (PiGGO) framework, a physics-informed, graph-based Bayesian state estimation approach in which a learned graph neural ordinary differential equation (GNODE) serves as the continuous-time state-transition model within an extended Kalman filter. The graph representation explicitly defines the system state-space, while physics-guided inductive biases encode known structural relationships and constrain the learning of nonlinear dynamics. By integrating graph-native learned dynamics with recursive Bayesian filtering, the proposed PiGGO framework enables online virtual sensing and uncertainty-aware state estimation for nonlinear systems with unknown model form, while maintaining generalisation across topologically similar structures. Numerical case studies demonstrate improved robustness to model uncertainty and measurement noise, outperforming both open-loop graph neural models and conventional filtering approaches in online prediction tasks.

Marcus Haywood-Alexander, Gregory Duthé, Eleni Chatzi

aDepartment of Civil, Environmental and Geomatic Engineering, ETH Zürich, Wolfgang-Pauli Strasse 5, 8049, Zürich, Switzerland · bFuture Resilient Systems, Singapore-ETH Centre, Singapore, Singapore

State Estimation Extended Kalman Filter Physics-Guided Machine Learning Nonlinear Dynamics Neural Ordinary Differential Equations Uncertainty

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

Change-Point Detection Nonlinear Dynamics Dynamical Systems Inverse Design

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

Safety Constraints Control Barrier Functions Nonlinear Dynamics Affine Black Box

Apr 21, 2026physics.flu-dyn

A neural operator framework for data-driven discovery of stability and receptivity in physical systems

Understanding how complex systems respond to perturbations, such as whether they will remain stable or what their most sensitive patterns are, is a fundamental challenge across science and engineering. Traditional stability and receptivity (resolvent) analyses are powerful but rely on known equations and linearization, limiting their use in nonlinear or poorly modeled systems. Here, we introduce a data-driven framework that automatically identifies stability properties and optimal forcing responses from observation data alone, without requiring governing equations. By training a neural network as a dynamics emulator and using automatic differentiation to extract its Jacobian, we can compute eigenmodes and resolvent modes directly from data. We demonstrate the method on both canonical chaotic models and high-dimensional fluid flows, successfully identifying dominant instability modes and input-output structures even in strongly nonlinear regimes. By leveraging a neural network-based emulator, we readily obtain a nonlinear representation of system dynamics while additionally retrieving intricate dynamical patterns that were previously difficult to resolve. This equation-free methodology establishes a broadly applicable tool for analyzing complex, high-dimensional datasets, with immediate relevance to grand challenges in fields such as climate science, neuroscience, and fluid engineering.

Chengyun Wang, Liwei Chen, Nils Thuerey

School of Computation, Information and Technology, Technical University of Munich, Boltzmannstrasse 3, D-85748 Garching bei München, Germany · Beijing Institute of Astronautical Systems Engineering, Beijing, China

Nonlinear Dynamics Neural Operators

Apr 20, 2026cs.LG

AC-SINDy: Compositional Sparse Identification of Nonlinear Dynamics

We present AC-SINDy, a compositional extension of the Sparse Identification of Nonlinear Dynamics (SINDy) framework that replaces explicit feature libraries with a structured representation based on arithmetic circuits. Rather than enumerating candidate basis functions, the proposed approach constructs nonlinear features through compositions of linear functions and multiplicative interactions, yielding a compact and scalable parameterization and enabling sparsity to be enforced directly over the computational graph. We also introduce a formulation that separates state estimation from dynamics identification by combining latent state inference with shared dynamics and multi-step supervision, improving robustness to noise while preserving interpretability. Experiments on nonlinear and chaotic systems demonstrate that the method recovers accurate and interpretable governing equations while scaling more favorably than standard SINDy.

Peter Racioppo

Independent Researcher Los Angeles, CA 90731

System Identification Nonlinear Dynamics Nonlinearity Latent States

Apr 20, 2026cs.RO

Locomotion of an Elastic Snake Robot via Natural Dynamics

Nature suggests that exploiting the elasticities and natural dynamics of robotic systems could increase their locomotion efficiency. Prior work on elastic snake robots supports this hypothesis, but has not fully exploited the nonlinear dynamic behavior of the systems. Recent advances in eigenmanifold theory enable a better characterization of the natural dynamics in complex nonlinear systems. This letter investigates if and how the nonlinear natural dynamics of a kinematic elastic snake robot can be used to design efficient gaits. Two types of gaits based on natural dynamics are presented and compared to a state-of-the-art approach using dynamics simulations. The results reveal that a gait generated by switching between two nonlinear normal modes does not improve the locomotion efficiency of the robot. In contrast, gaits based on non-brake periodic trajectories (non-brake orbits) are perfectly efficient in the energy-conservative case. Further simulations with friction reveal that, in a more realistic scenario, non-brake orbit gaits achieve higher efficiency compared to the baseline gait on the rigid system. Overall, the investigation offers promising insights into the design of gaits based on natural dynamics, fostering further research.

