Dynamical Systems

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Overview

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

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

Mar 13, 2026

Fractals made Practical: Denoising Diffusion as Partitioned Iterated Function Systems

What is a diffusion model actually doing when it turns noise into a photograph? We show that the deterministic DDIM reverse chain operates as a Partitioned Iterated Function System (PIFS) and that this framework serves as a unified design language for denoising diffusion model schedules, architectures, and training objectives. From the PIFS structure we derive three computable geometric quantities: a per-step contraction threshold Lt∗L^*_t, a diagonal expansion function ft(λ)f_t(λ) and a global expansion threshold λ∗∗λ^{**}. These quantities require no model evaluation and fully characterize the denoising dynamics. They structurally explain the two-regime behavior of diffusion models: global context assembly at high noise via diffuse cross-patch attention and fine-detail synthesis at low noise via patch-by-patch suppression release in strict variance order. Self-attention emerges as the natural primitive for PIFS contraction. The Kaplan-Yorke dimension of the PIFS attractor is determined analytically through a discrete Moran equation on the Lyapunov spectrum. Through the study of the fractal geometry of the PIFS, we derive three optimal design criteria and show that four prominent empirical design choices (the cosine schedule offset, resolution-dependent logSNR shift, Min-SNR loss weighting, and Align Your Steps sampling) each arise as approximate solutions to our explicit geometric optimization problems tuning theory into practice.

Ann Dooms

Department of Mathematics & Data Science Vrije Universiteit Brussel Pleinlaan 2, 1050 Brussels, BELGIUM

Diffusion-Based DenoisingDiffusion ModelsDiffusion SamplingDiffusion Timestep SchedulingDynamical Systems
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Mar 12, 2026

Chemical Reaction Networks Learn Better than Spiking Neural Networks

We mathematically prove that chemical reaction networks without hidden layers can solve tasks for which spiking neural networks require hidden layers. Our proof uses the deterministic mass-action kinetics formulation of chemical reaction networks. Specifically, we prove that a certain reaction network without hidden layers can learn a classification task previously proved to be achievable by a spiking neural network with hidden layers. We provide analytical regret bounds for the global behavior of the network and analyze its asymptotic behavior and Vapnik-Chervonenkis dimension. In a numerical experiment, we confirm the learning capacity of the proposed chemical reaction network for classifying handwritten digits in pixel images, and we show that it solves the task more accurately and efficiently than a spiking neural network with hidden layers. This provides a motivation for machine learning in chemical computers and a mathematical explanation for how biological cells might exhibit more efficient learning behavior within biochemical reaction networks than neuronal networks.

Sophie Jaffard, Ivo F. Sbalzarini

Max Planck Institute of Molecular Cell Biology and Genetics Center for Systems Biology Dresden Dresden, Germany · Dresden University of Technology, Faculty of Computer Science Max Planck Institute of Molecular Cell Biology and Genetics Center for Systems Biology Dresden Dresden, Germany

ClassificationDynamical SystemsSpiking Neural Networks
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Mar 12, 2026

Feedback-Coupled Memory Systems: A Dynamical Model for Adaptive Coordination

This paper develops a dynamical framework for adaptive coordination in systems of interacting agents referred to here as Feedback-Coupled Memory Systems (FCMS). Instead of framing coordination as equilibrium optimization or agent-centric learning, the model describes a closed-loop interaction between agents, incentives, and a persistent environment. The environment stores accumulated coordination signals, a distributed incentive field transmits them locally, and agents update in response, generating a feedback-driven dynamical system. Three main results are established. First, under dissipativity, the closed-loop system admits a bounded forward-invariant region, ensuring dynamical viability independently of global optimality. Second, when incentives depend on persistent environmental memory, coordination cannot be reduced to a static optimization problem. Third, within the FCMS class, coordination requires a bidirectional coupling in which memory-dependent incentives influence agent updates, while agent behavior reshapes the environmental state. Numerical analysis of a minimal specification identifies a Neimark-Sacker bifurcation at a critical coupling threshold (βcβ_c), providing a stability boundary for the system. Near the bifurcation threshold, recovery time diverges and variance increases, yielding a computable early warning signature of coordination breakdown in observable time series. Additional simulations confirm robustness under nonlinear saturation and scalability to populations of up to N=106N = 10^{6} agents making it more relevant for real-world applications. The proposed framework offers a dynamical perspective on coordination in complex systems, with potential extensions to multi-agent systems, networked interactions, and macro-level collective dynamics.

Stefano Grassi

Bangkok University, Phahonyothin Rd, Khlong Nueng, Khlong Luang District, Pathum Thani 12120, Thailand

Critical Slowing DownMulti-Agent CoordinationDynamical Systems
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Mar 11, 2026

Factorized Neural Implicit DMD for Parametric Dynamics

A data-driven, model-free approach to modeling the temporal evolution of physical systems mitigates the need for explicit knowledge of the governing equations. Even when physical priors such as partial differential equations are available, such systems often reside in high-dimensional state spaces and exhibit nonlinear dynamics, making traditional numerical solvers computationally expensive and ill-suited for real-time analysis and control. Consider the problem of learning a parametric flow of a dynamical system: with an initial field and a set of physical parameters, we aim to predict the system's evolution over time in a way that supports long-horizon rollouts, generalization to unseen parameters, and spectral analysis. We propose a physics-coded neural field parameterization of the Koopman operator's spectral decomposition. Unlike a physics-constrained neural field, which fits a single solution surface, and neural operators, which directly approximate the solution operator at fixed time horizons, our model learns a factorized flow operator that decouples spatial modes and temporal evolution. This structure exposes underlying eigenvalues, modes, and stability of the underlying physical process to enable stable long-term rollouts, interpolation across parameter spaces, and spectral analysis. We demonstrate the efficacy of our method on a range of dynamics problems, showcasing its ability to accurately predict complex spatiotemporal phenomena while providing insights into the system's dynamic behavior.

