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44 papers in the last four weeks, up 159% on the four weeks before. 0.4% of all new papers.

Jul 13Week of Sep 28

Latest papers 319

Jul 2, 2026cs.LG

A Memory Efficient Unified Algorithm for Online Learning of Linear Dynamical Systems

Motivated by the challenge of stabilizing a general unknown linear dynamical system (LDS) from observations, we study the natural prerequisite of online prediction. Our goal is to achieve sublinear regret with a memory footprint that adapts to the intrinsic complexity of the dynamics rather than the full hidden-state dimension. We focus on the practically central regime of systems with low instability complexity -- eigenvalues outside the real stable interval that do not decay rapidly, together with non-semisimple modes -- potentially embedded in an otherwise stable real spectrum of much higher dimension; we write

k

for this count. This regime is the primary setting in which stabilization is plausible: we show that many systems with high instability complexity cannot be stabilized without exponentially large controls. Thus, prediction is meaningful for stabilization precisely when the instability complexity is small. Within this regime, we introduce a unified online algorithm that handles every LDS (including non-diagonalizable systems with complex or exploding modes) with a learnable parameter count of

\widetilde{O}(k)

. Finally, we prove a lower bound showing that

k

is a valid complexity measure: any filter-based predictor needs at least

k

filters. Experiments corroborate our theory: on a high-dimensional system, our predictor sharply outperforms prior methods at an equal parameter budget.

Yuval Ran-Milo, Angelos Assos, Elad Hazan

Tel Aviv University · Princeton University

Dynamical Systems Online System Identification State Space Models

Jul 1, 2026stat.ML

From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operators

We establish approximation and learning guarantees for Fourier neural operators (FNOs) applied to time-

T

solution operators of dissipative evolution equations. The analysis builds on the premise that FNOs can efficiently approximate and learn solution operators whenever these operators admit stable and accurate spectral discretizations. To formalize this idea, we introduce classes of evolution operators defined through spectral methods and derive FNO approximation bounds and polynomial sample complexity guarantees for these classes. For equations with polynomial nonlinearities, the learning rates depend primarily on the smoothness of the input space and the dimension of the physical domain. Our results hold uniformly over broad families of dissipative equations, rather than for a single fixed PDE, and apply in particular to the Navier--Stokes, Allen--Cahn, and Cahn--Hilliard equations. For equations with non-polynomial smooth nonlinearities, we prove that polynomial sample complexity still holds with rates that now additionally depend on the smoothness of the nonlinear terms and the dissipation strength. Overall, we connect classical spectral approximation theory with modern operator learning and explain when FNOs can learn nonlinear evolution operators efficiently.

Nisha Chandramoorthy, Daniel Sanz-Alonso, Nathan Waniorek

University of Chicago

Dynamical Systems Fourier Neural Operator PDE Operator Learning Neural Network Approximation Theory Neural Operators

Jun 30, 2026cs.CG

Guaranteed Escape for a Bouncing Robot in Pipe Chains

We study the symmetric bouncing of a point robot within orthogonally-joined rectangles with equal width, which we refer to as pipes. We provide an exhaustive case analysis of every trajectory pattern inside a single rectangular pipe segment, identifying the conditions under which the robot exits. We then extend the analysis to L-shaped pipes and, more generally, to linear chains of

k

orthogonally connected pipe segments. We prove exit guarantees for the special angle

α= π/4

. Furthermore, these results extend to pipes with curved joints.

Ahmad Kamaludeen, Somnath Kundu, Yeganeh Bahoo

Toronto Metropolitan University, Toronto, ON M5B 2K3, Canada

Dynamical Systems Robot Motion Planning

Jun 29, 2026cs.RO

Data-Driven Modeling and Control for Tethered Space Systems with Koopman-Informed Graphs

Modeling tethered space systems is critical for advanced orbital operations. Flexible components such as tethers and space nets are integral to these systems but present significant control challenges due to their high dimensional, strongly coupled, and nonlinear dynamics. While data driven methods offer alternative modeling approaches, they frequently struggle with long term predictive stability and spatial generalization. To address this, we propose the Koopman Graph Dynamics (KGD) framework to learn the structural dynamics by integrating the global linear evolution of the Koopman operator with the local topological priors of Graph Neural Networks. Building upon this representation, we develop a KGD based Model Predictive Control strategy for tethered space systems. Subsequently, the ground experiments on flexible tether and space net demonstrate the high precision modeling capabilities of the proposed method. Crucially, the framework exhibits exceptional capacity for spatial transfer without retraining. Models trained exclusively on small configurations successfully predict and control significantly larger, unseen physical scales. Furthermore, the orbit simulations within a physics engine verify the effectiveness of the proposed approach for tethered space systems.

