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.

Feb 16, 2026

On the Stability of Nonlinear Dynamics in GD and SGD: Beyond Quadratic Potentials

The dynamical stability of the iterates during training plays a key role in determining the minima obtained by optimization algorithms. For example, stable solutions of gradient descent (GD) correspond to flat minima, which have been associated with favorable features. While prior work often relies on linearization to determine stability, it remains unclear whether linearized dynamics faithfully capture the full nonlinear behavior. Recent work has shown that GD may stably oscillate near a linearly unstable minimum and still converge once the step size decays, indicating that linear analysis can be misleading. In this work, we explicitly study the effect of nonlinear terms. Specifically, we derive an exact criterion for stable oscillations of GD near minima in the multivariate setting. Our condition depends on high-order derivatives, generalizing existing results. Extending the analysis to stochastic gradient descent (SGD), we show that nonlinear dynamics can diverge in expectation even if a single batch is unstable. This implies that stability can be dictated by a single batch that oscillates unstably, rather than an average effect, as linear analysis suggests. Finally, we prove that if all batches are linearly stable, the nonlinear dynamics of SGD are stable in expectation.

Rotem Mulayoff, Sebastian U. Stich

CISPA Helmholtz Center for Information Security, Saarbrücken, Germany

Gradient Descent Dynamics Stochastic Gradient Descent Dynamical Systems Nonconvex Optimization

Feb 16, 2026

Preliminary sonification of ENSO using traditional Javanese gamelan scales

Sonification -- the mapping of data to non-speech audio -- offers an underexplored channel for representing complex dynamical systems. We treat El Niño-Southern Oscillation (ENSO), a canonical example of low-dimensional climate chaos, as a test case for culturally-situated sonification evaluated through complex systems diagnostics. Using parameter-mapping sonification of the Niño 3.4 sea surface temperature anomaly index (1870--2024), we encode ENSO variability into two traditional Javanese gamelan pentatonic systems (pelog and slendro) across four composition strategies, then analyze the resulting audio as trajectories in a two-dimensional acoustic phase space. Recurrence-based diagnostics, convex hull geometry, and coupling analysis reveal that the sonification pipeline preserves key dynamical signatures: alternating modes produce the highest trajectory recurrence rates, echoing ENSO's quasi-periodicity; layered polyphonic modes explore the broadest phase space regions; and the two scale families induce qualitatively distinct coupling regimes between spectral brightness and energy -- predominantly anti-phase in pelog but near-independent in slendro. Phase space trajectory analysis provides a rigorous geometric framework for comparing sonification designs within a complex systems context. Perceptual validation remains necessary; we contribute the dynamical systems methodology for evaluating such mappings.

Sandy Hardian Susanto Herho, Rusmawan Suwarman, Nurjanna Joko Trilaksono +2

Department of Earth and Planetary Sciences, University of California, Riverside, CA 92521, USA · School of Systems Science and Industrial Engineering, State University of New York (SUNY), Binghamton, NY 13902, USA · Ronin Institute for Independent Scholarship 2.0, Sacramento, CA 95816, USA +3

Sonification Chaotic Dynamical Systems Dynamical Systems

Feb 9, 2026

Is Flow Matching Just Trajectory Replay for Sequential Data?

Flow matching (FM) is increasingly used in scientific domains for time series generation and forecasting, where data often arise from underlying dynamical systems. However, it is not well-understood whether it learns transferable dynamical structure or simply performs an effective ''trajectory replay''. We study this question by deriving the optimal velocity field targeted by the empirical FM objective on sequential data. For the Gaussian conditional paths commonly used in practice, we show that the resulting sampler is an ODE whose dynamics describe a nonparametric, memory-augmented continuous-time dynamical system. The driving velocity field admits a closed-form expression as a weighted mixture of instantaneous velocities determined by observed transitions. This characterization provides an interpretable view of neural FM models as parametric estimation of an underlying nonparametric target and naturally leads to FreeFM: a closed-form, training-free realization of this target. FreeFM provides a transparent reference for studying sequential FM while yielding effective probabilistic forecasts directly from historical transitions.

