Dynamical Systems

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Dynamical systems describe how a system's state changes over time and explain behaviors such as equilibria, oscillations, attractors, bifurcations, and chaos. Researchers analyze these behaviors mathematically and learn models from data using approaches such as Koopman operators, neural ODEs, physics-informed networks, and structure-preserving Hamiltonian models. Applications range from forecasting Lorenz and pendulum dynamics to modeling reaction-diffusion processes, neural activity, and robot motion, often with control and safety guarantees. A central challenge is that accurate short-term predictions do not ensure faithful long-term behavior: learned models can drift, violate physical constraints, or fail to generalize to unseen regimes.

Latest papers

482 papers56 papers in the last four weeks, up 19% on the four weeks before. 0.3% of all new papers.

Mar 31, 2026

Design and Aerodynamic Modeling of MetaMorpher: A Hybrid Rotary andFixed-Wing Morphing UAV

In this paper, we present a generalized, comprehensive nonlinear mathematical model and conceptual design for the MetaMorpher, a metamorphic Unmanned Aerial Vehicle (UAV) designed to bridge the gap between vertical takeoff and landing agility and fixed-wing cruising efficiency. Building on the successful design of the spincopter platform, this work introduces a simplified mechanical architecture using lightweight materials and a novel wing-folding strategy. Unlike traditional rigid-body approximations, we derive a nonlinear flight dynamics model that enables arbitrary force distributions across a segmented wing structure. This modularity allows for testing different airfoils, mass distributions, and chord lengths in a single environment. As part of this work, various flight modes were specifically tested and analyzed in the Simulink environment. The results show that the model behaves predictably under different structural configurations, demonstrating its reliability as a tool for rapid design evaluation.

Anja Bosak, Dorian Erić, Ana Milas +1

All authors are with the University of Zagreb, Faculty of Electrical Engineering and Computing, LARICS Laboratory for Robotics and Intelligent Control Systems, Unska 3, 10000 Zagreb, Croatia

Aerial Robotics Dynamical Systems

Mar 31, 2026

Predictor-Based Output-Feedback Control of Linear Systems with Time-Varying Input and Measurement Delays via Neural-Approximated Prediction Horizons

Due to simplicity and strong stability guarantees, predictor feedback methods have stood as a popular approach for time delay systems since the 1950s. For time-varying delays, however, implementation requires computing a prediction horizon defined by the inverse of the delay function, which is rarely available in closed form and must be approximated. In this work, we formulate the inverse delay mapping as an operator learning problem and study predictor feedback under approximation of the prediction horizon. We propose two approaches: (i) a numerical method based on time integration of an equivalent ODE, and (ii) a data-driven method using neural operators to learn the inverse mapping. We show that both approaches achieve arbitrary approximation accuracy over compact sets, with complementary trade-offs in computational cost and scalability. Building on these approximations, we then develop an output-feedback predictor design for systems with delays in both the input and the measurement. We prove that the resulting closed-loop system is globally exponentially stable when the prediction horizon is approximated with sufficiently small error. Lastly, numerical experiments validate the proposed methods and illustrate their trade-offs between accuracy and computational efficiency.

Luke Bhan, Miroslav Krstic, Yuanyuan Shi

Department of Electrical and Computer Engineering, University of California San Diego, La Jolla, CA 92093-0411, USA · Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0411, USA

Feedback Control Dynamical Systems

Mar 28, 2026

Interpretable Physics Extraction from Data for Linear Dynamical Systems using Lie Generator Networks

When the system is linear, why should learning be nonlinear? Linear dynamical systems, the analytical backbone of control theory, signal processing and circuit analysis, have exact closed-form solutions via the state transition matrix. Yet when system parameters must be inferred from data, recent neural approaches offer flexibility at the cost of physical guarantees: Neural ODEs provide flexible trajectory approximation but may violate physical invariants, while energy preserving architectures do not natively represent dissipation essential to real-world systems. We introduce Lie Generator Networks (LGN), which learn a structured generator A and compute trajectories directly via matrix exponentiation. This shift from integration to exponentiation preserves structure by construction. By parameterizing A = S - D (skew-symmetric minus positive diagonal), stability and dissipation emerge from the underlying architecture and are not introduced during training via the loss function. LGN provides a unified framework for linear conservative, dissipative, and time-varying systems. On a 100-dimensional stable RLC ladder, standard derivative-based least-squares system identification can yield unstable eigenvalues. The unconstrained LGN yields stable but physically incorrect spectra, whereas LGN-SD recovers all 100 eigenvalues with over two orders of magnitude lower mean eigenvalue error than unconstrained alternatives. Critically, these eigenvalues reveal poles, natural frequencies, and damping ratios which are interpretable physics that black-box networks do not provide.

Shafayeth Jamil, Rehan Kapadia

Department of Electrical and Computer Engineering, University of Southern California

Structure-Preserving Neural Networks Dynamical Systems System Identification

Mar 27, 2026

Comparing Physics-Informed and Neural ODE Approaches for Modeling Nonlinear Biological Systems: A Case Study Based on the Morris-Lecar Model

Physics-Informed Neural Networks (PINNs) and Neural Ordinary Differential Equations (NODEs) represent two distinct machine learning frameworks for modeling nonlinear neuronal dynamics. This study systematically evaluates their performance on the two-dimensional Morris-Lecar model across three canonical bifurcation regimes: Hopf, Saddle-Node on Limit Cycle, and homoclinic orbit. Synthetic time-series data are generated via numerical integration under controlled conditions, and training is performed using collocation points for PINNs and adaptive solvers for NODEs (Dormand-Prince method). PINNs incorporate the governing differential equations into the loss function using automatic differentiation, which enforces physical consistency during training. In contrast, NODEs learn the system's vector field directly from data, without prior structural assumptions or inductive bias toward physical laws. Model performance is assessed using standard regression metrics, including Mean Squared Error (MSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and the coefficient of determination. Results indicate that PINNs tend to achieve higher accuracy and robustness in scenarios involving stiffness or sensitive bifurcations, owing to their embedded physical structure. NODEs, while more expressive and flexible, operate as black-box approximators without structural constraints, which can lead to reduced interpretability and stability in these regimes. Although advanced variants of NODEs (e.g., ANODEs, latent NODEs) aim to mitigate such limitations, their performance under stiff dynamics remains an open question. These findings emphasize the trade-offs between physics-informed models, which embed structure and interpretability, and purely data-driven approaches, which prioritize flexibility at the cost of physical consistency.