Siyuan Chen, Zhecheng Wang, Yixin Chen +4

Department of Computer Science, University of British Columbia, Vancouver, Canada · Department of Computer Science, University of Toronto, Toronto, Canada

Dynamical SystemsNeural Surrogate ModelingDynamic Mode DecompositionKoopman Operator LearningImplicit Neural Representations
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Mar 11, 2026

MCMC Informed Neural Emulators for Uncertainty Quantification in Dynamical Systems

Neural networks are a commonly used approach to replace physical models with computationally cheap surrogates. Parametric uncertainty quantification can be included in training, assuming that an accurate prior distribution of the model parameters is available. Here we study the common opposite situation, where direct screening or random sampling of model parameters leads to exhaustive training times and evaluations at unphysical parameter values. Our solution is to decouple uncertainty quantification from network architecture. Instead of sampling network weights, we introduce the model-parameter distribution as an input to network training via Markov chain Monte Carlo (MCMC). In this way, the surrogate achieves the same uncertainty quantification as the underlying physical model, but with substantially reduced computation time. The approach is fully agnostic with respect to the neural network choice. In our examples, we present a quantile emulator for prediction and a novel autoencoder-based ODE network emulator that can flexibly estimate different trajectory paths corresponding to different ODE model parameters. Moreover, we present a mathematical analysis that provides a transparent way to relate potential performance loss to measurable distribution mismatch.

Heikki Haario, Zhi-Song Liu, Martin Simon +1

Lappeenranta-Lahti University of Technology LUT, Lahti, Finland · Frankfurt University of Applied Sciences, Frankfurt am Main, Germany · University of Huddersfield, Huddersfield, United Kingdom

Ordinary Differential EquationsUncertainty QuantificationNeural Surrogate ModelingDynamical Systems
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Mar 11, 2026

JEDI: Jointly Embedded Inference of Neural Dynamics

Animal brains flexibly and efficiently achieve many behavioral tasks with a single neural network. A core goal in modern neuroscience is to map the mechanisms of the brain's flexibility onto the dynamics underlying neural populations. However, identifying task-specific dynamical rules from limited, noisy, and high-dimensional experimental neural recordings remains a major challenge, as experimental data often provide only partial access to brain states and dynamical mechanisms. While recurrent neural networks (RNNs) directly constrained neural data have been effective in inferring underlying dynamical mechanisms, they are typically limited to single-task domains and struggle to generalize across behavioral conditions. Here, we introduce JEDI, a hierarchical model that captures neural dynamics across tasks and contexts by learning a shared embedding space over RNN weights. This model recapitulates individual samples of neural dynamics while scaling to arbitrarily large and complex datasets, uncovering shared structure across conditions in a single, unified model. Using simulated RNN datasets, we demonstrate that JEDI accurately learns robust, generalizable, condition-specific embeddings. By reverse-engineering the weights learned by JEDI, we show that it recovers ground truth fixed point structures and unveils key features of the underlying neural dynamics in the eigenspectra. Finally, we apply JEDI to motor cortex recordings during monkey reaching to extract mechanistic insight into the neural dynamics of motor control. Our work shows that joint learning of contextual embeddings and recurrent weights provides scalable and generalizable inference of brain dynamics from recordings alone.

Anirudh Jamkhandi, Ali Korojy, Olivier Codol +2

Mila - Quebec AI Institute · Université de Montréal

Latent Dynamics ModelingDynamical SystemsRecurrent Neural NetworksSystem Identification
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Mar 10, 2026

Walking on Rough Terrain with Any Number of Legs

Robotics would gain by replicating the remarkable agility of arthropods in navigating complex environments. Here we consider the control of multi-legged systems which have 6 or more legs. Current multi-legged control strategies in robots include large black-box machine learning models, Central Pattern Generator (CPG) networks, and open-loop feed-forward control with stability arising from mechanics. Here we present a multi-legged control architecture for rough terrain using a segmental robot with 3 actuators for every 2 legs, which we validated in simulation for robots with 6 to 16 legs. Segments have identical state machines, and each segment also receives input from the segment in front of it. Our design bridges the gap between WalkNet-like event cascade controllers and CPG-based controllers: it tightly couples to the ground when contact is present, but produces fictive locomotion when ground contact is missing. The approach may be useful as an adaptive and computationally lightweight controller for multi-legged robots, and as a baseline capability for scaffolding the learning of machine learning controllers.

Zhuoyang Chen, Xinyuan Wang, Shai Revzen

The authors are with the University of Michigan, Ann Arbor, MI 48109, USA.