Ao Jin, Yifeng Ma, Panfeng Huang +1

Research Center for Intelligent Robotics, School of Astronautics, Northwestern Polytechnical University, Xi’an 710072, China · Department of Mechanical and Automation Engineering, The Chinese University Hong Kong, 999077, China

Dynamical Systems Koopman Operator Learning Nonlinear System Identification Model Predictive Control

Jun 27, 2026math.CT

Compositional Dynamics in Learning and Mechanics

We give a single compositional setting in which gradient-based learning and Hamiltonian-style mechanics appear as functorial semantics. The syntax is an operad Arr whose objects are input-output interfaces (pairs of manifolds) and whose morphisms are smooth adaptive arrangements, which consist of a reactive parameter space, a lens given by smooth output and input maps, and a real-valued potential. The main technical result of the paper is what we call lens internalization, a lax symmetric monoidal functor Lens(C)

\to

C associated to any symmetric monoidal closed category C. Using it, we provide two functors

Φ_\text{phase}

,

Φ_\text{conf}

: Arr

\to

PC into the 2-category of polynomial coalgebras -- input-output discrete dynamical systems -- which we take as the semantics category.

Φ_\text{phase}

stores both position and momentum, whereas

Φ_\text{conf}

stores only position. When applied to a parameterized function,

Φ_\text{conf}

recovers the gradient descent training algorithm, with backpropagation as the lens' backward pass. When applied to harmonic particles wired together -- in series, or according to any finite directed graph -- one diagram yields two different regimes, both of which are governed by the graph Laplacian:

Φ_\text{phase}

gives the discrete wave equation, which is conservative and second-order, and

Φ_\text{conf}

gives the discrete heat equation, which is dissipative and first-order. They are two semantics of one adaptive arrangement, e.g. with the same potential in each case. And because Arr is an operad, such diagrams nest -- larger systems wired from smaller ones -- and each semantics assembles a system's dynamics functorially from its parts. These dynamics are moreover executable: a parameterized neural network and a graph of particles both compile, by the same construction, to explicit state machines one can run.

David I. Spivak

Dynamical Systems Hamiltonian Dynamics

Jun 26, 2026q-bio.NC

Heterogeneous synaptic motifs bridge microscale structure and macroscale nonlinear dynamics

Recent breakthroughs in synaptic-resolution network connectomics have revealed that brain circuits feature fine-scale structural connectivity, such as pairs of correlated synaptic couplings known as second-order motifs. Large-scale recordings of neuronal activity in networks containing nonlinear neurons reveal macroscopic heterogeneous population dynamics throughout the brain. These findings rekindle the inquiry into this intriguing question: Can microscale synaptic structures contribute to macroscopic heterogeneous dynamics and computations in ways that canonical brain circuit models cannot? To answer this question, we create random RNNs with various cell types, nonlinear non-negative neural responses, and arbitrary marginal and second-order correlated synaptic statistics. We derive mean-field low-rank equations for P-population networks in which the pre- and postsynaptic neuronal population identities determine the synaptic and motif strengths. Our framework requires 2P latent dynamic variables with P variables describing mean population activity and P variables capturing within-population variability. Theoretical and simulational results demonstrate that chain motifs induce correlations in synaptic variability, enabling microscopic fluctuations to be integrated and influence mesoscopic mean population dynamics. We apply this framework to reverse engineer network connectivity that recapitulates the heterogeneous activity across the population in the mouse primary visual cortex. By bridging the gap between synaptic organization and nonlinear heterogeneous population dynamics, our results offer a principled approach and testable predictions regarding the relationship between fine-scale connectivity, heterogeneous dynamics, and functional computations.

Meiyi Zhang, Jinjian Yu, Louis Tao +1

Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, China · Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing, China · School of Physics, Peking University, China +7

Dynamical Systems Recurrent Neural Networks Visual Neuroscience

Jun 25, 2026cs.LG

Symplectic Neural Networks for learning Generalized Hamiltonians

Hamiltonian Neural Networks (HNNs) integrate physical priors into neural models by learning a system's Hamiltonian, improving generalization and sample efficiency. Identifying the system Hamiltonian from noisy observations of state variables is a challenging task. For simulations to faithfully reflect the long-term behavior of Hamiltonian systems, especially energy conservation, it is essential to use symplectic integrators, which preserve the system's geometric structure. This fidelity comes at a cost: implicit symplectic integrators are more computationally intensive and make backpropagation through the ODE solver non-trivial. However, by leveraging the fact that symplectic discretizations of the adjoint system yield the same sensitivities associated by backpropagation, we obtain an efficient method of training the Neural Network parameters. In our work, we explore this alternate method of HNN training under noisy observation of trajectories with our HNN model based on an implicit symplectic integrator. Computationally, a predictor-corrector based ODE solver and fixed point iteration help to mitigate the computational cost of the implicit timestepping, resulting in more efficient generation of gradient updates. We showcase the numerical advantage, in experiments, in system identification and energy preservation on a range of non-separable, chaotic systems and the efficient computation and memory complexity of our method. We also observe that the post-processing of the learned Hamiltonian using backward error analysis yields a modified Hamiltonian that is a more accurate approximation of the true Hamiltonian without the need to use more accurate discretizations of the flow map.