Soon Hoe Lim, Shizheng Lin, Michael W. Mahoney +1

Department of Mathematics, KTH Royal Institute of Technology · Nordita, KTH Royal Institute of Technology and Stockholm University · Department of Statistics, University of California at Berkeley +2

Flow Matching Time Series Generation Probabilistic Time Series Forecasting Dynamical Systems

Jan 30, 2026

SplineFlow: Flow Matching for Dynamical Systems with B-Spline Interpolants

Flow matching is a scalable generative framework for characterizing continuous normalizing flows with wide-range applications. However, current state-of-the-art methods are not well-suited for modeling dynamical systems, as they construct conditional paths using linear interpolants that may not capture the underlying state evolution, especially when learning higher-order dynamics from irregular sampled observations. Constructing unified paths that satisfy multi-marginal constraints across observations is challenging, since naïve higher-order polynomials tend to be unstable and oscillatory. We introduce SplineFlow, a theoretically grounded flow matching algorithm that jointly models conditional paths across observations via B-spline interpolation. Specifically, SplineFlow exploits the smoothness and stability of B-spline bases to learn the complex underlying dynamics in a structured manner while ensuring the multi-marginal requirements are met. Comprehensive experiments across various deterministic and stochastic dynamical systems of varying complexity, as well as on cellular trajectory inference tasks, demonstrate the strong improvement of SplineFlow over existing baselines. Our code is available at: https://github.com/santanurathod/SplineFlow.

Santanu Subhash Rathod, Pietro Liò, Xiao Zhang

CISPA Helmholtz Center for Information Security, Saarbrücken, Germany · Department of Computer Science and Technology, University of Cambridge, UK

Flow Matching Conditional Flow Matching Dynamical Systems B-Spline Trajectory Representation

Jan 29, 2026

Learning Hamiltonian Flow Maps: Mean Flow Consistency for Large-Timestep Molecular Dynamics

Simulating the long-time evolution of Hamiltonian systems is limited by the small timesteps required for stable numerical integration. To overcome this constraint, we introduce a framework to learn Hamiltonian Flow Maps by predicting the mean phase-space evolution over a chosen time span, enabling stable large-timestep updates far beyond the stability limits of classical integrators. To this end, we impose a Mean Flow consistency condition for time-averaged Hamiltonian dynamics. Unlike prior approaches, this allows training on independent phase-space samples without access to future states, avoiding expensive trajectory generation. Validated across diverse Hamiltonian systems, our method in particular improves upon molecular dynamics simulations using machine-learned force fields (MLFF). Our models maintain comparable training and inference cost, but support significantly larger integration timesteps while trained directly on widely-available trajectory-free MLFF datasets.

Winfried Ripken, Michael Plainer, Gregor Lied +5

Technical University Berlin · BIFOLD · Free University of Berlin +6

Dynamical Systems Flow Map Learning Mean Flow Molecular Dynamics Simulation Physics-Based Simulation Hamiltonian Dynamics

Jan 27, 2026

Model-Free Output Feedback Stabilization via Policy Gradient Methods

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

Ankang Zhang, Ming Chi, Xiaoling Wang +1

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

Dynamical Systems Partially Observable RL Feedback Control Policy Gradient Methods Model-Free RL

Jan 22, 2026

Neural Particle Automata: Learning Self-Organizing Particle Dynamics

We introduce Neural Particle Automata (NPA), a Lagrangian generalization of Neural Cellular Automata (NCA) from static lattices to dynamic particle systems. Unlike classical Eulerian NCA where cells are pinned to pixels or voxels, NPA model each cell as a particle with a continuous position and internal state, both updated by a shared, learnable neural rule. This particle-based formulation yields clear individuation of cells, allows heterogeneous dynamics, and concentrates computation only on regions where activity is present. At the same time, particle systems pose challenges: neighborhoods are dynamic, and a naive implementation of local interactions scale quadratically with the number of particles. We address these challenges by replacing grid-based neighborhood perception with differentiable Smoothed Particle Hydrodynamics (SPH) operators backed by memory-efficient, CUDA-accelerated kernels, enabling scalable end-to-end training. Across tasks including morphogenesis, point-cloud classification, and particle-based texture synthesis, we show that NPA retain key NCA behaviors such as robustness and self-regeneration, while enabling new behaviors specific to particle systems. Together, these results position NPA as a compact neural model for learning self-organizing particle dynamics.

Hyunsoo Kim, Ehsan Pajouheshgar, Sabine Süsstrunk +2

KAIST, South Korea · EPFL, Switzerland

Neural Cellular Automata Dynamical Systems

Jan 10, 2026

The Replicator-Optimization Mechanism: A Scale-Relative Formalism for Persistence-Conditioned Dynamics with a Conditional Consent-Friction Instantiation