Nikolaos M. Matzakos, Chrisovalantis Sfyrakis

Department of Education, School of Pedagogical and Technological Education (ASPETE), 141 21 Athens, Greece. · Department of Mechanical Engineering Educators, School of Pedagogical and Technological Education (ASPETE), 151 22 Athens, Greece.

Physics-Informed ML Neural ODEs Dynamical Systems

Mar 26, 2026

Conchordal: Emergent Harmony via Direct Cognitive Coupling in a Psychoacoustic Landscape

This paper introduces Conchordal, a bio-acoustic instrument for generative composition whose sonic agents are governed by artificial life dynamics within a psychoacoustic fitness landscape. The system is built on Direct Cognitive Coupling (DCC), a design principle requiring that generative dynamics operate directly within a landscape derived from psychoacoustic observables and read from that landscape without symbolic harmonic rules. The environment integrates roughness and harmonicity into a continuous consonance field without presupposing discrete scales or explicit harmonic rules. Agents adjust pitch through local proposal-and-accept dynamics under a crowding penalty, regulate survival via consonance-dependent metabolism, and entrain temporally through Kuramoto-style phase coupling. Four experiments are reported: (1) consonance search produces structured polyphony with enriched consonant intervals; (2) consonance-dependent metabolism yields survival differentials that vanish when recharge is disabled; (3) a minimal hereditary adaptation assay shows that parent-guided respawn plus metabolic selection can accumulate more structured polyphony without adult hill-climbing; and (4) a shared oscillatory scaffold organizes rhythmic timing under external forcing. A supplementary mechanism check reports one possible composer-configurable bridge by which spectral state can modulate temporal coupling. These findings show that a psychoacoustically derived landscape serves as an effective artificial-life terrain, yielding self-organization, selection, synchronization, and lineage-level accumulation in a non-traditional computational medium. At the level of the model, the same landscape therefore functions both as ecological terrain and as an internal proxy for musical coherence.

Koichi Takahashi

Institute for Advanced Biosciences and Graduate School of Media and Governance, Keio University

Music Generation Emergent Behavior in Multi-Agent Systems Artificial Life Dynamical Systems

Mar 24, 2026

Dynamical Systems Theory Behind a Hierarchical Reasoning Model

Current large language models (LLMs) primarily rely on linear sequence generation and massive parameter counts, yet they severely struggle with complex algorithmic reasoning. While recent reasoning architectures, such as the Hierarchical Reasoning Model (HRM) and Tiny Recursive Model (TRM), demonstrate that compact recursive networks can tackle these tasks, their training dynamics often lack rigorous mathematical guarantees, leading to instability and representational collapse. We propose the Contraction Mapping Model (CMM), a novel architecture that reformulates discrete recursive reasoning into continuous Neural Ordinary and Stochastic Differential Equations (NODEs/NSDEs). By explicitly enforcing the convergence of the latent phase point to a stable equilibrium state and mitigating feature collapse with a hyperspherical repulsion loss, the CMM provides a mathematically grounded and highly stable reasoning engine. On the Sudoku-Extreme benchmark, a 5M-parameter CMM achieves a state-of-the-art accuracy of 93.7 %, outperforming the 27M-parameter HRM (55.0 %) and 5M-parameter TRM (87.4 %). Remarkably, even when aggressively compressed to an ultra-tiny footprint of just 0.26M parameters, the CMM retains robust predictive power, achieving 85.4 % on Sudoku-Extreme and 82.2 % on the Maze benchmark. These results establish a new frontier for extreme parameter efficiency, proving that mathematically rigorous latent dynamics can effectively replace brute-force scaling in artificial reasoning.

Vasiliy A. Es'kin, Mikhail E. Smorkalov

Department of Radiophysics, University of Nizhny Novgorod, Nizhny Novgorod 603950, Russia · Huawei Nizhny Novgorod Research Center, Nizhny Novgorod 603006, Russia · Skolkovo Institute of Science and Technology, Moscow 121205, Russia

Sudoku Solving Neural Algorithmic Reasoning Algorithmic Reasoning Dynamical Systems Neural ODEs

Mar 24, 2026

Generalizing Dynamics Modeling More Easily from Representation Perspective

Learning system dynamics from observations is a critical problem in many applications over various real-world complex systems, e.g., climate, ecology, and fluid systems. Recently, neural dynamics modeling method have become a prevalent solution that embeds the object's observations into a latent space before learning dynamics using neural methods such as neural Ordinary Differential Equations (ODE). Existing dynamics modeling methods induce a specific model for each observation of different complex systems, resulting in poor generalization across systems. Inspired by the great success of pre-trained models, we conduct a generalized Pre-trained Dynamics EncoDER (PDEDER) which can embed the original state observations into a latent space where the dynamics can be captured more easily. To conduct the generalized PDEDER, we pre-train any Pre-trained Language Model (PLM) by minimizing the Lyapunov exponent objective, which constrains the chaotic behavior of governing dynamics learned in the latent space. By penalizing the divergence of embedded observations, our PDEDER promotes locally stable and well-structured latent dynamics, thereby facilitating more effective dynamics modeling than in the original observation space. In addition, we incorporate reconstruction and forecasting objectives to mitigate the risk of obtaining an over-smoothed latent space. Specifically, we collect 152 sets of real-world and synthetic observations from 23 complex systems as pre-training corpora and employ them to pre-train PDEDER. Given any future dynamic observation, we can fine-tune PDEDER with any specific dynamics modeling method. We evaluate PDEDER on 12 dynamic systems by short/long-term forecasting under both in-domain and cross-domain settings, and the empirical results indicate the effectiveness and generalizability of PDEDER.