Harsh Choudhary, Vyacheslav Kungurtsev, Chandan Gupta +2

Computer Science Department, Indraprasth Institute of Information Technology, New Delhi, India 110020 · Athena Research and Innovation Center, Marousi, Greece 15125 · HSBC Holdings, Singapore

Dynamical Systems Hamiltonian Neural Networks System Identification Neural ODEs Hamiltonian Dynamics

Jun 24, 2026cs.AI

Feedback-Coupled Memory Systems in Continuous Time

The Feedback-Coupled Memory Systems (FCMS) architecture formalizes closed-loop coordination through four abstract operators, two of which - the agent update operator

f_i

and the environmental update operator

Ψ

- are left axiomatically undefined in the original framework. To address this,

f_i

is defined by Mechanism-Based Intelligence (MBI), where agents update locally through a decentralized price mechanism and economic principles, and

Ψ

is defined by the Coupled Memory Graph Process (CMGP), a non-Markovian framework where the environment is treated as a physical substrate that records and responds to trajectory history coherently without external forcing. The resulting continuous-time FCMS instantiation achieves Lyapunov global dissipativity governed by the computable threshold

4β^2 < 2ημγ^2

. This generalizes both the discrete FCMS stability condition

4ηβ^2 < γ

and CMGP's physical bifurcation threshold

α_c = 1/K

, confirming that memory dissipation must outpace feedback gain as a universal organizing principle. Numerical simulation with

N=2

agents and mean-field validation at

N=10^6

confirm the stability threshold and the self-reinforcing coordination cascade that emerges when it is violated.

Stefano Grassi

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

Dynamical Systems Multi-Agent Coordination Lyapunov Stability Feedback Control

Jun 24, 2026cs.LG

Implementation of reinforcement learning in chemical reaction networks: application to phototaxis as curiosity-driven exploration

Living systems navigate environments using noisy and incomplete sensory signals. In unicellular algae, phototaxis is often modeled as a mechanistic run--tumble process driven by stimulus--response rules. However, such descriptions overlook how organisms actively sample their environment to reduce sensory ambiguity. From a minimal cognition perspective, we reframe this navigation as a subjective, information-driven sensorimotor process. To this end, we propose a framework linking a Partially Observable Markov Decision Process (POMDP) with biochemical reaction dynamics. Environmental variables are hidden, while the cell updates a minimal internal state from each observation through a memoryless Bayesian step. These internal dynamics balance orienting toward light with exploratory reorientation and can be implemented through Chemical-Reaction-Network Ordinary Differential Equations (CRN--ODEs). Our model includes a biophysical observation process for photoreception and a chemically computable polynomial bound on information gain. Using Inverse Reinforcement Learning (IRL) on 30 experimentally recorded Chlamydomonas trajectories, we infer the behavioral objective consistent with observed phototactic motion and benchmark the resulting dynamics with standard Stochastic Simulation Algorithm (SSA) baselines. Our model reproduces the empirical alignment-to-light distribution, comparable to objective SSA baselines on this dataset. Within this framework, run--tumble alternation emerges as an information-acquisition strategy: tumbling reorients the cell to sample new sensory configurations and resolve sensor ambiguity, demonstrating how intracellular biochemical networks can support adaptive information-seeking behavior in cellular navigation.

Ruyi Tang, Grégoire Sergeant-Perthuis, David Colliaux

LCQB-AG · 1CQSB, Sorbonne University, France · 2Sony CSL, France

Dynamical Systems Reinforcement Learning RL Exploration

Jun 23, 2026math.NA

Dirac-Frenkel dynamics with inertia for nonlinearly parametrized solutions of evolution problems

Even when Dirac-Frenkel dynamics determine a well-defined evolution in function space, the corresponding parameter dynamics can be non-unique or ill-conditioned for redundant nonlinear parametrizations such as neural networks or mixture models. We propose to add inertia to the Dirac-Frenkel dynamics and show that this allows useful parameter velocity information to persist from the past trajectory in directions that are weakly informed, while well-informed parameter velocity directions continue to follow the Dirac-Frenkel dynamics. We prove that the inertial formulation yields well-posed parameter dynamics and provide a posteriori error bounds. After time discretization, the method requires the solution of the same type of regularized linear least-squares problem as standard Dirac-Frenkel dynamics, but with the previous velocity appearing as an anchor. Numerical experiments demonstrate the increased robustness obtained with inertia.

Matteo Raviola, Benjamin Peherstorfer

Scientific Computing and Uncertainty Quantification - CADMOS Chair, EPFL · Courant Institute of Mathematical Sciences, New York University

Dynamical Systems Neural PDE Solvers

Jun 23, 2026cs.LG

Swarm-Inspired Generation of Collective Behaviors in Graph Dynamical Systems

Collective behavior arises when locally interacting units produce coordinated global organization, from synchronization in dynamical systems to task-relevant information flow on graphs. The central challenge is not only to explain how collective behavior emerges, but to design local interaction rules that can produce desired global organization and generalize across graphs, dynamics and tasks.To address this challenge, we introduce the Swarm-Inspired Emergent Synchronizer (SIES), a graph-dynamical framework that learns generalizable local-interaction laws for controllable collective organization. Each node is an agent-like dynamical unit with a state and task cue, and signed source-target-conditioned attention acts as an adaptive coupling term inside an explicit evolution model. Therefore, SIES combines an explicit dynamical engine with local agent intelligence, similar to biological swarms. For synchronization control, SIES learns a generalizable coupling operator that produces prescribed synchronization patterns for CDSs across untrained network scales, target phase relations, and intrinsic node dynamics without retraining. The learned operator also reaches gait-related modes faster than three oscillator baselines and generalizes synchronization-driven locomotion to simulated multi-legged robots of different scales and a physical hexapod after leg disablement. For graph representation learning, SIES applies the same signed interaction principle to message passing and achieves the highest performance among the compared methods on heterophilous node-classification benchmarks. Together, these results position SIES as a generalizable and learnable graph-dynamical interaction framework with promise for synchronization control, adaptive robot coordination, and heterophilous graph representation learning.