Legged RoboticsRobot LocomotionTerrain-Aware Robot LocomotionDynamical Systems
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Mar 10, 2026

Exploring Collatz Dynamics with Human-LLM Collaboration

We present a comprehensive structural analysis of the Collatz conjecture through ~1014 computational experiments yielding 630 formal results. By systematically deploying 29 distinct mathematical paradigms--including transfer operator spectral theory, S-unit equations, p-adic interpolation, martingale methods, modular sieving, formal language theory, cascade algebra, discrete logarithm obstruction, and Diophantine approximation--we establish a Paradigm Exhaustion Theorem: every known framework for promoting distributional convergence ("almost all orbits descend") to pointwise convergence ("all orbits descend") encounters an irreducible structural obstruction when applied to the Syracuse map. On the unconditional side, we prove: (i) the Syracuse transfer operator has a uniform spectral gap for all M, implying equidistribution modulo any power of 2; (ii) any nontrivial cycle of length L satisfies D > 2^F for all L >= 3, giving ord_D(2) > F and F+1 distinct residues mod D; (iii) divergent starting points have natural density 0 and Hausdorff dimension ~0.68; (iv) the formal language of divergent-compatible v-sequences is not context-free; (v) cylinder-averaged density-1 convergence is proved unconditionally via spectral contraction on the invariant core I_2; (vi) a discrete logarithm triple filter achieves 100% cycle blockage for all L tested. We identify the Distributional-to-Pointwise Gap as the irreducible core and prove it equivalent to the divergence component. The modular sieve is permanently nonempty via the Mersenne Bypass. The present work is not a proof of the Collatz conjecture; it characterizes why the conjecture resists proof. The 29-paradigm exhaustion constitutes the most comprehensive structural survey of Collatz attack surfaces to date. Produced through human-LLM collaboration; see Section 12.

Edward Y. Chang

Computer Science, Stanford University · QuadriumAI

Dynamical Systems
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Mar 9, 2026

STRIDE: Structured Lagrangian and Stochastic Residual Dynamics via Flow Matching

Robotic systems operating in unstructured environments must operate under significant uncertainty arising from intermittent contacts, frictional variability, and unmodeled compliance. While recent model-free approaches have demonstrated impressive performance, many deployment settings still require predictive models that support planning, constraint handling, and online adaptation. Analytical rigid-body models provide strong physical structure but often fail to capture complex interaction effects, whereas purely data-driven models may violate physical consistency, exhibit data bias, and accumulate long-horizon drift. In this work, we propose STRIDE, a dynamics learning framework that explicitly separates conservative rigid-body mechanics from uncertain, effectively stochastic non-conservative interaction effects. The structured component is modeled using a Lagrangian Neural Network (LNN) to preserve energy-consistent inertial dynamics, while residual interaction forces are represented using Conditional Flow Matching (CFM) to capture multi-modal interaction phenomena. The two components are trained jointly end-to-end, enabling the model to retain physical structure while representing complex stochastic behavior. We evaluate STRIDE on systems of increasing complexity, including a pendulum, the Unitree Go1 quadruped, and the Unitree G1 humanoid. Results show 20% reduction in long-horizon prediction error and 30% reduction in contact force prediction error compared to deterministic residual baselines, supporting more reliable model-based control in uncertain robotic environments.

Prakrut Kotecha, Ganga Nair B, Shishir Kolathaya

the Robert Bosch Center for Cyber-Physical Systems, Indian Institute of Science, Bengaluru · the Robert Bosch Center for Cyber-Physical Systems and the Department of Computer Science & Automation, Indian Institute of Science, Bengaluru

Lagrangian Neural NetworksRobot Dynamics LearningDynamical SystemsResidual Dynamics Learning
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Mar 7, 2026

Learning Quadruped Walking from Seconds of Demonstration

Quadruped locomotion provides a natural setting for understanding when model-free learning can outperform model-based control design, by exploiting data patterns to bypass the difficulty of optimizing over discrete contacts and the combinatorial explosion of mode changes. We give a principled analysis of why imitation learning with quadrupeds can be inherently effective in a small data regime, based on the structure of its limit cycles, Poincaré return maps, and local numerical properties of neural networks. The understanding motivates a new imitation learning method that regulates the alignment between variations in a latent space and those over the output actions. Hardware experiments confirm that a few seconds of demonstration is sufficient to train various locomotion policies from scratch entirely offline with reasonable robustness.

Ruipeng Zhang, Hongzhan Yu, Ya-Chien Chang +3

University of California, San Diego

Robot Imitation LearningData-Efficient Robot LearningQuadruped LocomotionDynamical Systems
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Mar 6, 2026

VertiAdaptor: Online Kinodynamics Adaptation for Vertically Challenging Terrain

Autonomous driving in off-road environments presents significant challenges due to the dynamic and unpredictable nature of unstructured terrain. Traditional kinodynamic models often struggle to generalize across diverse geometric and semantic terrain types, underscoring the need for real-time adaptation to ensure safe and reliable navigation. We propose VertiAdaptor (VA), a novel online adaptation framework that efficiently integrates elevation with semantic embeddings to enable terrain-aware kinodynamic modeling and planning via function encoders. VA learns a kinodynamic space spanned by a set of neural ordinary differential equation basis functions, capturing complex vehicle-terrain interactions across varied environments. After offline training, the proposed approach can rapidly adapt to new, unseen environments by identifying kinodynamics in the learned space through a computationally efficient least-squares calculation. We evaluate VA within the Verti-Bench simulator, built on the Chrono multi-physics engine, and validate its performance both in simulation and on a physical Verti-4-Wheeler platform. Our results demonstrate that VA improves prediction accuracy by up to 23.9% and achieves a 5X faster adaptation time, advancing the robustness and reliability of autonomous robots in complex and evolving off-road environments.