Persistence models often conflate propagation, survival, and cross-scale loss. The Replicator-Optimization Mechanism (ROM) is a replicator-mutator template separating baseline weight, bounded survival, and a transfer kernel at a declared scale. Its equation conserves mass but guarantees neither invariance, convergence, a potential, nor a preferred scale. For finite static density-independent continuous time, an irreducible weighted kernel yields a unique positive Perron-Frobenius composition; discrete-time power convergence needs primitivity. The componentwise ranking proved here is guaranteed under exact uniform-residual transfer. Strong lumpability gives universal first-order transfer closure, and blockwise effective fitness gives an exact quotient. An institutional instantiation uses normalized stakes, signed preference-decision alignment, information loss, and descriptive effective voice. It specifies conditional survival, not legitimacy or normative authority. A companion mixed-motive MARL battery reports exploratory evidence against the implemented proxy ratio in its environment: a positive signed target-coordinate-correlation effect survives held-out evaluation under shared-state contention, while a reduced feasible-centred frozen-policy crossing reverses the predicted correlation-noise interaction. The treatment varies ideal-point correlation inside a fixed reward family, not objective- or reward-function alignment. Lean checks mapped algebraic identities and scalar monotonicities, not the stationary theorem, empirical mapping, or normative bridge. ROM is an assumptions ledger and model-construction discipline, not a cross-substrate law.

Murad Farzulla

Dissensus, London, UK · King’s College London, London, UK

Dynamical Systems Replicator Dynamics Evolutionary Game Theory

Jan 7, 2026

Minimum distance classification for nonlinear dynamical systems

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

Dominique Martinez

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

Dynamical Systems Time Series Classification Classification Kernel Methods

Jan 1, 2026

Deep Neural Networks as Discrete Dynamical Systems: Implications for Physics-Informed Learning

We revisit the analogy between feed-forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation (PDE) forms. A comparative analysis between the numerical/exact solutions of the Burgers' and Eikonal equations, and the same obtained via PINNs is presented. We show that PINN learning provides a different computational pathway compared to standard numerical discretization in approximating essentially the same underlying dynamics of the system. Within this framework, DNNs can be interpreted as discrete dynamical systems whose layer-wise evolution approaches attractors, and multiple parameter configurations may yield comparable solutions, reflecting the degeneracy of the inverse mapping. In contrast to the structured operators associated with finite-difference (FD) procedures, PINNs learn dense parameter representations that are not directly associated with classical discretization stencils. This distributed representation generally involves a larger number of parameters, leading to reduced interpretability and increased computational cost. However, the additional flexibility of such representations may offer advantages in high-dimensional settings where classical grid-based methods become impractical.

Abhisek Ganguly, Santosh Ansumali, Sauro Succi

Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore, 560064, Karnataka, India · Italian Institute of Technology, Viale Regina Elena 291, 00161 Rome, Italy · University of Roma Tre, Italy +2

Dynamical Systems Neural PDE Solvers Physics-Informed ML Neural Network Training Dynamics

Dec 14, 2025

Exact Flow Linear Attention: Exact Solution from Continuous-Time Dynamics

In this paper, we introduce Exact Flow Linear Attention~(EFLA), an exact-flow formulation of delta-rule linear attention. We show that the delta-rule update can be interpreted as an explicit Euler discretization of an underlying continuous-time system. EFLA replaces this first-order update with the exact closed-form flow. By exploiting the rank-1 structure of the dynamics matrix, both the matrix exponential and the input integral collapse to a simple update that preserves delta-rule linear attention's algebraic structure, parameter count, linear-time complexity, and chunkwise parallelism. This attention mechanism removes the Euler discretization error of the delta-rule dynamics without introducing additional parameters. Experiments on robustness tests, language modeling benchmarks, and the MAD synthetic benchmark show that EFLA improves stability under corrupted and high-energy inputs, reduces perplexity, and achieves stronger downstream performance compared to SSM and Euler-style baselines. These results establish exact-flow integration as a principled and scalable update mechanism for delta-rule linear attention.

Jingdi Lei, Di Zhang, Soujanya Poria

Nanyang Technological University · Fudan University

Dynamical Systems Linear Attention Delta-Rule Linear Attention Attention Mechanisms

Dec 11, 2025

Tracking large chemical reaction networks and rare events by neural networks

Chemical reaction networks are widely used to model stochastic dynamics in biophysical processes. Their exact probabilistic description via the chemical master equation becomes intractable for high-dimensional systems due to the exponential growth of the state space. This difficulty is further exacerbated when metastable transitions and rare events dominate the dynamics. Here, we present an advanced neural-network approach that effectively tackles large reaction networks and rare-event dynamics within a unified framework. Building on autoregressive representations of the full joint probability distribution, we achieve efficient training through second-order optimization methods, including natural gradient descent and the time-dependent variational principle. We leverage enhanced-sampling strategies that enable accurate resolution of low-probability regions governing rare transitions. We demonstrate the method on a multistable genetic toggle switch and a mitogen-activated protein kinase (MAPK) cascade consisting of 16 species and 35 reactions with dense coupling. We further quantify rare events in spatially extended Schlögl models in both one- and two-dimensional lattices, reaching system sizes with extremely large state spaces up to

3\times10^{15}

. These results highlight the potential of neural modeling as a flexible and scalable framework for studying high-dimensional chemical reaction networks and rare events.