Ji Chen, Song Chen, Chengzhang Gong +2

College of Control Science and Engineering, Zhejiang University, Hangzhou, 310000, Zhejiang, China. · Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, 119076, Singapore. · Huzhou Institute of Zhejiang University, Huzhou, 313000, Zhejiang, China. +3

Dynamical Systems Graph Representation Learning Multi-Robot Coordination

Jun 23, 2026cs.LG

When Do Conservation Laws Survive Learned Representations? Certified Horizons for Latent World Models

We ask a representation-learning question about physical world models: when does a conservation law remain certifiable after a model learns a latent representation? A certified horizon bounds -- in advance, from measurable model defects -- how many steps a rollout provably stays on a physical invariant's level set. The key design choice is what is certified: not a learned latent Hamiltonian or a learned scalar witness (a model can conserve either while drifting in true energy), but the decoded physical invariant obtained by decoding the latent state and evaluating the known invariant. Around this object we derive shell-horizon certificates whose budget decomposes into representation, readout, and latent-dynamics defects, with a monotone alignment bridge through which a soft learned witness yields a certified horizon for the decoded invariant, and test them across state, learned-lift, and pixel observations on conservative systems. Conservation certificates can survive learned representation, but not all geometric priors survive equally. Hard canonical symplectic structure yields the longest horizons in known phase coordinates yet does not cross a learned chart, whereas a controlled-Lipschitz-aligned soft invariant survives in the nonlinear learned-representation settings we test -- two lift systems, with the gain growing with nonlinearity, and pixels. Pixel certification is recovered on a readout-stable sub-tube, and the Kepler problem exposes a geometric boundary. The central object is therefore not a latent Hamiltonian, but a decoded physical invariant whose robustness to representation learning can be measured, certified, and falsified.

Hongbo Wang

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA

Dynamical Systems Latent World Models Certified Robustness Hamiltonian Dynamics

Jun 22, 2026cs.LG

Topological Out-of-Domain Generalization in Dynamical Systems Reconstruction

Predicting the behavior of dynamical systems (DS) beyond the dynamical and parameter regimes observed in training is a pivotal and essentially unresolved problem in scientific ML. It is central to any good scientific theory, which we expect to be able to make predictions about regimes not covered by currently available data. Recent hierarchical and hyper-network guided approaches for DS reconstruction (DSR) enable training on many DS simultaneously, and revealed that extracted latent features are often related to crucial control parameters of the underlying DS that varied across the training corpus. However, true out-of-domain forecasting abilities of these models, e.g., across tipping points, remain limited, and fine-tuning, or even full model retraining, on time series from the new dynamical regime is usually required. Here, we mathematically analyze the root of these limitations in previous model formulations and identify three core shortcomings rooted in a mismatch between structural assumptions of the reconstruction model and typical properties of physical systems. We propose a combination of remedies for these shortcomings, most importantly feature splitting, and furthermore derive a closed-form bound on the reliable extrapolation range. We demonstrate empirically that our techniques allow for accurate zero-shot prediction into new dynamical regimes, outside the observed training regime, as, e.g., encountered across tipping points.

Georg Trede, Charlotte Ricarda Doll, Elias Weber +1

Dept. of Theoretical Neuroscience, Central Institute of Mental Health, Mannheim, Germany · Faculty of Physics and Astronomy, Heidelberg University, Germany · Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Germany

Dynamical Systems Zero-Shot Time Series Forecasting OOD Generalization Nonlinear System Identification

Jun 20, 2026cs.AI

Attractor Domain Theory: A Mathematical Framework for Cardiovascular Attractor Analysis with Wearable Photoplethysmography (PPG) Validation

The cardiovascular system evolves along a bounded trajectory in physiological state space that converges to a compact geometric object: the cardiac attractor. A wearable photoplethysmograph (PPG) or electrocardiograph (ECG) observes a one-dimensional projection of this attractor; by Takens' embedding theorem, delay coordinates reconstruct its full geometry. Three decades of nonlinear cardiac dynamics have extracted Lyapunov exponents, recurrence statistics, and sample entropy from reconstructed attractors, yet no principled account exists of which attractor properties capture which cardiovascular quantities, or why, leaving feature selection as a search problem and negative results uninterpretable. We introduce Attractor Domain Theory (ADT), which proves that the reconstructed attractor's information partitions into three mutually non-redundant domains: the Geometry Domain G (delay embedding; native capability: artifact rejection), the Ergodic Domain S (asymptotic statistical invariants; native capability: stability estimation), and the Variational Domain V (finite-time Lyapunov exponent field; native capability: hemodynamic inference). We prove a Domain Sufficiency Theorem (the Parseval analog for attractor information) and establish that three domains are necessary and sufficient. Geometry Domain validation via the SCSI framework across 176,742 PPG segments from four datasets yields AUC = 0.757 [0.686-0.828] and NPV = 0.966 after correcting three systematic evaluation artifacts (+0.179 net inflation). Ablation confirms C_NL as the dominant Geometry Domain component (Delta AUC = -0.413) and intra-domain redundancy across five components.