Yiming Wang, Zhengnan Zhang, Genghe Zhang +7

College of Computer Science and Technology, and Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, China · College of Software and Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, China · School of Computing, Belfast, Northern Ireland, UK +1

Dynamical Systems Latent Dynamics Modeling State Representation Learning Representation Learning

Mar 23, 2026

Identifiability and amortized inference limitations in Kuramoto models

Bayesian inference is a powerful tool for parameter estimation and uncertainty quantification in dynamical systems. However, for nonlinear oscillator networks such as Kuramoto models, widely used to study synchronization phenomena in physics, biology, and engineering, inference is often computationally prohibitive due to high-dimensional state spaces and intractable likelihood functions. We present an amortized Bayesian inference approach that learns a neural approximation of the posterior from simulated phase dynamics, enabling fast, scalable inference without repeated sampling or optimization. Applied to synthetic Kuramoto networks, the method shows promising results in approximating posterior distributions and capturing uncertainty, with computational savings compared to traditional Bayesian techniques. These findings suggest that amortized inference is a practical and flexible framework for uncertainty-aware analysis of oscillator networks.

Emma Hannula, Jana de Wiljes, Matthew T. Moores +2

LUT University, Yliopistonkatu 34, Lappeenranta, 53850, Finland · TU Ilmenau, Ehrenbergstraße 29, Ilmenau, 98693, Germany · University of Cambridge, Herschel Road, Cambridge, CB3 9AL, United Kingdom

Kuramoto Model Amortized Inference Dynamical Systems Neural Posterior Estimation Simulation-Based Inference Parameter Identifiability+1

Mar 23, 2026

Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approach

We present a numerical framework for the stability and bifurcation analysis of nonlinear partial differential equations (PDEs) in which the solutions are sought in the function space spanned by physics-informed random projection neural networks (PI-RPNNs), and discretized via a collocation approach. These are shallow, single-hidden-layer networks with randomly sampled and fixed a priori hidden-layer weights; only the linear output layer weights are optimized, reducing training to a single least-squares solve. This linear output structure enables the explicit generalized eigenvalue formulation governing the linear stability of stationary solutions, separating the physical domain interior dynamics from the algebraic constraints imposed by boundary conditions (BCs), at no additional training cost and without requiring additional PDE solves. However, the random projection collocation matrix is inherently numerically rank-deficient, rendering naive eigenvalue computation unreliable and contaminating the true eigenvalue spectrum with spurious near-zero modes. To overcome this limitation, we introduce a matrix-free shift-invert Krylov-Arnoldi method that operates directly in weight space, avoiding explicit inversion of the numerically rank-deficient collocation matrix and enabling the reliable computation of several leading eigenpairs of the physical Jacobian - the discretized Frechet derivative of the PDE operator with respect to the solution field, whose eigenvalue spectrum determines linear stability. We further prove that the PI-RPNN-based generalized eigenvalue problem is almost surely regular, guaranteeing solvability with standard eigensolvers, and that the singular values of the random projection collocation matrix decay exponentially for analytic activation functions, rigorously explaining the observed ill-conditioning.

Gianluca Fabiani, Michail E. Kavousanakis, Constantinos Siettos +1

Hopkins Extreme Materials Institute, Johns Hopkins University, Baltimore, 21218, MD, USA · Dept. of Chemical and Biomolecular Engineering, Johns Hopkins University, Baltimore, 21218, MD, USA · School of Chemical Engineering, National Technical University of Athens, 9 Iroon Polytechniou Street, Athens, 15772, Greece +2

Dynamical Systems Neural PDE Solvers Physics-Informed ML

Mar 22, 2026

FluidWorld: Reaction-Diffusion Dynamics as a Predictive Substrate for World Models

World models learn to predict future states of an environment, enabling planning and mental simulation. Current approaches default to Transformer-based predictors operating in learned latent spaces. This comes at a cost: O(N^2) computation and no explicit spatial inductive bias. This paper asks a foundational question: is self-attention necessary for predictive world modeling, or can alternative computational substrates achieve comparable or superior results? I introduce FluidWorld, a proof-of-concept world model whose predictive dynamics are governed by partial differential equations (PDEs) of reaction-diffusion type. Instead of using a separate neural network predictor, the PDE integration itself produces the future state prediction. In a strictly parameter-matched three-way ablation on unconditional UCF-101 video prediction (64x64, ~800K parameters, identical encoder, decoder, losses, and data), FluidWorld is compared against both a Transformer baseline (self-attention) and a ConvLSTM baseline (convolutional recurrence). While all three models converge to comparable single-step prediction loss, FluidWorld achieves 2x lower reconstruction error, produces representations with 10-15% higher spatial structure preservation and 18-25% more effective dimensionality, and critically maintains coherent multi-step rollouts where both baselines degrade rapidly. All experiments were conducted on a single consumer-grade PC (Intel Core i5, NVIDIA RTX 4070 Ti), without any large-scale compute. These results establish that PDE-based dynamics, which natively provide O(N) spatial complexity, adaptive computation, and global spatial coherence through diffusion, are a viable and parameter-efficient alternative to both attention and convolutional recurrence for world modeling.