Tong Xu, Chenhui Pan, Aniket Datar +1

Department of Computer Science, George Mason University

Dynamical SystemsOnline System IdentificationOff-Road Robot NavigationAutonomous DrivingTerrain-Aware Robot LocomotionAutonomous Driving Planning
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Mar 6, 2026

Frequency-Separable Hamiltonian Neural Network for Multi-Timescale Dynamics

While Hamiltonian mechanics provides a powerful inductive bias for neural networks modeling dynamical systems, Hamiltonian Neural Networks and their variants often fail to capture complex temporal dynamics spanning multiple timescales. This limitation is commonly linked to the spectral bias of deep neural networks, which favors learning low-frequency, slow-varying dynamics. Prior approaches have sought to address this issue through symplectic integration schemes that enforce energy conservation or by incorporating geometric constraints to impose structure on the configuration-space. However, such methods either remain limited in their ability to fully capture multiscale dynamics or require substantial domain specific assumptions. In this work, we exploit the observation that Hamiltonian functions admit decompositions into explicit fast and slow modes and can be reconstructed from these components. We introduce the Frequency-Separable Hamiltonian Neural Network (FS-HNN), which parameterizes the system Hamiltonian using multiple networks, each governed by Hamiltonian dynamics and trained on data sampled at distinct timescales. We further extend this framework to partial differential equations by learning a state- and boundary-conditioned symplectic operators. Empirically, we show that FS-HNN improves long-horizon extrapolation performance on challenging dynamical systems and generalizes across a broad range of ODE and PDE problems.

Yaojun Li, Yulong Yang, Christine Allen-Blanchette

Princeton University, USA

Hamiltonian Neural NetworksDynamical SystemsNeural PDE Solvers
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Mar 6, 2026

Random Quadratic Form on a Sphere: Synchronization by Common Noise

We introduce the Random Quadratic Form (RQF): a stochastic differential equation which formally corresponds to the gradient flow of a random quadratic functional on a sphere. While the one-point dynamics of the system is a Brownian motion and thus has no preferred direction, the two-point motion exhibits nontrivial synchronizing behaviour. In this work we study synchronization of the RQF, namely we give both distributional and path-wise characterizations of the solutions by studying invariant measures and random attractors of the system. The RQF model is motivated by the study of the role of linear layers in transformers and illustrates the synchronization by common noise phenomena arising in the simplified models of transformers. In particular, we provide an alternative (independent of self-attention) explanation of the clustering behaviour in deep transformers and show that tokens cluster even in the absence of the self-attention mechanism.

Maximilian Engel, Anna Shalova

Korteweg-de Vries Institute for Mathematics, University of Amsterdam · Department of Mathematics and Computer Science, FU Berlin

Stochastic Differential EquationsDynamical SystemsTransformer
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Mar 5, 2026

Behavior-dLDS: A decomposed linear dynamical systems model for neural activity partially constrained by behavior

Brain-wide recordings of large-scale networks of neurons now provide an unprecedented view into how the brain drives behavior. However, brain activity contains both information directly related to behavior as well as the potential for many internal computations. Moreover, observable behavior is executed not only by the brain, but also by the spinal cord and peripheral nervous system. Behavior is a coarse-grained product of neural activity, and we thus take the view that it can be best represented by lower-dimensional latent neural dynamics. Capturing this indirect relationship while disambiguating behavior-generating networks from internal computations running in parallel requires new modeling approaches that can embody the parallel and distributed nature of large-scale neural populations. We thus present behavior-decomposed linear dynamical systems (b-dLDS) to disentangle simultaneously recorded subsystems and identify how the latent neural subsystems relate to behavior. We demonstrate the ability of b-dLDS to decouple behavioral vs. internal computations on controlled, simulated data, showing improvements over a state-of-the-art model that uses behavior to supervise all dynamics based on behavior. We also demonstrate b-dLDS's interpretability benefits on a task-driven RNN dataset featuring a nonlinear relationship between behavior and activations. We then show that b-dLDS can further scale up to tens of thousands of neurons by applying our model to a large-scale recording of a zebrafish hindbrain during the complex positional homeostasis behavior, wherein b-dLDS highlights asymmetry in behavior-related dynamic connectivity networks.

Eva Yezerets, En Yang, Misha B. Ahrens +1

Department of Biomedical Engineering Center for Imaging Science Johns Hopkins University · Department of Biology University of North Carolina · Janelia Research Campus Howard Hughes Medical Institute

Latent Dynamics ModelingDynamical SystemsBrain Network Analysis
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Mar 5, 2026

Curve-Induced Dynamical Systems on Riemannian Manifolds and Lie Groups

Deploying robots in household environments requires safe, adaptable, and interpretable behaviors that respect the geometric structure of tasks. Often represented on Lie groups and Riemannian manifolds, this includes poses on SE(3) or symmetric positive definite matrices encoding stiffness or damping matrices. In this context, dynamical system-based approaches offer a natural framework for generating such behavior, providing stability and convergence while remaining responsive to changes in the environment. We introduce Curve-induced Dynamical systems on Smooth Manifolds (CDSM), a real-time framework for constructing dynamical systems directly on Riemannian manifolds and Lie groups. The proposed approach constructs a nominal curve on the manifold, and generates a dynamical system which combines a tangential component that drives motion along the curve and a normal component that attracts the state toward the curve. We provide a stability analysis of the resulting dynamical system and validate the method quantitatively. On an S2 benchmark, CDSM demonstrates improved trajectory accuracy, reduced path deviation, and faster generation and query times compared to state-of-the-art methods. Finally, we demonstrate the practical applicability of the framework on both a robotic manipulator, where poses on SE(3) and damping matrices on SPD(n) are adapted online, and a mobile manipulator.