Timothy Oladunni, Farouk Ganiyu Adewumi

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

Dynamical Systems Photoplethysmography

Jun 19, 2026cs.NE

Soliton-like Waves in a Two-Dimensional Recurrent Spiking Neural Network with Weighted Spike-Timing-Dependent Plasticity

We construct a minimal but biologically plausible spiking neuron model operating in discrete time, combining multiplicative spike-timing-dependent plasticity (WSTDP), divisive normalization of synaptic integration, homeostatic threshold adaptation, and a one-step refractory period. We show that this normalization admits a biologically plausible dendritic implementation in which each binary junction operates using only locally available information. Assembling excitatory-inhibitory pairs of such neurons into a two-dimensional recurrent network and applying periodic localized stimulation, we find that the network spontaneously gives rise to stable, self-propagating wave packets with the properties of dissipative solitons: they maintain a stable spatial profile, propagate at constant speed, and annihilate upon frontal collision. Their emergence requires a geometric asymmetry between excitatory and inhibitory connection radii, and initial inhibitory synapses stronger than excitatory ones. WSTDP engraves the direction of propagation into the synaptic weight profile, so that the network learns by itself to sustain propagation in one direction while suppressing the reverse. When two sources are active simultaneously, the resulting waves annihilate upon collision, defining a semi-persistent boundary whose position encodes the relative phase and frequency of the two sources. These results provide a minimal computational framework for studying the emergence of cortical traveling waves, activity zone delimitation, and spatial memory from local plasticity rules alone.

Ch. Meessen

Independent researcher, computational neuroscience, France

Dynamical Systems Recurrent Neural Networks Spiking Neural Networks Spike-Timing-Dependent Plasticity

Jun 19, 2026cs.LG

Topological Neural Dynamics: A Neuron-wise Framework for Sequence Modeling

Existing sequence models, including RNNs, LSTMs, continuous-time networks, and Transformers, share a common structural principle: layer-wise dynamics, where all neurons in the same layer co-evolve through a shared parameterized operator, leaving individual neurons no freedom to evolve independently. Yet in many complex dynamical systems, rich global behavior emerges precisely from locally evolving units interacting through structured connectivity. Inspired by this principle, we introduce Topological Neural Dynamics (TND), a sequence modeling framework that shifts computation from layer-wise to neuron-wise dynamics. TND represents a neural system as a directed neuron graph, an interaction operator, and a local dynamics function, where each neuron evolves independently and collective computation emerges from interactions through the explicit graph topology. We instantiate TND as a discrete-time graph-coupled dynamical system and evaluate it as a case study on a behavior cloning task in single-player Pong. Compared with Vanilla RNN, Sparse RNN, LSTM, Closed-form continuous-time neural network (CfC), and Transformer baselines, TND achieves the best catch rate and a mean of 17.47 consecutive catches per round, more than three times that of the strongest baseline. These results suggest that shifting from layer-wise to neuron-wise dynamics provides an effective inductive bias for sequence modeling.

Borui Cai, Yao Zhao

Hangzhou International Innovation Institute, Beihang University, China · Victoria University, Melbourne, Australia

Dynamical Systems Dynamic Neural Networks Sequence Modeling

Jun 19, 2026cs.AI

Closure of Self-Determining System Based on Causal and Constitutive Relations

A self-determining system is defined as one in which causes originating within the system influence the system itself. This definition raises the question of how to specify system boundaries. Although the concept of "closure" is commonly used for this purpose, defining boundaries solely in terms of causal relations introduce challenges, such as how to handle external causes and circular causality. To address this issue, we introduce two types of asymmetric relations: causal and constitutive. We propose that system boundaries can be defined as closures of loops formed by these relations, referred to as causal-constitutive loops. By constraining constitutive relations, the resulting system necessarily includes internal causes and thereby satisfies self-determination. Furthermore, to prevent reduction to supervenience, constitutive relations must involve at least two independent variables. This minimal requirement leads to two interdependent loops, which implies a dual-process organization.

Yoshiyuki Ohmura, Earnest Kota Carr, Yasuo Kuniyoshi

The University of Tokyo, 7-3-1 Hongo Bunkyo-ku, Tokyo, Japan

Dynamical Systems

Jun 18, 2026cs.RO

Learning-Based Modeling of Soft Robots via Cosserat Rod Theory

Modeling soft robot dynamics is challenging due to their continuum structure and typically nonlinear dynamics. Creating models based on first-order principles is typically time-demanding, and their expressiveness is limited, whereas data-driven models lack interpretability and physical consistency. This work aims to overcome these challenges by introducing a port-Hamiltonian Gaussian Process Regression framework for learning and simulating the dynamics of planar, rod-like soft robots. In detail, the proposed model integrates Cosserat rod theory and Hamiltonian physics with data-driven inference to preserve the system's energy structure while accurately learning the rod dynamics. Numerical simulations show that we can achieve accurate and energy-consistent representations of a rod-like soft robot, showing the potential for a robust and interpretable pathway for modeling complex continuum mechanics.