Fabien Polly

Independent Researcher

Reaction-Diffusion Systems World Model Learning Video World Models Video Prediction Dynamical Systems

Mar 21, 2026

LLM-ODE: Data-driven Discovery of Dynamical Systems with Large Language Models

Discovering the governing equations of dynamical systems is a central problem across many scientific disciplines. As experimental data become increasingly available, automated equation discovery methods offer a promising data-driven approach to accelerate scientific discovery. Among these methods, genetic programming (GP) has been widely adopted due to its flexibility and interpretability. However, GP-based approaches often suffer from inefficient exploration of the symbolic search space, leading to slow convergence and suboptimal solutions. To address these limitations, we propose LLM-ODE, a large language model-aided model discovery framework that guides symbolic evolution using patterns extracted from elite candidate equations. By leveraging the generative prior of large language models, LLM-ODE produces more informed search trajectories while preserving the exploratory strengths of evolutionary algorithms. Empirical results on 91 dynamical systems show that LLM-ODE variants consistently outperform classical GP methods in terms of search efficiency and Pareto-front quality. Overall, our results demonstrate that LLM-ODE improves both efficiency and accuracy over traditional GP-based discovery and offers greater scalability to higher-dimensional systems compared to linear and Transformer-only model discovery methods.

Amirmohammad Ziaei Bideh, Jonathan Gryak

The Graduate Center, CUNY New York, New York, USA · Queens College and The Graduate Center, CUNY New York, New York, USA

ODE Discovery Dynamical Systems Large Language Model-Guided Scientific Discovery Symbolic Regression Genetic Programming

Mar 20, 2026

From Data to Laws: Neural Discovery of Conservation Laws Without False Positives

Conservation laws are fundamental to understanding dynamical systems, but discovering them from data remains challenging due to parameter variation, non-polynomial invariants, local minima, and false positives on chaotic systems. We introduce NGCG, a neural-symbolic pipeline that decouples dynamics learning from invariant discovery and systematically addresses these challenges. A multi-restart variance minimiser learns a near-constant latent representation; system-specific symbolic extraction (polynomial Lasso, log-basis Lasso, explicit PDE candidates, and PySR) yields closed-form expressions; a strict constancy gate and diversity filter eliminate spurious laws. On a benchmark of nine diverse systems including Hamiltonian and dissipative ODEs, chaos, and PDEs, NGCG achieves consistent discovery (DR=1.0, FDR=0.0, F1=1.0) on all four systems with true conservation laws, with constancy two to three orders of magnitude lower than the best baseline. It is the only method that succeeds on the Lotka--Volterra system, and it correctly outputs no law on all five systems without invariants. Extensive experiments demonstrate robustness to noise (

σ= 0.1

), sample efficiency (50--100 trajectories), insensitivity to hyperparameters, and runtime under one minute per system. A Pareto analysis shows that the method provides a range of candidate expressions, allowing users to trade complexity for constancy. NGCG achieves strong performance relative to prior methods for data-driven conservation-law discovery, combining high accuracy with interpretability.

Rahul D Ray

Department of Electronics and Electrical Engineering, BITS Pilani, Hyderabad Campus

Scientific Discovery Dynamical Systems Neuro-Symbolic Reasoning

Mar 20, 2026

Formulation and Learning Of The Forecasting Problem

Machine learning has opened new frontiers in purely data-driven forecasting of dynamical systems; the resulting methods are showing considerable promise in some application domains. However, in contrast to model-driven algorithms, analysis of these data-driven methods is poorly developed. In this paper we address this issue, developing a theory to underpin data-driven methods to solve forecasting problems. The theoretical framework comprises two key components: (i) establishing the existence of a mapping defining the forecasting problem; (ii) showing that it is possible to approximate this mapping by supervised learning techniques, to arbitrary accuracy. We work in the continuous-time and noise-free setting as this defines a simple canonical setting in which to establish an underpinning theory; the continuous-time setting leads to an operator formulation of the problem in which the underlying forecasting map is defined on pathspace. As a byproduct of our framework we show that analogous existence and approximation results may be obtained for the smoothing problem of recovering unobserved components of dynamical systems, from observed components, using supervised learning. The theoretical results, for both forecasting and smoothing, are illustrated with experiments studying the Lorenz 63, Lorenz 96 and Kuramoto-Sivashinsky dynamical systems.

Edoardo Calvello, Elizabeth Carlson, Nikola Kovachki +2

California Institute of Technology, Pasadena, CA · Oregon State University, Corvallis, OR · NVIDIA Corporation, Santa Clara, CA +1

Data Assimilation Operator Learning Dynamical Systems Neural Operators

Mar 20, 2026

Structural Controllability of Large-Scale Hypergraphs

Controlling real-world networked systems, including ecological, biomedical, and engineered networks that exhibit higher-order interactions, remains challenging due to inherent nonlinearities and large system scales. Despite extensive studies on graph controllability, the controllability properties of hypergraphs remain largely underdeveloped. Existing results focus primarily on exact controllability, which is often impractical for large-scale hypergraphs. In this article, we develop a structural controllability framework for hypergraphs by modeling hypergraph dynamics as polynomial dynamical systems. In particular, we extend classical notions of accessibility and dilation from linear graph-based systems to polynomial hypergraph dynamics and establish a hypergraph-based criterion under which the topology guarantees satisfaction of classical Lie-algebraic and Kalman-type rank conditions for almost all parameter choices. We further derive a topology-based lower bound on the minimum number of driver nodes required for structural controllability and leverage this bound to design a scalable driver node selection algorithm combining dilation-aware initialization via maximum matching with greedy accessibility expansion. We demonstrate the effectiveness and scalability of the proposed framework through numerical experiments on hypergraphs with tens to thousands of nodes and higher-order interactions.