Saray Bakker, Martin Schonger, Tobias Löw +2

Delft University of Technology, The Netherlands · Idiap Research Institute, Switzerland · École Polytechnique Fédérale de Lausanne (EPFL), Switzerland

Dynamical SystemsRobot Motion GenerationRobotic Control
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Mar 4, 2026

Latent Particle World Models: Self-supervised Object-centric Stochastic Dynamics Modeling

We introduce Latent Particle World Model (LPWM), a self-supervised object-centric world model scaled to real-world multi-object datasets and applicable in decision-making. LPWM autonomously discovers keypoints, bounding boxes, and object masks directly from video data, enabling it to learn rich scene decompositions without supervision. Our architecture is trained end-to-end purely from videos and supports flexible conditioning on actions, language, and image goals. LPWM models stochastic particle dynamics via a novel latent action module and achieves state-of-the-art results on diverse real-world and synthetic datasets. Beyond stochastic video modeling, LPWM is readily applicable to decision-making, including goal-conditioned imitation learning, as we demonstrate in the paper. Code, data, pre-trained models and video rollouts are available: https://taldatech.github.io/lpwm-web

Tal Daniel, Carl Qi, Dan Haramati +5

Carnegie Mellon University · UT Austin · Brown University +2

World Model LearningObject-Centric World ModelsVideo Object-Centric LearningObject-Centric Representation LearningDynamical Systems
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Mar 4, 2026

Lyapunov Stability of Stochastic Vector Optimization: Theory and Numerical Implementation

The use of stochastic differential equations in multi-objective optimization has been limited, in practice, by two persistent gaps: incomplete stability analyses and the absence of accessible implementations. We revisit a drift--diffusion model for unconstrained vector optimization in which the drift is induced by a common descent direction and the diffusion term preserves exploratory behavior. The main theoretical contribution is a self-contained Lyapunov analysis establishing global existence, pathwise uniqueness, and non-explosion under a dissipativity condition, together with positive recurrence under an additional coercivity assumption. We also derive an Euler--Maruyama discretization and implement the resulting iteration as a pymoo-compatible algorithm -- pymoo being an open-source Python framework for multi-objective optimization -- with an interactive PymooLab front-end for reproducible experiments. Empirical results on DTLZ2 with objective counts from three to fifteen indicate a consistent trade-off: compared with established evolutionary baselines, the method is less competitive in low-dimensional regimes but remains a viable option under restricted evaluation budgets in higher-dimensional settings. Taken together, these observations suggest that stochastic drift--diffusion search occupies a mathematically tractable niche alongside population-based heuristics -- not as a replacement, but as an alternative whose favorable properties are amenable to rigorous analysis.

Thiago Santos, Sebastiao Xavier

Department of Mathematics, Federal University of Ouro Preto, UFOP, Ouro Preto, Brazil

Multi-Objective OptimizationStochastic Differential EquationsDynamical Systems
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Mar 3, 2026

A Dynamical Theory of Sequential Retrieval in Input-Driven Hopfield Networks

Reasoning is the ability to integrate internal states and external inputs in a meaningful and semantically consistent flow. Contemporary machine learning (ML) systems increasingly rely on such sequential reasoning, from language understanding to multi-modal generation, often operating over dictionaries of prototypical patterns reminiscent of associative memory models. Understanding retrieval and sequentiality in associative memory models provides a powerful bridge to gain insight into ML reasoning. While the static retrieval properties of associative memory models are well understood, the theoretical foundations of sequential retrieval and multi-memory integration remain limited, with existing studies largely relying on numerical evidence. This work develops a dynamical theory of sequential reasoning in Hopfield networks. We consider the recently proposed input-driven plasticity (IDP) Hopfield network and analyze a two-timescale architecture coupling fast associative retrieval with slow reasoning dynamics. We derive explicit conditions for self-sustained memory transitions, including gain thresholds, escape times, and collapse regimes. Together, these results provide a principled mathematical account of sequentiality in associative memory models, bridging classical Hopfield dynamics and modern reasoning architectures.

Simone Betteti, Giacomo Baggio, Sandro Zampieri

The Italian Institute of Artificial Intelligence for Industry · Department of Information Engineering, Università degli Studi di Padova

Hopfield NetworksAssociative MemoryDynamical Systems
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Mar 2, 2026

Graph neural network force fields for adiabatic dynamics of lattice Hamiltonians

Scalable and symmetry-consistent force-field models are essential for extending quantum-accurate simulations to large spatiotemporal scales. While descriptor-based neural networks can incorporate lattice symmetries through carefully engineered features, we show that graph neural networks (GNNs) provide a conceptually simpler and more unified alternative in which discrete lattice translation and point-group symmetries are enforced directly through local message passing and weight sharing. We develop a GNN-based force-field framework for the adiabatic dynamics of lattice Hamiltonians and demonstrate it for the semiclassical Holstein model. Trained on exact-diagonalization data, the GNN achieves high force accuracy, strict linear scaling with system size, and direct transferability to large lattices. Enabled by this scalability, we perform large-scale Langevin simulations of charge-density-wave ordering following thermal quenches, revealing dynamical scaling and anomalously slow sub--Allen--Cahn coarsening. These results establish GNNs as an elegant and efficient architecture for symmetry-aware, large-scale dynamical simulations of correlated lattice systems.

