Nonlinear System Identification

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5 papers in the last four weeks, up 67% on the four weeks before. 0.0% of all new papers.

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Latest papers 107

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 25, 2026cs.LG

Recovering Governing Equations from Solution Data: Identifiability Bounds for Linear and Nonlinear ODEs

Learning governing equations from observed solution data is a fundamental challenge in scientific machine learning, yet the theoretical conditions under which a ground-truth ODE can be uniquely and stably identified from multiple solution observations remain largely undeveloped, and no quantitative analysis of the sample complexity of such learning tasks exists in the literature. To address this gap, we introduce the Hausdorff distance on solution sets as the natural metric for comparing differential equations, since it captures the worst-case separation between two equations over all admissible initial conditions and thus encodes the minimax structure of the identification problem. We establish identifiability bounds for governing ODEs across a wide class of structure equations--ranging from linear ODEs to nonlinear classes with Lipschitz (Hölder)-continuous vector fields--characterizing precisely when two distinct equations can be distinguished from solution data. Using this metric, we derive metric entropy estimates for the relevant ODE classes and analyze sample complexity bounds, quantifying how many solution observations are needed to reliably recover the governing equation.

Yang Pan, Helmut Bölcskei

Parameter Identifiability Sample Complexity Nonlinear System Identification Statistical Learning Theory ODE Discovery

Jun 23, 2026cs.LG

LLM-ACES: Closed-Loop Discovery of Dynamical Systems with LLM-Guided Adaptive Search

Recovering governing Ordinary Differential Equations (ODEs) from data is a central challenge in modeling dynamical systems across scientific domains. Existing approaches cast discovery as a static inference problem over fixed datasets, assuming that the observed trajectories are sufficiently informative. However, dynamical systems evolve over large state spaces, and limited data can make multiple equations observationally indistinguishable, leading to identifiability gaps and the recovery of incorrect governing equations. To address this, we introduce LLM-ACES, or LLM-guided Active Closed-loop Equation Search, a closed-loop framework that jointly optimizes symbolic hypothesis construction and adaptive data acquisition. In LLM-ACES, a large language model (LLM) proposes operator priors that partition the large search space into distinct regions, within which candidate equations are fit to the observed data. The disagreement among these candidates guides the acquisition of informative trajectories, creating a feedback loop that iteratively refines both the hypothesis space and the discovered dynamics. On 122 ODE systems spanning ODEBench and ODEBase, LLM-ACES achieves the lowest median NMSE, outperforming state-of-the-art baselines by several orders of magnitude while achieving a high symbolic accuracy of 46.2% and 52.4%, respectively. Our analysis further shows that LLM-ACES is sample-efficient, achieving better performance with one-tenth the data. Furthermore, LLM-ACES's feedback-driven data acquisition makes it robust to noise and recovers the correct symbolic structure, while baselines introduce spurious terms that fit the data locally but obscure the true governing relationships.

Nikhil Abhyankar, Sha Li, Sanchit Kabra +3

Virginia Tech

Nonlinear System Identification System Identification Autonomous Scientific Discovery Large Language Model-Guided Scientific Discovery ODE Discovery Symbolic Regression

Jun 23, 2026cs.LG

Learning Dynamical Systems from Multiple Sparse Datasets: A Hierarchical Bayesian Modeling Approach

Estimating parameters of dynamical systems from sparse, noisy, and irregularly sampled data is often severely ill-conditioned. When multiple related datasets are available, they provide additional information if the shared structure and variability are properly modeled. We propose a hierarchical Bayesian framework for probabilistic meta-learning in dynamical systems, modeling dataset-specific parameters as draws from a shared population distribution. A numerical ODE solver is embedded within gradient-based MCMC to enable efficient posterior inference of the shared population and dataset-specific parameter distribution. Experiments show improved predictive performance over unpooled methods, highlighting the potential for data-efficient system identification in settings with sparse data.