Mohammad Ali, Nithin Senthur Kumar, Eric J. Barth +1

Department of Computer Science, Vanderbilt University, Nashville, TN 37235, USA · Department of Mechanical Engineering, Vanderbilt University, Nashville, TN 37235, USA

Dynamical Systems Port-Hamiltonian Systems Soft Robotics Gaussian Process Regression

Jun 18, 2026q-fin.RM

Optimal Order of Multi-Agent and General Many-Body Systems

This paper develops a general framework for analyzing multi-agent systems with feedback loops between agents actions and collective observations. The framework is built on two fundamental agent-level variables: power, which measures agent influence on collective outcomes, and response functions, which determine how agents react to observations. We derive how macroscopic properties, including total power, useful power, entropy, order, fragility, and mobility, emerge from these two variables of heterogeneous agents. To study the trade off between growth and resilience, we introduce a system-level utility function parameterized by a risk-appetite coefficient and derive an optimal degree of order that balances productivity, stability, and adaptability. The analysis suggests that stronger synchronization can increase collective output but may also increase systemic fragility and reduce mobility. We further argue that order, entropy, information, and useful energy are task-dependent and system-relative concepts whose meanings depend on the objectives of the system. By measuring and designing agent power distributions and response functions, it may be possible to better understand, predict, and optimize collective behavior and identify the conditions under which collective intelligence and optimal order emerge.

Jake J. Xia

Harvard Management Company · Research Affiliate at Massachusetts Institute of Technology

Dynamical Systems Multi-Agent System Optimization Multi-Agent Coordination Multi-Agent Systems Emergent Behavior in Multi-Agent Systems

Jun 18, 2026cs.RO

Stable Transformer-Actor-Critic Model Predictive Control: A Contraction Analysis Approach

Actor-Critic Model Predictive Control (MPC) effectively addresses complex, non-convex control problems, but guaranteeing the closed-loop stability of sequence-based learning models within these pipelines remains challenging. This paper introduces a novel Transformer-Actor-Critic MPC architecture with formal robustness guarantees. First, we prove that Transformer networks can satisfy global incremental Input-to-State Stability (

δ

ISS). We then leverage Riemannian contraction theory to analyze the interconnected dynamics between the physical plant and the predictive neural network. Finally, we integrate these theoretical bounds as a training regularizer to yield a certifiably robust policy. The framework is validated on a nonlinear 3D drone model executing target-reaching and obstacle-avoidance maneuvers.

Antonio Marino, Valerio Modugno, Marco Cognetti

CAM · University of Cambridge · UCL +3

Dynamical Systems Nonlinear MPC Robust Control

Jun 18, 2026cs.LG

Kolmogorov-Arnold Reservoir Computing

Reservoir computing offers a lightweight framework for forecasting dynamical systems but may struggle to capture long-range dependencies due to limited representational capacity. Conventional reservoir computing recurrently uses fixed reservoirs with hyperparameter sensitivity, while the next generation reservoir computing removes recurrence at the cost of rapidly growing feature dimensions. Here, we develop Kolmogorov-Arnold Reservoir Computing (KARC), which replaces reservoirs with explicit basis-function expansions inspired by the Kolmogorov-Arnold representation theorem. We rigorously show that KARC is a lightweight design of Kolmogorov-Arnold networks (KANs), preserving the potential expressive capacity of KANs while admitting efficient closed-form training of reservoir computing. At comparable cost, KARC outperforms existing reservoir computing methods on challenging benchmarks including partial differential equations. It can also be integrated with generative diffusion models for facilitating text-to-image generation. This work thus establishes a principled bridge between reservoir computing and KANs, yielding a unified framework for efficient dynamical forecasting and generative modeling.

Juntian Huang, Jürgen Kurths, Ying Tang

Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China · School of Physics, University of Electronic Science and Technology of China, Chengdu 611731, China · Key Laboratory of Quantum Physics and Photonic Quantum Information, Ministry of Education, University of Electronic Science and Technology of China, Chengdu 611731, China +4

Dynamical Systems Time Series Forecasting Reservoir Computing Kolmogorov-Arnold Networks

Jun 18, 2026math.OC

Semiglobal Input-Delay Tolerance Algorithm for Distributed Nonconvex Optimization of Networked Nonlinear Systems

This paper studies a class of distributed optimization problems in networked nonlinear systems (NNSs) subject to input delays and consensus constraints. It introduces input-delay tolerant semiglobal convergence (IDTSC), meaning that for any prescribed compact initial set there exists an admissible delay bound under which the optimal solution is computed within consensus constraints and all node states converge to the solution. Building on a hierarchical design and input-to-state stability analysis, a new semiglobal input-delay tolerant (SIDT) algorithm is developed that practically achieves IDTSC for distributed optimization under the coupling between input delays and nonlinear dynamics. Further, by relaxing strict convexity requirements through the Polyak-Łojasiewicz condition, the SIDT algorithm broadens its applicability to nonconvex optimization. Finally, numerical experiments corroborate the theory on NNSs with input delays.