Joshua Pickard, Xin Mao, Can Chen

Eric and Wendy Schmidt Center, Broad Institute of MIT and Harvard, Cambridge, MA 02142, USA · School of Data Science and Society, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA · Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA +1

Dynamical Systems

Mar 20, 2026

K-GMRF: Kinetic Gauss-Markov Random Field for First-Principles Covariance Tracking on Lie Groups

Tracking non-stationary covariance matrices is fundamental to vision yet hindered by existing estimators that either neglect manifold constraints or rely on first-order updates, incurring inevitable phase lag during rapid evolution. We propose K-GMRF, an online, training-free framework for covariance tracking that reformulates the problem as forced rigid-body motion on Lie groups. Derived from the Euler-Poincaré equations, our method interprets observations as torques driving a latent angular velocity, propagated via a structure-preserving symplectic integrator. We theoretically prove that this second-order dynamics achieves zero steady-state error under constant rotation, strictly superior to the proportional lag of first-order baselines. Validation across three domains demonstrates robust tracking fidelity: (i) on synthetic ellipses, K-GMRF reduces angular error by 30x compared to Riemannian EMA while maintaining stability at high speeds; (ii) on SO(3) stabilization with 20% dropout, it decreases geodesic error from 29.4° to 9.9°; and (iii) on OTB motion-blur sequences, it improves loU from 0.55 to 0.74 on BlurCar2 with a 96% success rate. As a fully differentiable symplectic module, K-GMRF provides a plug-and-play geometric prior for data-constrained scenarios and an interpretable layer within modern deep architectures.

ZhiMing Li

School of Computer Science and Technology, Tianjin University

Covariance Estimation Visual Object Tracking Dynamical Systems

Mar 19, 2026

Extending SSMs with the Exponentially Weighted Signature

We introduce the exponentially weighted signature (EWS), a continuous-time model that computes iterated integrals of a path, where each increment is weighted by the matrix exponential of a learnable generator over elapsed clock time. We prove that it solves a linear controlled differential equation, keeps the group-like structure and the universality of the signature, and satisfies a modified Chen identity, enabling a parallel scan. At depth one the EWS is a state-space model (SSM), and we map linear time-invariant SSMs, Mamba channels and Mamba-

2

heads to it in closed form. The EWS extends SSMs through an arbitrary matrix generator, a clock that generalises the step size to causal functionals of the input, and higher truncation depths that are non-linear in the path within a single layer. Empirically, the EWS achieves the highest average accuracy and rank on six long time-series classification datasets, where depth generally helps. Learned clocks prove necessary for state tracking on formal language tasks, and at depth one, the EWS matches or exceeds competing SSMs on regression and forecasting with far fewer parameters.

Alexandre Bloch, Benjamin Walker, Joël Mouterde +3

Mathematical Institute, University of Oxford · SKF · Department of Mathematics, Imperial College London

Dynamical Systems Time Series Classification Time Series Forecasting State Space Models

Mar 19, 2026

Computationally Efficient Density-Driven Optimal Control via Analytical KKT Reduction and Contractive MPC

Efficient coordination for collective spatial distribution is a fundamental challenge in multi-agent systems. Prior research on Density-Driven Optimal Control (D2OC) established a framework to match agent trajectories to a desired spatial distribution. However, implementing this as a predictive controller requires solving a large-scale Karush-Kuhn-Tucker (KKT) system, whose computational complexity grows cubically with the prediction horizon. To resolve this, we propose an analytical structural reduction that transforms the T-horizon KKT system into a condensed quadratic program (QP). This formulation achieves O(T) linear scalability, significantly reducing the online computational burden compared to conventional O(T^3) approaches. Furthermore, to ensure rigorous convergence in dynamic environments, we incorporate a contractive Lyapunov constraint and prove the Input-to-State Stability (ISS) of the closed-loop system against reference propagation drift. Numerical simulations verify that the proposed method facilitates rapid density coverage with substantial computational speed-up, enabling long-horizon predictive control for large-scale multi-agent swarms.

Julian Martinez, Kooktae Lee

Department of Mechanical and Aerospace Engineering, Texas Tech University, Lubbock, TX 79409, USA

Input-to-State Stability Multi-Agent Control Optimal Control Model Predictive Control Dynamical Systems

Mar 18, 2026

Towards Infinitely Long Neural Simulations: Self-Refining Neural Surrogate Models for Dynamical Systems

Recent advances in autoregressive neural surrogate models have enabled orders-of-magnitude speedups in simulating dynamical systems. However, autoregressive models are generally prone to distribution drift: compounding errors in autoregressive rollouts that severely degrade generation quality over long time horizons. Existing work attempts to address this issue by implicitly leveraging the inherent trade-off between short-time accuracy and long-time consistency through hyperparameter tuning. In this work, we introduce a unifying mathematical framework that makes this tradeoff explicit, formalizing and generalizing hyperparameter-based strategies in existing approaches. Within this framework, we propose a robust, hyperparameter-free model implemented as a conditional diffusion model that balances short-time fidelity with long-time consistency by construction. Our model, Self-refining Neural Surrogate model (SNS), can be implemented as a standalone model that refines its own autoregressive outputs or as a complementary model to existing neural surrogates to ensure long-time consistency. We also demonstrate the numerical feasibility of SNS through high-fidelity simulations of complex dynamical systems over arbitrarily long time horizons.

Qi Liu, Laure Zanna, Joan Bruna

Courant Institute School of Mathematics, Computing, and Data Science, New York University, USA.

Neural Surrogate Modeling Autoregressive Modeling Dynamical Systems Diffusion Models

Mar 18, 2026

Data-driven model order reduction for structures with piecewise linear nonlinearity using dynamic mode decomposition

Piecewise-linear nonlinear systems appear in many engineering disciplines. Prediction of the dynamic behavior of such systems is of great importance from practical and theoretical viewpoint. In this paper, a data-driven model order reduction method for piecewise-linear systems is proposed, which is based on dynamic mode decomposition (DMD). The overview of the concept of DMD is provided, and its application to model order reduction for nonlinear systems based on Galerkin projection is explained. The proposed approach uses impulse responses of the system to obtain snapshots of the state variables. The snapshots are then used to extract the dynamic modes that are used to form the projection basis vectors. The dynamics described by the equations of motion of the original full-order system are then projected onto the subspace spanned by the basis vectors. This produces a system with much smaller number of degrees of freedom (DOFs). The proposed method is applied to two representative examples of piecewise linear systems: a cantilevered beam subjected to an elastic stop at its end, and a bonded plates assembly with partial debonding. The reduced order models (ROMs) of these systems are constructed by using the Galerkin projection of the equation of motion with DMD modes alone, or DMD modes with a set of classical constraint modes to be able to handle the contact nonlinearity efficiently. The obtained ROMs are used for the nonlinear forced response analysis of the systems under harmonic loading. It is shown that the ROMs constructed by the proposed method produce accurate forced response results.