Yunhao Fan, Gia-Wei Chern

Department of Physics, University of Virginia, Charlottesville, VA 22904, USA

Graph Neural NetworksHamiltonian DynamicsDynamical SystemsEquivariant GNNs
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Mar 2, 2026

CHLU: The Causal Hamiltonian Learning Unit as a Symplectic Primitive for Deep Learning

Current deep learning primitives dealing with temporal dynamics suffer from a fundamental dichotomy: they are either discrete and unstable (LSTMs) \citep{pascanu_difficulty_2013}, leading to exploding or vanishing gradients; or they are continuous and dissipative (Neural ODEs) \citep{dupont_augmented_2019}, which destroy information over time to ensure stability. We propose the Causal Hamiltonian Learning Unit (pronounced: clue), a novel Physics-grounded computational learning primitive. By enforcing a Relativistic Hamiltonian structure and utilizing symplectic integration, a CHLU strictly conserves phase-space volume, as an attempt to solve the memory-stability trade-off. We show that the CHLU is designed for infinite-horizon stability, as well as controllable noise filtering. We then demonstrate a CHLU's generative ability using the MNIST dataset as a proof-of-principle.

Pratik Jawahar, Maurizio Pierini

University of Manchester · CERN

Symplectic RNNsHamiltonian Neural NetworksDynamical Systems
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Mar 1, 2026

Path Integral Particle Filtering for Hybrid Systems via Saltation Matrices

State estimation for hybrid systems that undergo intermittent contact with their environments, such as extraplanetary robots and satellites undergoing docking operations, is difficult due to the discrete uncertainty propagation during contact. To handle this propagation, this paper presents an optimal-control-based particle filtering method that leverages saltation matrices to map out uncertainty propagation during contact events. By exploiting a path integral filtering framework that exploits the duality between smoothing and optimal control, the resulting state estimation algorithm is robust to outlier effects, flexible to non-Gaussian noise distributions, and handles challenging contact dynamics in hybrid systems. To evaluate the validity and consistency of the proposed approach, this paper tests it against strong baselines on the stochastic dynamics generated by a bouncing ball and spring loaded inverted pendulum.

Karthik Shaji, Sreeranj Jayadevan, Bo Yuan +2

Graduate Student, Daniel Guggenheim School of Aerospace Engineering · MathWorks · PlusAI +1

Particle FilteringRobot State EstimationDynamical Systems
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Feb 25, 2026

Solving stiff dark matter equations via Jacobian Normalization with Physics-Informed Neural Networks

Stiff differential equations pose a major challenge for Physics-Informed Neural Networks (PINNs), often causing poor convergence. We propose a simple, hyperparameter-free method to address stiffness by normalizing loss residuals with the Jacobian. We provide theoretical indications that Jacobian-based normalization can improve gradient descent and validate it on benchmark stiff ordinary differential equations. We then apply it to a realistic system: the stiff Boltzmann equations (BEs) governing weakly interacting massive particle (WIMP) dark matter (DM). Our approach achieves higher accuracy than attention mechanisms previously proposed for handling stiffness, recovering the full solution where prior methods fail. This is further demonstrated in an inverse problem with a single experimental data point - the observed DM relic density - where our inverse PINNs correctly infer the cross section that solves the BEs in both Standard and alternative cosmologies.

M. P. Bento, H. B. Câmara, J. R. Rocha +1

Departamento de Física and CFTP, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal · Institute of Experimental and Applied Physics, Czech Technical University in Prague, Prague 160 00, Czech Republic

Physics-Informed MLDynamical SystemsHigh-Energy Physics
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Feb 21, 2026

Systematic Analysis of Coupling Effects on Closed-Loop and Open-Loop Performance in Aerial Continuum Manipulators

This paper investigates two distinct approaches to the dynamic modeling of aerial continuum manipulators (ACMs): the decoupled and the coupled formulations. Both open-loop and closed-loop behaviors of a representative ACM are analyzed. The primary objective is to determine the conditions under which the decoupled model attains accuracy comparable to the coupled model while offering reduced computational cost under identical numerical conditions. The system dynamics are first derived using the Euler--Lagrange method under the piecewise constant curvature (PCC) assumption, with explicit treatment of the near-zero curvature singularity. A decoupled model is then obtained by neglecting the coupling terms in the ACM dynamics, enabling systematic evaluation of open-loop responses under diverse actuation profiles and external wrenches. To extend the analysis to closed-loop performance, a novel dynamics-based proportional-derivative sliding mode image-based visual servoing (DPD-SM-IBVS) controller is developed for regulating image feature errors in the presence of a moving target. The controller is implemented with both coupled and decoupled models, allowing a direct comparison of their effectiveness. The open-loop simulations reveal pronounced discrepancies between the two modeling approaches, particularly under varying torque inputs and continuum arm parameters. Conversely, the closed-loop experiments demonstrate that the decoupled model achieves tracking accuracy on par with the coupled model (within subpixel error) while incurring lower computational cost.