Jiayu Weng, Xinyi Zhu, Jing Liu +3

Institute of Data Science, University of Hong Kong, Hong Kong · Department of Systems Science, Faculty of Arts and Sciences, Beijing Normal University, Zhuhai 519087, China · School of Physics, University of Electronic Science and Technology of China, Chengdu 611731, China +6

Dynamical Systems Neural Surrogate Modeling Markov Models

Dec 5, 2025

Symmetric Linear Dynamical Systems are Learnable from Few Observations

We consider the problem of learning the parameters of a

N

-dimensional stochastic linear dynamics under both full and partial observations from a single trajectory of time

T

. We introduce and analyze a new estimator that achieves a small maximum element-wise error on the recovery of symmetric dynamic matrices using only

T=\mathcal{O}(\log N)

observations, irrespective of whether the matrix is sparse or dense. This estimator is based on the method of moments and does not rely on problem-specific regularization. This is especially important for applications such as structure discovery.

Minh Vu, Andrey Y. Lokhov, Marc Vuffray

Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Dynamical Systems Parameter Estimation Stochastic Linear Dynamical Systems System Identification

Nov 12, 2025

When is a System Discoverable from Data? Discovery Requires Chaos

The deep learning revolution has enabled major AI advances in physical sciences, particularly in discovering dynamical systems from observational data. Yet the reliability of learned surrogates and symbolically discovered models is often undermined by non-uniqueness: models may fit observed trajectories perfectly, but lack genuine predictive power. We show that chaos, despite its association with unpredictability, is often what makes governing equations discoverable. We prove that the classical Lorenz system is analytically discoverable and verify experimentally that for many systems out-of-domain generalization emerges precisely at chaos onset. Conservative systems, by contrast, are inherently non-discoverable from data alone. The prevalence of chaotic systems in benchmarks may have inadvertently obscured this limitation. These findings help explain why AI excels at weather forecasting, yet struggles in many engineering applications: the systems most amenable to discovery (chaotic ones) are often least amenable to control.

Zakhar Shumaylov, Peter Zaika, Philipp Scholl +3

University of Cambridge · LMU Munich and Munich Center for Machine Learning (MCML) · IBM Research

Dynamical Systems Nonlinear System Identification System Identification Chaotic Dynamical Systems

Oct 29, 2025

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

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

α

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

α

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

α

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

Jingzhou Liu

Shallow Neural Networks Dynamical Systems Neural Network Optimization Neural Network Activation Functions Neural Network Training Dynamics

Oct 18, 2025

Simulation-free Structure Learning for Stochastic Population Dynamics

Modeling dynamical systems and unraveling their underlying structural dependencies is central to many domains in the natural sciences. Various physical systems, such as those arising in cell biology, are inherently high-dimensional and stochastic in nature, and admit only partial, noisy state measurements. Our primary motivating setting is single-cell biology, where destructive measurements yield unpaired population snapshots rather than longitudinal trajectories of the same cells. This poses a significant challenge for addressing the problems of modeling the underlying dynamics and inferring the network structure of these systems. Existing methods are typically tailored either for structure learning or modeling dynamics at the population level, but are limited in their ability to address both problems together. In this work, we address both problems simultaneously: we present StructureFlow, a novel and principled simulation-free training approach for jointly learning the structure and stochastic population dynamics of physical systems. We showcase the utility of StructureFlow for the tasks of structure learning from interventions and dynamical (trajectory) inference of conditional population dynamics. We empirically evaluate our approach on high-dimensional synthetic systems, a set of biologically plausible simulated systems, and an experimental single-cell dataset. We show that StructureFlow can learn the structure of underlying systems while simultaneously modeling their conditional population dynamics --- a key step toward model-based mechanistic understanding of systems behavior.

Noah El Rimawi-Fine, Adam Stecklov, Lucas Nelson +4

McGill University · Mila – Quebec AI Institute · AITHYRA +5

Dynamical Systems Causal Structure Learning Conditional Generative Modeling

Oct 2, 2025

Neural non-canonical Hamiltonian dynamics for long-time simulations

This work focuses on learning non-canonical Hamiltonian dynamics from data, where long-term predictions require the preservation of structure both in the learned model and in numerical schemes. Previous research focused on either facet, respectively with a potential-based architecture and with degenerate variational integrators, but new issues arise when combining both. In experiments, the learnt model is sometimes numerically unstable due to the gauge dependency of the scheme, rendering long-time simulations impossible. In this paper, we identify this problem and propose two different training strategies to address it, either by directly learning the vector field or by learning a time-discrete dynamics through the scheme. Several numerical test cases assess the ability of the methods to learn complex physical dynamics, like the guiding center from gyrokinetic plasma physics.