Tristan Ehlert, Arne Sachtler, Annika Schmidt +2

German Aerospace Center (DLR), Robotics and Mechatronics Center (RMC), Münchner Str. 20, 82234 Weßling, Germany · Technical University of Munich (TUM), Department of Computer Engineering; Boltzmannstraße 3, 85748 Garching, Germany

Locomotion Nonlinear Dynamics Rigid Objects Robot Systems

Apr 18, 2026quant-ph

Q-SINDy: Quantum-Kernel Sparse Identification of Nonlinear Dynamics with Provable Coefficient Debiasing

Quantum feature maps offer expressive embeddings for classical learning tasks, and augmenting sparse identification of nonlinear dynamics (SINDy) with such features is a natural but unexplored direction. We introduce \textbf{Q-SINDy}, a quantum-kernel-augmented SINDy framework, and identify a specific failure mode that arises: \emph{coefficient cannibalization}, in which quantum features absorb coefficient mass that rightfully belongs to the polynomial basis, corrupting equation recovery. We derive the exact cannibalization-bias formula

Δξ_P = (P^\top P)^{-1}P^\top Q\,\hatξ_Q

and prove that orthogonalizing quantum features against the polynomial column space at fit time eliminates this bias exactly. The claim is verified numerically to machine precision (

<10^{-12}

) on multiple systems. Empirically, across six canonical dynamical systems (Duffing, Van der Pol, Lorenz, Lotka-Volterra, cubic oscillator, Rössler) and three quantum feature map architectures (ZZ-angle encoding, IQP, data re-uploading), orthogonalized Q-SINDy consistently matches vanilla SINDy's structural recovery while uncorrected augmentation degrades true-positive rates by up to 100%. A refined dynamics-aware diagnostic,

R^2_Q

for

\dot X

, predicts cannibalization severity with statistical significance (Pearson

r=0.70

,

p=0.023

). An RBF classical-kernel control across 20 hyperparameter configurations fails more severely than any quantum variant, ruling out feature count as the cause. Orthogonalization remains robust under depolarizing hardware noise up to 2% per gate, and the framework extends without modification to Burgers' equation.

Samrendra Roy, Syed Bahauddin Alam

Department of Nuclear, Plasma, and Radiological Engineering University of Illinois at Urbana-Champaign Urbana, IL, USA · Department of Nuclear, Plasma, and Radiological Engineering University of Illinois at Urbana-Champaign National Center for Supercomputing Applications Urbana, IL, USA

System Identification Nonlinear Dynamics Coefficients Incentives

Apr 16, 2026cs.LG

One-shot learning for the complex dynamical behaviors of weakly nonlinear forced oscillators

Extrapolative prediction of complex nonlinear dynamics remains a central challenge in engineering. This study proposes a one-shot learning method to identify global frequency-response curves from a single excitation time history by learning governing equations. We introduce MEv-SINDy (Multi-frequency Evolutionary Sparse Identification of Nonlinear Dynamics) to infer the governing equations of non-autonomous and multi-frequency systems. The methodology leverages the Generalized Harmonic Balance (GHB) method to decompose complex forced responses into a set of slow-varying evolution equations. We validated the capabilities of MEv-SINDy on two critical Micro-Electro-Mechanical Systems (MEMS). These applications include a nonlinear beam resonator and a MEMS micromirror. Our results show that the model trained on a single point accurately predicts softening/hardening effects and jump phenomena across a wide range of excitation levels. This approach significantly reduces the data acquisition burden for the characterization and design of nonlinear microsystems.