Akira Saito, Masato Tanaka

Department of Mechanical Engineering, Meiji University, Kawasaki, Kanagawa 214-8571, Japan. · Toyota Research Institute of North America Toyota Motor North America, Inc. Ann Arbor, Michigan, 48105, USA

Reduced-Order Modeling Dynamical Systems Structural Mechanics

Mar 18, 2026

Mathematical Modeling of Cancer-Bacterial Therapy: Analysis and Numerical Simulation via Physics-Informed Neural Networks

Bacterial cancer therapy exploits anaerobic bacteria's ability to target hypoxia tumor regions, yet the interactions among tumor growth, bacterial colonization, oxygen levels, immunosuppressive cytokines, and bacterial communication remain poorly quantified. We present a mathematical model of five coupled nonlinear reaction-diffusion equations in a two-dimensional tissue domain. We proved the global well-posedness of the model and identified its steady states to analyze stability. Furthermore, a physics-informed neural network (PINN) solves the system without a mesh and without requiring extensive data. It provides convergence guarantees by combining residual stability and Sobolev approximation error bounds. This results in an overall error rate of O(n^-2 ln^4(n) + N^-1/2), which depends on the network width n and the number of collocation points N. We conducted several numerical experiments, including predicting the tumor's response to therapy. We also performed a sensitivity analysis of certain parameters. The results suggest that long-term therapeutic efficacy may require the maintenance of hypoxia regions in the tumor, or using bacteria that tolerate oxygen better, may be necessary for long-lasting tumor control.

Ayoub Farkane, David Lassounon

TICLab, International University of Rabat, Rabat, 11100, Morocco · Université de Lorraine, CNRS, CRAN, F-54000 Nancy, France · Université de Lorraine, CNRS, IECL, F-54000, Nancy, France +1

Reaction-Diffusion Systems Oncology Dynamical Systems

Mar 17, 2026

Unifying Optimization and Dynamics to Parallelize Sequential Computation: A Guide to Parallel Newton Methods for Breaking Sequential Bottlenecks

Massively parallel hardware (GPUs) and long sequence data have made parallel algorithms essential for machine learning at scale. Yet dynamical systems, like recurrent neural networks and Markov chain Monte Carlo, were thought to suffer from sequential bottlenecks. Recent work showed that dynamical systems can in fact be parallelized across the sequence length by reframing their evaluation as a system of nonlinear equations, which can be solved with Newton's method using a parallel associative scan. However, these parallel Newton methods struggled with limitations, primarily inefficiency, instability, and lack of convergence guarantees. This thesis addresses these limitations with methodological and theoretical contributions, drawing particularly from optimization. Methodologically, we develop scalable and stable parallel Newton methods, based on quasi-Newton and trust-region approaches. The quasi-Newton methods are faster and more memory efficient, while the trust-region approaches are significantly more stable. Theoretically, we unify many fixed-point methods into our parallel Newton framework, including Picard and Jacobi iterations. We establish a linear convergence rate for these techniques that depends on the method's approximation accuracy and stability. Moreover, we give a precise condition, rooted in dynamical stability, that characterizes when parallelization provably accelerates a dynamical system and when it cannot. Specifically, the sign of the Largest Lyapunov Exponent of a dynamical system determines whether or not parallel Newton methods converge quickly. In sum, this thesis unlocks scalable and stable methods for parallelizing sequential computation, and provides a firm theoretical basis for when such techniques will and will not work. This thesis also serves as a guide to parallel Newton methods for researchers who want to write the next chapter in this ongoing story.

Xavier Gonzalez

STANFORD UNIVERSITY

Lyapunov Exponent Estimation Dynamical Systems Quasi-Newton Methods

Mar 17, 2026

Data-driven forced response analysis with min-max representations of nonlinear restoring forces

This paper discusses a novel data-driven nonlinearity identification method for mechanical systems with nonlinear restoring forces such as polynomial, piecewise-linear, and general displacement-dependent nonlinearities. The proposed method is built upon the universal approximation theorem that states that a nonlinear function can be approximated by a linear combination of activation functions in artificial neural network framework. The proposed approach utilizes piecewise linear springs with initial gaps to act as the activation functions of the neurons of artificial neural networks. A library of piecewise linear springs with initial gaps are constructed, and the contributions of the springs on the nonlinear restoring force are determined by solving the linear regression problems. The piecewise linear springs are realized by combinations of min and max functions with biases. The proposed method is applied to a Duffing oscillator with cubic stiffness, and a piecewise linear oscillator with a gap and their nonlinearities are successfully determined from their free responses. The obtained models are then used for conducting forced response analysis and the results match well with those of the original system. The method is then applied to experimentally-obtained free response data of a cantilevered plate that is subjected to magnetic restoring force, and successfully finds the piecewise linear representation of the magnetic force. It is also shown that the obtained model is capable of accurately capturing the steady-state response of the system subject to harmonic base excitation.

Akira Saito, Hiromu Fujita

Department of Mechanical Engineering, Meiji University, Kawasaki, Kanagawa 214-8571, Japan.