Clémentine Courtès, Emmanuel Franck, Michael Kraus +2

IRMA, MACARON · Institut de Recherche Mathématique Avancée, UMR 7501 Université de Strasbourg et CNRS, 7 rue René Descartes 672000 Strasbourg, France. · INRIA Nancy-Grand Est, MACARON Project, Strasbourg, France. +5

Dynamical Systems Nonlinear System Identification Hamiltonian Neural Networks Physics-Informed ML Hamiltonian Dynamics

Sep 21, 2025

Data-efficient Kernel Methods for Learning Hamiltonian Systems

Hamiltonian dynamics describe a wide range of physical systems. As such, data-driven simulations of Hamiltonian systems are important for many scientific and engineering problems. In this work, we propose kernel-based methods for identifying and forecasting Hamiltonian systems directly from trajectory data. We present two approaches: a 2-step method that reconstructs trajectories before learning the Hamiltonian, and a 1-step method that jointly infers both. Across several benchmark systems, including mass-spring dynamics, a nonlinear pendulum, and the Henon-Heiles system, we demonstrate that our framework achieves accurate, data-efficient predictions and outperforms 2-step kernel-based baselines, particularly in scarce-data regimes, while preserving the Hamiltonian structure. Moreover, we prove a priori error estimates, ensuring reliability of the learned models. We also provide a more general, problem-agnostic numerical framework that goes beyond Hamiltonian systems and can be used for data-driven learning of arbitrary dynamical systems.

Yasamin Jalalian, Mostafa Samir, Boumediene Hamzi +2

Department of Computing and Mathematical Sciences, Caltech, CA, USA. · Beyond Limits, CA, USA. · The Alan Turing Institute, London, UK. +1

Dynamical Systems Nonlinear System Identification System Identification Kernel Methods Hamiltonian Dynamics

Aug 16, 2025

Universal Learning of Nonlinear Dynamics

We study the fundamental problem of one-step prediction of a marginally stable unknown nonlinear dynamical system. We describe an algorithm for this problem, based on the technique of spectral filtering, which learns a mapping from past observations to the next based on a spectral representation of the system. Using techniques from online convex optimization, we prove vanishing prediction error for any nonexpansive nonlinear dynamical system with finitely many marginally stable modes, with rates governed by a novel quantitative control-theoretic notion of learnability. The main technical component of our method is a new spectral filtering algorithm for linear dynamical systems, which incorporates past observations and applies to general noisy and marginally stable systems. This generalizes the original spectral filtering algorithm to both asymmetric dynamics as well as incorporating noise correction, and is of independent interest.

Evan Dogariu, Anand Brahmbhatt, Elad Hazan

Courant Institute of Mathematical Sciences, New York University · Princeton University

Dynamical Systems Nonlinear System Identification Spectral Methods

Jul 30, 2025

Diagrams-to-Dynamics (D2D): Exploring Causal Loop Diagram Leverage Points under Uncertainty

Background: Causal loop diagrams (CLDs) are widely used in health and environmental research to represent hypothesized causal structures underlying complex problems. However, as qualitative and static representations, CLDs are limited in their ability to support dynamic analysis and inform intervention strategies. We propose Diagrams-to-Dynamics (D2D), a method for converting CLDs into exploratory system dynamics models in the absence of empirical data. With minimal user input - following a protocol to label variables as stocks, flows or auxiliaries, and constants - D2D utilizes the structural information already encoded in CLDs, namely the existence and polarity of causal connections, to simulate hypothetical interventions and explore potentially influential places to intervene, known as 'leverage points,' under uncertainty. Results: D2D helps distinguish between high- and low-ranked leverage points. We compare D2D to a calibrated system dynamics model constructed from the same CLD and variable labels. D2D showed greater consistency with the calibrated model than did static network centrality analysis, while also providing uncertainty estimates and guidance for future data collection. Conclusions: The D2D method is implemented in an open-source Python package and a web-based application to support further testing and to lower the barrier to dynamic modeling for researchers working with CLDs. Future studies could help establish the approach's utility across a broad range of cases and domains.