Jing-Zhe Xu, Zhi-Wei Liu, Ming-Feng Ge +2

School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan 430074, China · School of Mechanical Engineering and Electronic Information, China University of Geosciences, Wuhan 430074, China

Dynamical Systems Optimization Convergence Analysis Distributed Optimization Nonconvex Optimization

Jun 18, 2026cs.RO

A Differentiable Composite Approximation Framework for Autonomous Underwater Vehicle Maneuvering Modeling from Sea-Trial Data

Field-based modeling from onboard measurements can produce autonomous underwater vehicle (AUV) maneuvering models that reflect real operating characteristics. From an approximation perspective, conventional maneuvering models use predefined constraint polynomial bases, whereas data-driven models use data-adaptive bases. Motivated by this basis-function view, this paper presents a differentiable composite-approximation formulation, in which the polynomial-basis component and the data-adaptive basis component are treated as differentiable parts of a single predictor and calibrated jointly. A gradient-based co-calibration method is developed for full-scale AUV maneuvering prediction, where a sensitivity-aware mechanism regulates bounded polynomial updates while the neural residual captures remaining nonlinear discrepancies under a shared prediction objective. To account for ocean-current effects in field data, a turning-motion-based current estimation and compensation procedure is incorporated to construct current-compensated learning targets for training and rollout. The framework is evaluated using sea-trial data collected from a 7-meter AUV under multiple maneuvering conditions. Results show that the proposed method improves recursive trajectory and velocity prediction compared with polynomial-only, neural-only, and frozen-prior hybrid baselines, demonstrating its applicability to field-data-based AUV maneuvering modeling.

Aobo Wang, Aifei Xia, Zihao Wang +1

College of Shipbuilding Engineering, Harbin Engineering University, Harbin, 150001, China · China Academy of Aerospace Aerodynamics, Beijing, 100074, China · Institute of Artificial Intelligence, Shanghai University, Shanghai, 200444, China +1

Dynamical Systems Underwater Robotics System Identification Trajectory Prediction

Jun 16, 2026math.OC

Horizon-Uniform Sensitivity Certificates for Finite-Horizon Pontryagin Systems

Finite-horizon optimal-control computations repeatedly solve two-point Pontryagin boundary value problems whose conditioning can deteriorate as the horizon grows. We give a verifiable data-level certificate under which it does not. Hyperbolicity of the reduced state--costate transition matrix, together with scaled stable--unstable boundary transversality, yields an endpoint-corrected Green inverse with horizon-independent constants and weighted contractions transfer this inverse to the nonlinear problem, so the original Pontryagin endpoint rows

x_0=x_{\rm in}

and

p_T=r_x(x_T,y)

carry a unique local stationary branch whose first-order expansion and Lipschitz constants are uniform in the horizon. Consequently the finite-horizon feedback map is horizon-uniformly Lipschitz, first-order expandable, and satisfies an exact shrinking-horizon consistency identity. Symplectic and Riccati criteria certify the hypotheses from matrix data: every stabilizable definite linear-quadratic system with invertible dynamics and a locally concave terminal Hessian at the reference qualifies. Reproducible computations illustrate both certificates.

Pyuyi Chufeng Huang, Zikang Song, Xingshu Chen

School of Cyber Science and Engineering, Sichuan University, Chengdu, Sichuan, China · School of Mathematics, Sichuan University, Chengdu, Sichuan, China

Dynamical Systems Feedback Control Linear Quadratic Regulator Optimal Control

Jun 15, 2026stat.ML

Dynestyx: A Probabilistic Programming Library for Dynamical Systems

State-space models (SSMs) are the standard formalism for Bayesian treatment of dynamical systems, with natural applications in statistics, signal processing, and machine learning. Despite their importance in both theory and application, dynamical systems have proven difficult to incorporate in modern probabilistic programming languages (PPLs), making state-of-the-art methods less accessible to practitioners and introducing friction in following the "Bayesian workflow." We introduce dynestyx, a probabilistic programming library with first-class support for SSMs, including state-of-the-art methods in the estimation of both states and parameters. Through a single, unified interface, users may specify arbitrary priors for discrete-time or continuous-time dynamical systems, perform inference over mixed-effect data, and make state and parameter estimates with principled uncertainty quantification.

Daniel Waxman, Dmitry Batenkov, John Feser +4

1Basis Research Institute · 2Massachusetts Institute of Technology

Dynamical Systems Parameter Estimation Bayesian Inference Probabilistic Programming State Space Models

Jun 15, 2026cs.ET

Neural dynamical systems on ferroelectric compute-in-memory for real-time forecasting

Neural dynamical systems are expressive temporal predictors that capture continuous-time dynamics through fine-grained state updates. However, this sequential structure maps poorly onto digital hardware optimized for dense matrix operations, a mismatch that analog neuromorphic computing, with its native continuous-time dynamics, can resolve. We introduce FerroNDS, a neuromorphic system built from two analog primitives: an integrator for temporal accumulation and an oscillator for frequency-selective filtering. We map this system onto compute-in-memory hardware based on multi-bit ferrodiodes. A 128-neuron instance of FerroNDS computes short-time Fourier transform and forecasts a 500-ms horizon for periodic, quasi-periodic, and chaotic signals. The system achieves sub-watt real-time operation with per-neuron per-inference energy of 1.64

μ

J (200 Hz) and 0.29

μ

J (10 kHz), 25-40

\times

area reduction over SRAM-based digital systems, and per-layer latency of 3.18 ms (200 Hz) and 63.87

μ

s (10 kHz). To our knowledge, this is the first end-to-end integration of a ferrodiode into a neuromorphic computational framework, establishing ferroelectric compute-in-memory as a practical substrate for analog neural dynamical systems.