Cristian Brugnara, Lea Multerer, Marco Forgione +1

SUPSI, Dalle Molle Institute for Artificial Intelligence (IDSIA USI-SUPSI), Lugano, Switzerland

Irregular Time-Series Modeling Parameter Estimation Meta-Learning Bayesian Inference Nonlinear System Identification Hierarchical Bayesian Modeling+1

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.LG

Frequency-Domain Neural ODEs for Modeling Non-Linear Dynamical Systems

Standard continuous-depth models, such as Neural Ordinary Differential Equations (NODEs), offer significant advantages in modeling physical systems by learning continuous vector fields rather than discrete temporal steps. However, when applied to complex dynamical systems, standard NODEs frequently struggle with highly nonlinear dynamics. This paper investigates the Frequency-domain Neural ODE (FNODE), an architecture that projects continuous temporal dynamics into the frequency domain using the Fast Fourier Transform (FFT). By operating in the frequency domain, the model provides better generalization to the dynamical system. The architecture is empirically evaluated against discrete models, specifically Gated Recurrent Units (GRUs) and Long Short-Term Memory (LSTMs), and other continuous-depth variants, including Augmented Neural ODE (ANODE), across four distinct dynamical systems: the Lotka-Volterra model, the forced Duffing oscillator, the Van der Pol oscillator, and the Lorenz system. To rigorously assess generalization and robustness, curriculum and ensemble learning are used to evaluate the model's convergence by estimating confidence intervals across different ensemble models. The empirical results demonstrate that the FNODE architecture achieves better generalization while exhibiting remarkable convergence stability.

Mohammed Ashraf, Ayman A. El-Badawy

Department of Mechatronics Engineering, German University in Cairo, Cairo, Egypt

Frequency-Domain Feature Learning Nonlinear System Identification Neural ODEs

Jun 19, 2026stat.ML

Orthogonal Discrepancy Kernels for Learning with Partial Physics

We introduce a semi-parametric framework for nonlinear system identification, which decouples discrepancy functions from physics-based components. Orthogonal Gaussian process regression balances sparse parameter selection (the white box) with discrepancy learning (the black box) to produce interpretable models from incomplete physics.

Swapnil Manna, Timothy J. Rogers, Lawrence Bull

IISER Pune, India · University of Sheffield, UK · University of Glasgow, UK

Semiparametric Inference Nonlinear System Identification Model Calibration Physics-Informed ML Gaussian Process Regression

Jun 17, 2026eess.SP

Explicit Interaction Architectures for Dynamical Learning: A Controlled Study of Structural Inductive Bias

We investigate a structure-first approach to dynamical learning in which the organization of stateful interactions is prescribed explicitly rather than left entirely to a generic recurrent parameterization. We introduce causal recurrent units built from an ordered sequence of local, state-modulated transformations. The construction is motivated by wave-based interaction models, but the units studied here do not impose scattering, passivity, or energy-balance constraints. Because fixed recurrent dynamics, designed reservoir topologies, readout-only learning, and recurrent depth are already well established, the empirical question is deliberately narrower: does the proposed interaction organization provide a useful inductive bias under controlled computational conditions? We compare a one-layer structured model, a two-layer structured model, and a generic echo-state network (ESN), all with 12 recurrent states and the same strictly linear ridge readout. Each model family receives the same random-search budget on calibration data that are disjoint from the final test data, after which the selected hyperparameters are frozen. On a custom nonlinear identification task, the one-layer structured model attains a mean validation NMSE of 2.76 x 10^{-4}, compared with 3.19 x 10^{-4} for the two-layer model and 3.94 x 10^{-4} for the ESN. On NARMA10 the ordering reverses: the ESN attains 0.312, compared with 0.348 and 0.357 for the one- and two-layer structured models. Thus, the proposed organization can be competitive and advantageous on one task, but it is not universally superior; moreover, recurrent depth does not provide a systematic benefit under matched state dimension. The results support a task-dependent interpretation of structural inductive bias and position the present architecture as a controlled precursor to stronger wave- and system-theoretic constructions.