Jeroen F. Uleman, Loes Crielaard, Leonie K. Elsenburg +4

Copenhagen Health Complexity Center, University of Copenhagen, Copenhagen, Denmark. · Department of Public and Occupational Health, Amsterdam UMC University of Amsterdam, Amsterdam, The Netherlands. · Center for Urban Mental Health, University of Amsterdam, Amsterdam, The Netherlands. +4

Dynamical Systems Causal Intervention

Jul 19, 2025

Artifacts of Numerical Integration in Learning Dynamical Systems

In many applications, one needs to learn a dynamical system from its solutions sampled at a finite number of time points. The learning problem is often formulated as an optimization problem over a chosen function class. However, in the optimization procedure, prediction data from generic dynamics requires a numerical integrator to assess the mismatch with the observed data. This paper reveals potentially serious effects of a chosen numerical scheme on the learning outcome. Specifically, the analysis demonstrates that a damped oscillatory system may be incorrectly identified as having "anti-damping" and exhibiting a reversed oscillation direction, even though it adequately fits the given data points. This paper shows that the stability region of the selected integrator will distort the nature of the learned dynamics. Crucially, reducing the step size or raising the order of an explicit integrator does not, in general, remedy this artifact, because higher-order explicit methods have stability regions that extend further into the right half complex plane. Furthermore, it is shown that the implicit midpoint method can preserve either conservative or dissipative properties from discrete data, offering a principled integrator choice even when the only prior knowledge is that the system is autonomous.

Bing-Ze Lu, Richard Tsai

National Chung Cheng University, Taiwan · The University of Texas at Austin, USA

Numerical Integration Dynamical Systems Nonlinear System Identification Ordinary Differential Equations

Jul 14, 2025

Dynamical stability for dense patterns in attractor neural networks

Recurrent neural networks are canonical models of biological memory. In these models, memories are represented by distributed patterns of neural activity that are stored in the recurrent connections between neurons, such that they become attractors of the network's dynamics. During memory recall, network dynamics thus converge toward one of these memory patterns when started from a noisy or partial cue. Therefore, memory performance critically hinges on the dynamical stability of the stored patterns. However, previous theoretical approaches only studied dynamical stability under highly restrictive conditions that do not readily apply to biological neural circuits. Here, we develop a theory of the local stability of discrete fixed points in a broad class of networks with graded neural activities and in the presence of noise. Using methods from random matrix theory, we analyze the bulk and outliers of the eigenvalue spectra of the Jacobians that characterize network dynamics around fixed points. We show that either all fixed points are stable or all of them are unstable, depending on whether their number is below a ``critical load for stability'', which is distinct from the classical critical capacity that measures the maximal number of achievable fixed points regardless of their stability. We further analyze the dependence of this critical load for stability on experimentally measurable quantities characterizing the statistics of memory patterns and the activation functions of neurons. Our analysis highlights the computational benefits of sparse-like patterns and threshold-linear activation functions and offers testable predictions for neural circuits supporting memory.

Uri Cohen, Máté Lengyel

Computational and Biological Learning Lab, Department of Engineering, University of Cambridge, Cambridge, UK · Center for Cognitive Computation, Department of Cognitive Science, Central European University, Budapest, Hungary

Neural Network Memorization Associative Memory Neural Processes Dynamical Systems Hopfield Networks Recurrent Neural Networks+1

Jun 25, 2025

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

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

Xinghao Dong, Huchen Yang, Jin-Long Wu

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

Dynamical Systems Latent Diffusion Models Conditional Diffusion Models Diffusion Model Inference Acceleration

May 16, 2025

Regularity and Stability Properties of Selective SSMs with Discontinuous Gating

Selective State-Space Models (SSMs) such as Mamba have become central to long-sequence modeling. Still, their stability is poorly understood: their state-space coefficients are modulated online by a token-dependent gating signal, making the recurrence neither linear time-invariant nor classically nonlinear. We study continuous-time selective SSMs through passivity, dissipativity, and Input-to-State Stability (ISS), explicitly separating the selection signal

x(\cdot)

from the driving input

u(\cdot)

. We obtain four results: exponential forgetting under strict dissipativity; a canonical

\mathrm{AUC}_{\mathrm{loc}}

quadratic storage for the frozen-selection subsystem that accommodates discontinuous gating; a parametric LMI together with universal kernel constraints and "irreversible forgetting" under universal quadratic storage; and sufficient conditions for global ISS uniformly over admissible selection schedules. We then bridge to practice by deriving a sampled block LMI for the Mamba selective-scan core, which is used as a differentiable training-time regularizer. Across seven standard time-series datasets and four prediction horizons, the regularizer reduces sampled Mamba-core LMI violations by roughly

92\%

in

28/28

pairs at a clean-MSE cost of less than

0.018\%

. It improves internal Mamba passivity and state-norm diagnostics under injected perturbations. Our results turn classical control-theoretic tools into verifiable structural and training criteria for selective SSMs, while honestly scoping which guarantees transfer to a deep selective-scan architecture.