Teng Ma, Luca Rosafalco, Wei Cui +2

System Identification Nonlinear Dynamics Oscillatory Dynamics Single Excitation Time History

Apr 16, 2026cs.LG

SOLIS: Physics-Informed Learning of Interpretable Neural Surrogates for Nonlinear Systems

Nonlinear system identification must balance physical interpretability with model flexibility. Classical methods yield structured, control-relevant models but rely on rigid parametric forms that often miss complex nonlinearities, whereas Neural ODEs are expressive yet largely black-box. Physics-Informed Neural Networks (PINNs) sit between these extremes, but inverse PINNs typically assume a known governing equation with fixed coefficients, leading to identifiability failures when the true dynamics are unknown or state-dependent. We propose \textbf{SOLIS}, which models unknown dynamics via a \emph{state-conditioned second-order surrogate model} and recasts identification as learning a Quasi-Linear Parameter-Varying (Quasi-LPV) representation, recovering interpretable natural frequency, damping, and gain without presupposing a global equation. SOLIS decouples trajectory reconstruction from parameter estimation and stabilizes training with a cyclic curriculum and \textbf{Local Physics Hints} windowed ridge-regression anchors that mitigate optimization collapse. Experiments on benchmarks show accurate parameter-manifold recovery and coherent physical rollouts from sparse data, including regimes where standard inverse methods fail.

Murat Furkan Mansur, Tufan Kumbasar

AI and Intelligent Systems Laboratory, Istanbul Technical University, 34469, Istanbul, Türkiye

Parametric Physics-Informed Neural Network System Identification Nonlinear Dynamics Neural Ordinary Differential Equations Surrogate Models Robotwin

Jan 27, 2026eess.SY

Model-Free Output Feedback Stabilization via Policy Gradient Methods

Stabilizing a dynamical system is a fundamental problem that serves as a cornerstone for many complex tasks in the field of control systems. The problem becomes challenging when the system model is unknown. Among the Reinforcement Learning (RL) algorithms that have been successfully applied to solve problems pertaining to unknown linear dynamical systems, the policy gradient (PG) method stands out due to its ease of implementation and can solve the problem in a model-free manner. However, most of the existing works on PG methods for unknown linear dynamical systems assume full-state feedback. In this paper, we take a step towards model-free learning for partially observed linear dynamical systems with output feedback and focus on the fundamental stabilization problem of the system. We propose an algorithmic framework that stretches the boundary of PG methods to the problem without global convergence guarantees. We show that by leveraging zeroth-order PG update based on system trajectories and its convergence to stationary points, the proposed algorithms return a stabilizing output feedback policy for discrete-time linear dynamical systems. We also explicitly characterize the sample complexity of our algorithm and verify the effectiveness of the algorithm using numerical examples.

Ankang Zhang, Ming Chi, Xiaoling Wang +1

School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan 430074, China · College of Automation, Nanjing University of Posts and Telecommunications, Nanjing 210023, China

Robust Control Stabilization Model-Free Reinforcement Learning Policy Gradient Nonlinear Dynamics

Jan 7, 2026cs.LG

Minimum distance classification for nonlinear dynamical systems

We address the problem of classifying trajectories or sequences generated by nonlinear dynamical systems, where each class corresponds to a distinct dynamical system. We propose Dynafit, a kernel-based method that learns a distance metric between training data and the underlying dynamics. New observations are assigned to the class whose dynamics best fit the observations according to the learned metric.The learning algorithm approximates the Koopman operator, which globally linearizes the dynamics in a (potentially infinite-dimensional) feature space associated with a kernel function. The distance metric is computed in the feature space independently of its dimensionality by exploiting the kernel trick commonly used in machine learning. The kernel function can be tailored to incorporate prior knowledge of the dynamics when available. We consider a classical test example, the logistic map as a discrete dynamical system, and derive analytically the kernel function from the polynomial Koopman basis that exactly linearizes the dynamics. Dynafit is applicable to a wide range of classification tasks involving nonlinear dynamical systems and sensors. We illustrate its effectiveness through three examples: chaos detection in the logistic map, recognition of handwritten dynamical patterns, and classification of visual dynamic textures.

Dominique Martinez

Aix-Marseille University, CNRS, ISM, 13009 Marseille, France

Nonlinear Dynamics Kernel Method Chaos Distance Approximate Nearest-Neighbor Search Physical Activities

Oct 29, 2025math.OC

Nonlinear Dynamics In Optimization Landscape of Shallow Neural Networks with Tunable Leaky ReLU

In this work, we study the nonlinear dynamics of a shallow neural network trained with mean-squared loss and leaky ReLU activation. Under Gaussian inputs and equal layer width k, (1) we establish, based on the equivariant gradient degree, a theoretical framework, applicable to any number of neurons k>= 4, to detect bifurcation of critical points with associated symmetries from global minimum as leaky parameter

α

varies. Typically, our analysis reveals that a multi-mode degeneracy consistently occurs at the critical number 0, independent of k. (2) As a by-product, we further show that such bifurcations are width-independent, arise only for nonnegative

α

and that the global minimum undergoes no further symmetry-breaking instability throughout the engineering regime

α

in range (0,1). An explicit example with k=5 is presented to illustrate the framework and exhibit the resulting bifurcation together with their symmetries.