Nonlinear System Identification Dynamical Systems

Mar 17, 2026

Trajectory-Optimized Time Reparameterization for Learning-Compatible Reduced-Order Modeling of Stiff Dynamical Systems

Stiff dynamical systems present a challenge for machine-learning reduced-order models (ML-ROMs), as explicit time integration becomes unstable in stiff regimes while implicit integration within learning loops is computationally expensive and often degrades training efficiency. Time reparameterization (TR) offers an alternative by transforming the independent variable so that rapid physical-time transients are spread over a stretched-time coordinate, enabling stable explicit integration on uniformly sampled grids. Although several TR strategies have been proposed, their effect on learnability in ML-ROMs remains incompletely understood. This work investigates time reparameterization as a stiffness-mitigation mechanism for neural ODE reduced-order modeling and introduces a trajectory-optimized TR (TOTR) formulation. The proposed approach casts time reparameterization as an optimization problem in arc-length coordinates, in which a traversal-speed profile is selected to penalize acceleration in stretched time. By targeting the smoothness of the training dynamics, this formulation produces reparameterized trajectories that are better conditioned and easier to learn than existing TR methods. TOTR is evaluated on three stiff problems: a parameterized stiff linear system, the van der Pol oscillator, and the HIRES chemical kinetics model. Across all cases, the proposed approach yields smoother reparameterizations and improved physical-time predictions under identical training regimens than other TR approaches. Quantitative results demonstrate loss reductions of one to two orders of magnitude compared to benchmark algorithms. These results highlight that effective stiffness mitigation in ML-ROMs depends critically on the regularity and learnability of the time map itself, and that optimization-based TR provides a robust framework for explicit reduced-order modeling of multiscale dynamical systems.

Joe Standridge, Daniel Livescu, Paul Cizmas

Texas A&M University, College Station, Texas 77840 · Los Alamos National Laboratory, Los Alamos, New Mexico 87545

Reduced-Order Modeling Dynamical Systems Neural ODEs

Mar 17, 2026

Early-Terminable Energy-Safe Iterative Coupling for Parallel Simulation of Partitioned Port-Hamiltonian Systems

Parallel simulation of robotic systems requires partitioning the dynamics into coupled subsystems. Finite-iteration coupling across the partition boundary can inject spurious energy, even when each subsystem is passive. We propose an early-terminable, energy-safe coupling interface for port-Hamiltonian subsystems based on Douglas--Rachford splitting in wave (scattering) coordinates. The wave-domain formulation reduces passivity to norm inequalities and coupling to orthogonality. Within this setting, the deep correspondence between monotone operator theory and discrete passivity can be exploited to construct a Douglas--Rachford inner iteration whose Fejér monotonicity provides algorithmic dissipation. Under passivity of the subsystem integrators and an impedance-tuning condition, the proposed method guarantees discrete passivity of the augmented storage for any finite inner-iteration budget and converges to the monolithic discretization as the budget increases. Experiments on a linear--Duffing coupled-oscillator benchmark support the finite-iteration energy inequality at numerical roundoff (1e-14 in double precision), with state-error metrics decreasing over the tested inner-iteration budgets.

Qi Wei, Jianfeng Tao, Hongyu Nie +1

School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

Dynamical Systems Port-Hamiltonian Systems Physics-Based Simulation Hamiltonian Dynamics

Mar 16, 2026

A scaled TW-PINN: A physics-informed neural network for traveling wave solutions of reaction-diffusion equations with general coefficients

We propose an efficient and generalizable physics-informed neural network (PINN) framework for computing traveling wave solutions of

n

-dimensional reaction-diffusion equations with various reaction and diffusion coefficients. By applying a scaling transformation with the traveling wave form, the original problem is reduced to a one-dimensional scaled reaction-diffusion equation with unit reaction and diffusion coefficients. This reduction leads to the proposed framework, termed scaled TW-PINN, in which a single PINN solver trained on the scaled equation is reused for different coefficient choices and spatial dimensions. We also prove a universal approximation property of the proposed PINN solver for traveling wave solutions. Numerical experiments in one and two dimensions, together with a comparison to the existing wave-PINN method, demonstrate the accuracy, flexibility, and superior performance of scaled TW-PINN. Finally, we explore an extension of the framework to the Fisher's equation with general initial conditions.

Seungwan Han, Kwanghyuk Park, Jiaxi Gu +1

Reaction-Diffusion Systems PDE Solving Physics-Informed ML Dynamical Systems

Mar 16, 2026

Storage and selection of multiple chaotic attractors in minimal reservoir computers

Modern predictive modeling increasingly calls for a single learned dynamical substrate to operate across multiple regimes. From a dynamical-systems viewpoint, this capability decomposes into the storage of multiple attractors and the selection of the appropriate attractor in response to contextual cues. In reservoir computing (RC), multi-attractor learning has largely been pursued using large, randomly wired reservoirs, on the assumption that stochastic connectivity is required to generate sufficiently rich internal dynamics. At the same time, recent work shows that minimal deterministic reservoirs can match random designs for single-system chaotic forecasting. Under which conditions can minimal topologies learn multiple chaotic attractors? In this paper, we find that minimal architectures can successfully store multiple chaotic attractors. However, these same architectures struggle with task switching, in which the system must transition between attractors in response to external cues. We test storage and selection on all 28 unordered system pairs formed from eight three-dimensional chaotic systems. We do not observe a robust dependence of multi-attractor performance on reservoir topology. Over the ten topologies investigated, we find that no single one consistently outperforms the others for either storage or cue-dependent selection. Our results suggest that while minimal substrates possess the representational capacity to model coexisting attractors, they may lack the robust temporal memory required for cued transitions.