Nikola Zubić, Davide Scaramuzza

Robotics and Perception Group University of Zurich

Dynamical Systems Selective SSMs Input-to-State Stability

Apr 11, 2025

A Hybrid ABM-PDE Framework for Real-World Infectious Disease Simulations

This paper presents a hybrid modeling approach that couples an Agent-Based Model (ABM) with a partial differential equation (PDE) model in an epidemic setting to simulate the spatial spread of infectious diseases using a compartmental structure with seven health states. The goal is to reduce the computational complexity of a full-ABM by introducing a coupled ABM-PDE model that offers significantly faster simulations while maintaining comparable accuracy. Our results demonstrate that the hybrid model not only reduces the overall simulation runtime (defined as the number of runs required for stable results multiplied by the duration of a single run) but also achieves smaller errors across both 25% and 100% population samples. The coupling mechanism ensures consistency at the model interface: agents crossing from the ABM into the PDE domain are removed and represented as density contributions, while surplus density in the PDE domain is used to generate agents with plausible trajectories derived from mobile phone data. We evaluate the hybrid model using real-world mobility and infection data for the Berlin-Brandenburg region in Germany, showing that it captures the core epidemiological dynamics while enabling efficient large-scale simulations. These results demonstrate that the proposed ABM-PDE framework provides a robust and computationally efficient alternative to full-scale agent-based simulations, making it suitable for realistic epidemic modeling and scenario analysis.

Kristina Kehrer, Tim O. F. Conrad

Zuse Institute Berlin Berlin, 14195

Dynamical Systems Agent-Based Modeling Spatial Epidemic Modeling

Feb 20, 2025

No Undesirable Equilibria, No Collisions: Combining Modulation and Control Barrier Functions for Liveness and Safety in Dynamic Collision Avoidance

Control Barrier Function Quadratic Programs (CBF-QPs) provide formal safety guarantees for general control-affine systems and accommodate actuation constraints, but do not inherently guarantee liveness, i.e., sustained progress toward and convergence to a goal. They may admit undesirable equilibria or dysfunctional circulation that impede goal convergence. In contrast, Modulation of Dynamical Systems (Mod-DS) promotes liveness through geometric reshaping of nominal vector fields, but is primarily suited to fully actuated systems and does not directly accommodate general actuation constraints. We establish theoretical connections between these paradigms, showing that normal Mod-DS is a special case of CBF-QP and that reference Mod-DS is linked to CBF-QP through a shared formulation. These connections motivate the Modulated CBF-QP (MCBF-QP) framework, which incorporates geometric guidance from Mod-DS into CBF-QPs to enhance liveness while retaining formal safety and control-affine generality. We develop reference MCBF-QP, which reduces undesirable equilibria and dysfunctional circulation for star-shaped obstacles, and on-manifold MCBF-QP, which eliminates undesirable equilibria and reduces dysfunctional circulation for arbitrarily shaped concave obstacles. Although no Lyapunov function is explicitly constructed, we prove that the tangent guidance constraint introduced into the CBF-QP formulation implicitly enforces the decrease condition of a hybrid Lyapunov function, yielding asymptotic goal convergence and thus liveness under the stated assumptions. Extensive simulation and hardware experiments demonstrate safe and live obstacle avoidance with the proposed MCBF-QP framework in static and dynamic environments.

Yifan Xue, Nadia Figueroa

the Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104 USA

Control Barrier Functions Safety-Critical Control Robot Collision Avoidance Dynamic Obstacle Avoidance Dynamical Systems

Feb 1, 2025

Active Learning with Bayesian Multi-Fidelity Laplace Neural Operators for Oscillatory Parametric PDEs

Surrogate models of parametric dynamical systems are essential for many-query and real-time predictions in engineering applications such as design optimization and digital twins. However, generating high-fidelity (HF) training data over a broad range of parameters and operating conditions remains computationally expensive. To address this challenge, we propose a Bayesian multi-fidelity Laplace neural operator (MF-LNO) for uncertainty-aware active learning of oscillatory parametric PDEs. Specifically, the proposed Bayesian MF-LNO iteratively calibrates the discrepancy between low- and high-fidelity data, where predictive uncertainty guides the adaptive acquisition of informative HF trajectories. Such predictive uncertainty is quantified via replica-exchange stochastic gradient Langevin dynamics (reSGLD), whose broad posterior exploration enables uncertainty to serve as an error indicator for adaptive HF sample acquisition. Numerical experiments on the Lorenz system, Duffing oscillator, and beam dynamics demonstrate that uncertainty-guided HF sample acquisition consistently outperforms random sampling, while the proposed Bayesian MF-LNO achieves higher prediction accuracy than MF-DeepONet with predictive uncertainty quantification. These results demonstrate that Bayesian multi-fidelity LNOs, combined with uncertainty-guided active learning, provide a data-efficient framework for operator learning in engineering dynamical systems.