Niloufar Amiri, Shayan Sepahvand, Iraj Mantegh +1

Department of Mechanical, Industrial, and Mechatronics Engineering, Toronto Metropolitan University, Toronto, ON M5B 2K3, Canada · Aerospace Research Center, National Research Council of Canada (NRC), Montreal, QC H3T 2B2, Canada

Continuum Robot ControlAerial ManipulationDynamical Systems
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Feb 21, 2026

Transformers for dynamical systems learn transfer operators in-context

Large-scale foundation models for scientific machine learning adapt to physical settings unseen during training, such as zero-shot transfer between turbulent scales. This phenomenon, in-context learning, challenges conventional understanding of learning and adaptation in physical systems. Here, we study in-context learning of dynamical systems in a minimal setting: we train a small two-layer, single-head transformer to forecast one dynamical system, and then evaluate its ability to forecast a different dynamical system without retraining. We discover an early tradeoff in training between in-distribution and out-of-distribution performance, which manifests as a secondary double descent phenomenon. We discover that attention-based models apply a transfer-operator forecasting strategy in-context. They (1) lift low-dimensional time series using delay embedding, to detect the system's higher-dimensional dynamical manifold, and (2) identify and forecast long-lived invariant sets that characterize the global flow on this manifold. Our results clarify the mechanism enabling large pretrained models to forecast unseen physical systems at test time without retraining, and they illustrate the unique ability of attention-based models to leverage global attractor information in service of short-term forecasts.

Anthony Bao, Jeffrey Lai, William Gilpin

Department of Electrical & Computer Engineering, The University of Texas at Austin, Austin, Texas 78712, USA · The Oden Institute, The University of Texas at Austin, Austin, Texas 78712, USA · Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA

In-Context LearningDynamical SystemsZero-Shot Time Series ForecastingTime Series TransformersOperator Learning
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Feb 20, 2026

Deepmechanics

Physics-informed deep learning models have emerged as powerful tools for learning dynamical systems. These models directly encode physical principles into network architectures. However, systematic benchmarking of these approaches across diverse physical phenomena remains limited, particularly in conservative and dissipative systems. In addition, benchmarking that has been done thus far does not integrate out full trajectories to check stability. In this work, we benchmark three prominent physics-informed architectures such as Hamiltonian Neural Networks (HNN), Lagrangian Neural Networks (LNN), and Symplectic Recurrent Neural Networks (SRNN) using the DeepChem framework, an open-source scientific machine learning library. We evaluate these models on six dynamical systems spanning classical conservative mechanics (mass-spring system, simple pendulum, double pendulum, and three-body problem, spring-pendulum) and non-conservative systems with contact (bouncing ball). We evaluate models by computing error on predicted trajectories and evaluate error both quantitatively and qualitatively. We find that all benchmarked models struggle to maintain stability for chaotic or nonconservative systems. Our results suggest that more research is needed for physics-informed deep learning models to learn robust models of classical mechanical systems.

Abhay Shinde, Aryan Amit Barsainyan, Jose Siguenza +3

Deep Forest Sciences, Mumbai, Maharashtra, India · Deep Forest Sciences, Bengaluru, Karnataka, India · Deep Forest Sciences, Palo alto, California, USA +1

Symplectic RNNsLagrangian Neural NetworksPhysics-Informed MLDynamical SystemsHamiltonian Neural NetworksRecurrent Neural Networks
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Feb 18, 2026

Position: A Dynamical Systems Perspective is Needed to Advance Time Series Modeling

Time series (TS) modeling has come a long way from early statistical, mainly linear, approaches to the current trend in TS foundation models. With a lot of hype and industrial demand in this field, it is not always clear how much progress there really is. To advance TS forecasting and analysis to the next level, here we argue that the field needs a dynamical systems (DS) perspective. TS of observations from natural or engineered systems almost always originate from some underlying DS, and arguably access to its governing equations would yield theoretically optimal forecasts. This is the promise of DS reconstruction (DSR), a class of ML/AI approaches that aim to infer surrogate models of the underlying DS from data. But models based on DS principles offer other profound advantages: Beyond short-term forecasts, they enable to predict the long-term statistics of an observed system, which in many practical scenarios may be the more relevant quantities. DS theory furthermore provides domain-independent theoretical insight into mechanisms underlying TS generation, and thereby will inform us, e.g., about upper bounds on performance of any TS model, generalization into unseen regimes as in tipping points, or potential control strategies. After reviewing some of the central concepts, methods, measures, and models in DS theory and DSR, we will discuss how insights from this field can advance TS modeling in crucial ways, enabling better forecasting with much lower computational and memory footprints. We conclude with a number of specific suggestions for translating insights from DSR into TS modeling.

Daniel Durstewitz, Christoph Jürgen Hemmer, Florian Hess +2

Dept. of Theoretical Neuroscience, Central Institute of Mental Health, Mannheim, Germany · Faculty of Physics & Astronomy, Heidelberg Univ., Heidelberg, Germany · Interdisciplinary Center for Scientific Computing, Heidelberg, Germany

Time Series ModelingTime Series ForecastingLatent Dynamics ModelingDynamical SystemsSurrogate Modeling
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Feb 17, 2026

Evolutionary Systems Thinking: From Equilibrium Models to Open-Ended Adaptive Dynamics

Complex change is often described as ``evolutionary'' in economics, policy, technology, and organizations, yet many system dynamics models represent behavior within a fixed set of stocks, flows, relationships, and transition rules. Such models can generate nonlinear, oscillatory, path-dependent, or chaotic behavior, but structural novelty must ordinarily be specified in advance. This paper argues that evolutionary dynamics should be treated as a core systems-thinking problem rather than as a biological metaphor. We introduce Stability-Driven Assembly (SDA), a minimal non-equilibrium framework in which stochastic interactions and differential persistence generate endogenous selection without genes, template-based replication, or an externally specified fitness function. Longer-lived configurations accumulate in the population and therefore become more likely to participate in subsequent interactions. This creates feedback among persistence, population composition, and future pattern formation. The resulting abundance-weighted sampling is equivalent to fitness-proportional selection, allowing SDA to be interpreted as a natural genetic algorithm driven by persistence-weighted population dynamics. SDA provides a conceptual basis for distinguishing fixed-state-space dynamics from evolving possibility spaces, in which persistent structures can reshape future flows, interactions, and available configurations. It also suggests that equilibrium should be treated as provisional: a quasi-stable regime may be reorganized when a more persistent configuration emerges. We conclude by outlining two ways to extend system dynamics practice: constructing an SDA-style population model alongside a stock-flow model, and using SDA perturbation analysis to examine the vulnerability of an existing regime to structural innovation.