Keshava Katti, Adithya Selvakumar, Pratik Chaudhari +1

Electrical & Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA

Efficient Neural Network Inference Dynamical Systems Compute-in-Memory Time Series Forecasting Neuromorphic Computing Chaotic Time-Series Prediction

Jun 15, 2026stat.ML

Attention is Just Another Name for Coupling? A Fast-Slow ODE Perspective on Hierarchical Pretraining

We re-interpret Transformer pretraining as a fast-slow, singularly perturbed flow along depth, with untied weights as its non-autonomous feature. The linearised dynamics is a depth-ordered product of layer maps. Along a token-homogeneous reference trajectory, the linearised layer factorises along the eigenbasis of a frozen attention kernel. Past a computable saturation depth, the flow factors through the block coarse-graining -- in other words, running the layers is running the coarse variable, dually. Weight perturbations supported on the decaying bundle move neither the persistent component of the distinguished trajectory nor the frozen kernel to first order, so the framework partitions parameter space into visible and invisible directions, with the cross-block coupling of the slow path sitting entirely on the visible side. How large a gate the slow path can carry is bounded by a stability margin. On the data side: if block emissions follow an exponential family, block-mean pooling captures all the information the slow path can use; but if neighbouring blocks carry no shared structure, no cross-block channel can help the prediction, and the gate amplitude is invisible in the prediction risk. Stability delimits what the architecture may do; the data decides what it will.

Zhengyuan Gao

Independent scholar

Dynamical Systems Transformer

Jun 14, 2026cs.LG

A Bifurcation Theory Framework for Gradient Descent on the Edge of Stability

The Edge of Stability (EoS) phenomenon, where gradient descent operates with sharpness exceeding the classical convergence threshold yet the loss decreases over long timescales, is ubiquitous in modern deep learning but remains poorly understood in realistic settings. Prior rigorous analyses have been largely confined to scalar or low-dimensional losses with specific structural forms. In this work, we develop a bifurcation theory framework for gradient descent on the edge of stability that applies directly to overparameterized neural networks. By decomposing the training dynamics into components normal and tangent to the manifold of minimizers, we show that stable EoS training arises from a flip bifurcation in the normal direction, governed by the sign of the first Lyapunov coefficient, while the tangent dynamics drift toward regions of decreasing sharpness. Under mild spectral and geometric assumptions on the loss landscape, we prove convergence to the minimizing manifold when training at the EoS threshold. As a corollary, we recover and unify prior results: we show that the product-stability condition of Gan (2026) is an instance of our framework.

Eric Gan

Independent Researcher.

Dynamical Systems Edge of Stability Neural Network Training Dynamics Gradient Descent Dynamics

Jun 12, 2026cs.LG

Deep Spectral Learning of Embedded Latent Transfer Operators for Stochastic Dynamical Systems

We propose a spectral learning method for stochastic nonlinear dynamical systems represented with embedded latent transfer operators in deep feature spaces. We instantiate the method as Deep Spectral Encoder (DSE), an operator-based latent state-space model in which a time-invariant neural encoder implements learnable nonlinear feature maps from observations, and these features define Markovian latent states whose temporal evolution and observation mapping are described by the transfer and observation operators, respectively. Functional canonical correlation analysis in a learnable Galerkin-projected feature space provides state coordinates from past and future observations, and the two linear operators are estimated on the state coordinates as ridge-regularized closed-form solutions that coincide with Galerkin projections of the associated covariance operators. On this representation, we generalize sequential Bayesian filtering and Koopman spectral mode decomposition in feature space. Experiments on several scenarios show stable and superior performance with sequential Bayesian filtering and dynamic mode decomposition baselines even under noise and partial observability.

Ryogo Tanaka, Yoshinobu Kawahara

Graduate School of Information Science and Technology, The University of Osaka, Osaka, Japan · Center for Advanced Intelligence Project, RIKEN, Tokyo, Japan

Dynamical Systems Stochastic Linear Dynamical Systems Koopman Operator Learning Latent Dynamics Modeling Operator Learning

Jun 11, 2026cs.MA

Tuning Agent-Based Predator-Prey Models Toward Lotka-Volterra Dynamics

Recent growth in compute power has made it increasingly feasible to use large-scale agent-based models to simulate complex adaptive systems. A central difficulty is that such models contain many local rules and parameters, where small changes can lead to runaway behaviour, population collapse, or saturation at artificial bounds. We study this problem in a continuous predator-prey system where sheep and wolves are active agents with local sensing, internal energy, and recurrent neural network-based controllers. We ask whether environmental and demographic parameters can be tuned so that the resulting population dynamics resemble classical Lotka-Volterra cycles. We optimise these parameters with a feature-based loss that rewards sustained oscillations, phase lag, bounded populations, and long-term persistence, first for random controllers and then for evolved controllers in a more naturalistic setting. The model is implemented in ABMax, a JAX-based agent-based modelling framework that enables efficient batched simulation on hardware accelerators.

Corinna Mandl, Siddharth Chaturvedi, Marcel van Gerven

Department of Machine Learning and Neural Computing Donders Institute for Brain, Cognition and Behaviour, Radboud University Nijmegen, the Netherlands

Dynamical Systems Agent-Based Modeling