Bongseok Kim, Haoyang Zheng, Michael Penwarden +1

School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, 47907, IN, USA. · Department of Mathematics, Purdue University, 150 North University Street, West Lafayette, 47907, IN, USA.

Dynamical Systems Surrogate Modeling Uncertainty Quantification Active Learning PDE Operator Learning Neural Operators

Oct 31, 2024

Projected Neural Differential Equations for Learning Constrained Dynamics

Neural differential equations offer a powerful approach for learning dynamical systems from data. However, they do not inherently respect known constraints, such as conservation laws, that should be obeyed by the learned dynamics. It is well known that enforcing constraints in data-driven models can enhance their generalizability and numerical stability. In this paper, we introduce projected neural differential equations (PNDEs), a method for constraining neural differential equations based on projection of the predicted velocities onto the tangent space of the manifold that fulfills the constraint. In tests on several examples from different fields, including chaotic dynamical systems and power grid models, PNDEs outperform existing methods for constraining learned dynamics, require fewer hyperparameters, and are computationally more efficient. The proposed approach demonstrates potential for enhancing the modeling of constrained dynamical systems, particularly in domains where accuracy and reliability are essential.

Alistair White, Anna Büttner, Maximilian Gelbrecht +4

Munich Climate Center and Earth System Modelling Group, Department of Aerospace and Geodesy, School of Engineering and Design, Technical University of Munich, Munich, Germany · Potsdam Institute for Climate Impact Research, Potsdam, Germany · California Institute of Technology, Pasadena, California, USA +4

Dynamical Systems Neural ODEs

Oct 14, 2024

Automated Discovery of Operable Dynamics from Videos

Dynamical systems form the foundation of scientific discovery, traditionally modeled with predefined state variables such as the angle and angular velocity, and differential equations such as the equation of motion for a single pendulum. We introduce a framework that automatically discovers a low-dimensional and operable representation of system dynamics, including a set of compact state variables that preserve the smoothness of the system dynamics and a differentiable vector field, directly from video without requiring prior domain-specific knowledge. The prominence and effectiveness of the proposed approach are demonstrated through both quantitative and qualitative analyses of a range of dynamical systems, including the identification of stable equilibria, the prediction of natural frequencies, and the detection of chaotic and limit cycle behaviors. The results highlight the potential of our data-driven approach to advance automated scientific discovery.

Kuang Huang, Dong Heon Cho, Boyuan Chen

Department of Applied Physics and Applied Mathematics, Columbia University, 500 W. 120th Street, New York, 10027, NY, USA. · Department of Computer Science, Duke University, 308 Research Dr., Durham, 27705, NC, USA. · Department of Mechanical Engineering and Material Science, Duke University, 121 Hudson Hall, Durham, 27708, NC, USA. +1

Dynamical Systems Latent Dynamics Modeling System Identification Autonomous Scientific Discovery

Jun 7, 2024

Gradient Descent on Logistic Regression with Non-Separable Data and Large Step Sizes

We study gradient descent (GD) dynamics on logistic regression problems with large, constant step sizes. For linearly-separable data, it is known that GD converges to the minimizer with arbitrarily large step sizes, a property which no longer holds when the problem is not separable. In fact, the behaviour can be much more complex -- a sequence of period-doubling bifurcations begins at the critical step size

2/λ

, where

λ

is the largest eigenvalue of the Hessian at the solution. Using a smaller-than-critical step size guarantees convergence if initialized nearby the solution: but does this suffice globally? In one dimension, we show that a step size less than

1/λ

suffices for global convergence. However, for all step sizes between

1/λ

and the critical step size

2/λ

, one can construct a dataset such that GD converges to a stable cycle. In higher dimensions, this is actually possible even for step sizes less than

1/λ

. Our results show that although local convergence is guaranteed for all step sizes less than the critical step size, global convergence is not, and GD may instead converge to a cycle depending on the initialization.

Si Yi Meng, Antonio Orvieto, Daniel Yiming Cao +1

Department of Computer Science Cornell University Ithaca, NY, USA · ELLIS Institute Tübingen MPI for Intelligent Systems Tübingen AI Center, Germany

Dynamical Systems Logistic Regression Optimization Convergence Analysis Gradient Descent Gradient Descent Dynamics