Jingzhou Liu

Rectified Linear Unit Neurons Nonlinear Dynamics Optimization Landscape

Oct 26, 2025cs.CE

Data-driven Sensor Placement for Predictive Applications: A Correlation-Assisted Attribution Framework (CAAF)

Optimal sensor placement (OSP) is critical for efficient, accurate monitoring, control, and inference in complex physical systems. We propose a machine-learning-based feature attribution (FA) framework to identify OSP for target predictions. FA quantifies input contributions to a model output; however, it struggles with highly correlated input data often encountered in practical applications for OSP. To address this, we propose a Correlation-Assisted Attribution Framework (CAAF), which introduces a clustering step on the candidate sensor locations before performing FA to reduce redundancy and enhance generalizability. We first illustrate the core principles of the proposed framework through a series of validation cases, then demonstrate its effectiveness in realistic dynamical systems such as structural health monitoring, airfoil lift prediction, and wall-normal velocity estimation for turbulent channel flow. The results show that the CAAF outperforms alternative approaches that typically struggle due to the presence of nonlinear dynamics, chaotic behavior, and multi-scale interactions, and enables the effective application of FA for identifying OSP in real-world environments.

Sze Chai Leung, Di Zhou, H. Jane Bae

Department of Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA 91125, USA. · Lynn Booth & Kent Kresa Department of Aerospace, California Institute of Technology, Pasadena, CA 91125, USA. · Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville, TN 37996, USA.

Optimal Sensor Placement Data-Driven Aerodynamics Nonlinear Dynamics

Jun 25, 2025cs.LG

Stochastic and Non-local Closure Modeling for Nonlinear Dynamical Systems via Latent Score-based Generative Models

We propose a latent score-based generative AI framework for learning stochastic, non-local closure models and constitutive laws in nonlinear dynamical systems of computational mechanics. This work addresses a key challenge of modeling complex multiscale dynamical systems without a clear scale separation, for which numerically resolving all scales is prohibitively expensive, e.g., for engineering turbulent flows. While classical closure modeling methods leverage domain knowledge to approximate subgrid-scale phenomena, their deterministic and local assumptions can be too restrictive in regimes lacking a clear scale separation. Recent developments of diffusion-based stochastic models have shown promise in the context of closure modeling, but their prohibitive computational inference cost limits practical applications in many real-world settings. This work addresses this limitation by jointly training convolutional autoencoders with conditional diffusion models in latent space, significantly reducing the dimensionality of the sampling process while preserving essential physical characteristics. Numerical results demonstrate that the joint training approach helps discover a proper latent space that not only guarantees small reconstruction errors but also ensures good performance of the diffusion model in the latent space. When integrated into numerical simulations, the proposed stochastic modeling framework via latent conditional diffusion models achieves significant computational acceleration while maintaining comparable predictive accuracy to standard diffusion models in physical space.

Xinghao Dong, Huchen Yang, Jin-Long Wu

Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, WI, 53706, USA

Multiscale Dynamics Latent Dynamics Diffusion Dynamics Nonlinear Dynamics Latent Space Stochastic+1

May 22, 2025cs.LG

Meta-reinforcement learning with minimum attention

Minimum attention applies the least action principle in changes of control concerning state and time, first proposed by Brockett. The involved regularization is highly relevant in emulating biological control, such as motor learning. We apply minimum attention in reinforcement learning (RL) as part of the rewards and investigate its connection to meta-learning and stabilization. Specifically, model-based meta-learning with minimum attention is explored in high-dimensional nonlinear dynamics. Ensemble-based model learning and gradient-based meta-policy learning are alternately performed. Empirically, minimum attention improves fast adaptation in few shots and reduces variance from perturbations of the model and environment, compared to model-free and model-based RL baseline, and yields consistent gain when integrated into modern world models (DreamerV3, MAMBA). Furthermore, the minimum attention demonstrates an improvement in energy efficiency.

Shashank Gupta, Pilhwa Lee

Department of EECS University of Michigan Ann Arbor, MI 48109 · Department of Mathematics Morgan State University Baltimore, MD 21251

Meta-Learning Reinforcement Learning Baseline Less Data Nonlinear Dynamics