Francesco Martinuzzi, Holger Kantz

Max Planck Institute for the Physics of Complex Systems, Dresden, Germany

Reservoir Computing Dynamical Systems Chaotic Dynamical Systems Chaotic Time-Series Prediction

Mar 16, 2026

Ultra-Early Prediction of Tipping Points: Integrating Dynamical Measures with Reservoir Computing

Complex dynamical systems-such as climate, ecosystems, and economics-can undergo catastrophic and potentially irreversible regime changes, often triggered by environmental parameter drift and stochastic disturbances. These critical thresholds, known as tipping points, pose a prediction problem of both theoretical and practical significance, yet remain largely unresolved. To address this, we articulate a model-free framework that integrates the measures characterizing the stability and sensitivity of dynamical systems with the reservoir computing (RC), a lightweight machine learning technique, using only observational time series data. The framework consists of two stages. The first stage involves using RC to robustly learn local complex dynamics from observational data segmented into windows. The second stage focuses on accurately detecting early warning signals of tipping points by analyzing the learned autonomous RC dynamics through dynamical measures, including the dominant eigenvalue of the Jacobian matrix, the maximum Floquet multiplier, and the maximum Lyapunov exponent. Furthermore, when these dynamical measures exhibit trend-like patterns, their extrapolation enables ultra-early prediction of tipping points significantly prior to the occurrence of critical transitions. We conduct a rigorous theoretical analysis of the proposed method and perform extensive numerical evaluations on a series of representative synthetic systems and eight real-world datasets, as well as quantitatively predict the tipping time of the Atlantic Meridional Overturning Circulation system. Experimental results demonstrate that our framework exhibits advantages over the baselines in comprehensive evaluations, particularly in terms of dynamical interpretability, prediction stability and robustness, and ultra-early prediction capability.

Xin Li, Qunxi Zhu, Chengli Zhao +4

College of Science, National University of Defense Technology, Changsha, Hunan 410073, China · Research Institute of Intelligent Complex Systems and MOE Frontiers Center for Brain Science, Fudan University, Shanghai 200433, China · School of Mathematical Sciences, SCMS, SCAM, and CCSB, Fudan University, Shanghai 200433, China +1

Early Warning Detection Dynamical Systems

Mar 15, 2026

Physically Accurate Rigid-Body Dynamics in Particle-Based Simulation

Robotics demands simulation that can reason about the diversity of real-world physical interactions, from rigid to deformable objects and fluids. Current simulators address this by stitching together multiple subsolvers for different material types, resulting in a compositional architecture that complicates physical reasoning. Particle-based simulators offer a compelling alternative, representing all materials through a single unified formulation that enables seamless cross-material interactions. Among particle-based simulators, position-based dynamics (PBD) is a popular solver known for its computational efficiency and visual plausibility. However, its lack of physical accuracy has limited its adoption in robotics. To leverage the benefits of particle-based solvers while meeting the physical fidelity demands of robotics, we introduce PBD-R, a revised PBD formulation that enforces physically accurate rigid-body dynamics through a novel momentum-conservation constraint and a modified velocity update. Additionally, we introduce a solver-agnostic benchmark with analytical solutions to evaluate physical accuracy. Using this benchmark, we show that PBD-R significantly outperforms PBD and achieves competitive accuracy with MuJoCo while requiring less computation.

Ava Abderezaei, Nataliya Nechyporenko, Joseph Miceli +2

Department of Computer Science, University of Colorado Boulder, CO, USA

Physics-Based Simulation Robotics Simulation Dynamical Systems

Mar 13, 2026

Verification and Forward Invariance of Control Barrier Functions for Differential-Algebraic Systems

Differential-algebraic equations (DAEs) arise in power networks, chemical processes, and multibody systems, where algebraic constraints encode physical conservation laws. The safety of such systems is critical, yet safe control is challenging because algebraic constraints restrict allowable state trajectories. Control barrier functions (CBFs) provide computationally efficient safety filters for ordinary differential equation (ODE) systems. However, existing CBF methods are not directly applicable to DAEs due to potential conflicts between the CBF condition and the constraint manifold. This paper introduces DAE-aware CBFs that incorporate the differential-algebraic structure through projected vector fields. We derive conditions that ensure forward invariance of safe sets while preserving algebraic constraints and extend the framework to higher-index DAEs. A systematic verification framework is developed, establishing necessary and sufficient conditions for geometric correctness and feasibility of DAE-aware CBFs. For polynomial systems, sum-of-squares certificates are provided, while for nonpolynomial and neural network candidates, satisfiability modulo theories are used for falsification. The approach is validated on wind turbine and flexible-link manipulator systems.

Hongchao Zhang, Mohamad H. Kazma, Meiyi Ma +2

Department of Computer Science, Vanderbilt University, Nashville, TN 37235 USA. · Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235 USA.

Control Barrier Functions Dynamical Systems Formal Verification

Mar 13, 2026

Clustering Astronomical Orbital Synthetic Data Using Advanced Feature Extraction and Dimensionality Reduction Techniques

The dynamics of Saturn's satellite system offer a rich framework for studying orbital stability and resonance interactions. Traditional methods for analysing such systems, including Fourier analysis and stability metrics, struggle with the scale and complexity of modern datasets. This study introduces a machine learning-based pipeline for clustering approximately 22,300 simulated satellite orbits, addressing these challenges with advanced feature extraction and dimensionality reduction techniques. The key to this approach is using MiniRocket, which efficiently transforms 400 timesteps into a 9,996-dimensional feature space, capturing intricate temporal patterns. Additional automated feature extraction and dimensionality reduction techniques refine the data, enabling robust clustering analysis. This pipeline reveals stability regions, resonance structures, and other key behaviours in Saturn's satellite system, providing new insights into their long-term dynamical evolution. By integrating computational tools with traditional celestial mechanics techniques, this study offers a scalable and interpretable methodology for analysing large-scale orbital datasets and advancing the exploration of planetary dynamics.

Eraldo Pereira Marinho, Nelson Callegari Junior, Fabricio Aparecido Breve +1

Institute of Geosciences and Exact Sciences (IGCE), Unesp, Rio Claro, Brazil.

Dynamical Systems