Dan Adler

Open-Ended EvolutionDynamical Systems
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Feb 17, 2026

Relative Geometry of Neural Forecasters: Linking Accuracy and Alignment in Learned Latent Geometry

Neural networks can accurately forecast complex dynamical systems, yet how they internally represent underlying latent geometry remains poorly understood. We study neural forecasters through the lens of representational alignment, introducing anchor-based, geometry-agnostic relative embeddings that remove rotational and scaling ambiguities in latent spaces. Applying this framework across seven canonical dynamical systems - ranging from periodic to chaotic - we reveal reproducible family-level structure: multilayer perceptrons align with other MLPs, recurrent networks with RNNs, while transformers and echo-state networks achieve strong forecasts despite weaker alignment. Alignment generally correlates with forecasting accuracy, yet high accuracy can coexist with low alignment. Relative geometry thus provides a simple, reproducible foundation for comparing how model families internalize and represent dynamical structure.

Deniz Kucukahmetler, Maximilian Jean Hemmann, Julian Mosig von Aehrenfeld +4

Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany · School of Embedded Composite Artificial Intelligence (SECAI), Dresden/Leipzig, Germany · Leipzig University, Leipzig, Germany +2

Neural Representation GeometryLatent Space AlignmentDynamical Systems
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Feb 17, 2026

Continuous-Time Piecewise-Linear Recurrent Neural Networks

In dynamical systems reconstruction (DSR) we aim to recover the dynamical system (DS) underlying observed time series. Specifically, we aim to learn a generative surrogate model which approximates the underlying, data-generating DS, and recreates its long-term properties (`climate statistics'). In scientific and medical areas, in particular, these models need to be mechanistically tractable -- through their mathematical analysis we would like to obtain insight into the recovered system's workings. Piecewise-linear (PL), ReLU-based RNNs (PLRNNs) have a strong track-record in this regard, representing SOTA DSR models while allowing mathematical insight by virtue of their PL design. However, all current PLRNN variants are discrete-time maps. This is in disaccord with the assumed continuous-time nature of most physical and biological processes, and makes it hard to accommodate data arriving at irregular temporal intervals. Neural ODEs are one solution, but they do not reach the DSR performance of PLRNNs and often lack their tractability. Here we develop theory for continuous-time PLRNNs (cPLRNNs): We present a novel algorithm for training and simulating such models, bypassing numerical integration by efficiently exploiting their PL structure. We further demonstrate how important topological objects like equilibria or limit cycles can be determined semi-analytically in trained models. We compare cPLRNNs to both their discrete-time cousins as well as Neural ODEs on DSR benchmarks, including systems with discontinuities which come with hard thresholds.

Alena Brändle, Lukas Eisenmann, Florian Götz +1

Dept. of Theoretical Neuroscience, Central Institute of Mental Health (CIMH), Mannheim, Germany · Faculty of Physics and Astronomy, Heidelberg University · Hector Institute for Artificial Intelligence in Psychiatry, CIMH +2

Recurrent Neural NetworksDynamical SystemsReLU Neural Networks
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Feb 16, 2026

Drift-Diffusion Matching: Embedding dynamics in latent manifolds of asymmetric neural networks

Recurrent neural networks (RNNs) provide a theoretical framework for understanding computation in biological neural circuits, yet classical results, such as Hopfield's model of associative memory, rely on symmetric connectivity that restricts network dynamics to gradient-like flows. In contrast, biological networks support rich time-dependent behaviour facilitated by their asymmetry. Here we introduce a general framework, which we term drift-diffusion matching, for training continuous-time RNNs to represent arbitrary, nonlinear stochastic differential equations (SDEs), with given drift and diffusion coefficients, within a low-dimensional latent subspace. Allowing asymmetric connectivity, we show that RNNs can faithfully embed the drift and diffusion of a given SDE, including nonlinear and nonequilibrium dynamics such as chaotic attractors. As an application, we construct RNN realisations of stochastic systems that transiently explore various attractors through both input-driven switching and autonomous transitions driven by nonequilibrium currents, which we interpret as models of associative and sequential (episodic) memory. To elucidate how these dynamics are encoded in the network, we introduce decompositions of the RNN based on its asymmetric connectivity and its time-irreversibility. Our results extend attractor neural network theory beyond equilibrium, showing that asymmetric neural populations can implement a broad class of dynamical computations within low-dimensional manifolds, unifying ideas from associative memory, nonequilibrium statistical mechanics, and neural computation.

Ramón Nartallo-Kaluarachchi, Renaud Lambiotte, Alain Goriely

Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, United Kingdom · Centre for Eudaimonia and Human Flourishing, University of Oxford, 7 Stoke Pl, Oxford, OX3 9BX, United Kingdom · Complexity Science Hub, Metternichgasse 8, Vienna, 1030, Austria

Recurrent Neural NetworksDynamical SystemsLatent Dynamics ModelingStochastic Differential EquationsAssociative Memory
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