Augusto Sarti

Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB), Politecnico di Milano, Piazza L. Da Vinci 32, 20133 Italy.

Dynamic Neural Networks Nonlinear System Identification Recurrent Neural Networks

Jun 15, 2026stat.AP

Regularized Machine Learning for System Identification of Ship Free-Running Manoeuvres from CFD-Based Synthetic Data: A Comparative Study

This study investigates supervised machine learning techniques for identifying ship hydrodynamic coefficients from CFD-generated data from free-running simulations. Specifically, ordinary least squares and regularized regression methods are applied to Abkowitz-type manoeuvring models. Training and validation datasets are derived from URANS simulations of zig-zag and turning circle manoeuvres, which are validated against experimental benchmark data. The analysis evaluates the effects of coefficient set size, minimum training length required for predictive model training, and manoeuvre combinations on model performance. Results demonstrate the suitability of large-angle zig-zag manoeuvres for hydrodynamic system identification, provided that multicollinearity is addressed through appropriate coefficient selection, regression models, or input data variability. Larger coefficient sets offer greater model flexibility for variable conditions but are more prone to multicollinearity. Regularized regression techniques effectively mitigate multicollinearity and notably enhance prediction accuracy, as does incorporating more diverse manoeuvring data. Among tested models, Ridge regression provided the best compromise between computational efficiency and prediction accuracy.

R. F. Suárez, J. C. Berndt, M. Abdel-Maksoud

Hamburg University of Technology (TUHH), Institute for Fluid Dynamics and Ship Theory (M8), Am Schwarzenberg-Campus 4 (C), 21073, Hamburg, Germany

Nonlinear System Identification System Identification Ridge Regression

Jun 14, 2026cs.LG

Multi-Fidelity SINDy: Sparse Discovery of Nonlinear Dynamical Systems with Fidelity-Weighted Measurements

Data from simulations and experiments are rarely noise-free and often exhibit heterogeneous levels of fidelity. Measurement uncertainty may vary across repeated observations, sensing devices, or even within a single experiment. This work addresses the problem of discovering nonlinear dynamical systems from such inhomogeneous data. We extend the Sparse Identification of Nonlinear Dynamical Systems (SINDy) framework to account for variable noise levels by combining Ensemble SINDy and Weak SINDy within a weighted regression formulation derived from generalized least squares. A statistical justification for the weighting strategy is also provided. The methodology is validated on several benchmark systems, including ordinary and partial differential equations. In addition, we show the benefit of multi-fidelity integration for forecasting the dynamics of a double pendulum system. The results confirm that the proposed approach mitigates the adverse effects of heteroscedastic noise and that repeated, low-cost, low-quality measurements can improve model recovery, in some cases matching or outperforming reconstructions obtained using only high-fidelity data.

Filippo Zacchei, Ana Larrañaga, Attilio Frangi +2

MOX – Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy · Department of Mechanical Engineering, University of Washington, Seattle, Washington 98195, United States · Department of Civil and Environmental Engineering, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

Sparse Identification of Nonlinear Dynamics Nonlinear System Identification Multifidelity Modeling Heteroscedastic Regression

Jun 13, 2026cs.RO

Covariance-Regulated Recursive Koopman Learning for Nonlinear Systems with Uncertain Time-Varying Dynamics

Offline models for autonomous robots often fail under time-varying dynamics outside their training distribution. Koopman operator theory offers a linear representation of nonlinear dynamics via lifting, but its transition to real-time recursive estimation may suffer numerical vulnerabilities: covariance windup under low excitation when using exponential forgetting, and vanishing gain without forgetting. This paper introduces a Covariance-Regulated Recursive Koopman Learning (CR-RKL) framework with two complementary strategies--error dead-zone gating and constant-trace normalization--each independently capable of preventing covariance explosion and parameter freezing, with the latter additionally preserving the geometric structure of uncertainty. Validated on a non-holonomic differential-drive robot with wheel slip and Stribeck friction and on a 26-gram butterfly-inspired flapping-wing micro aerial vehicle, CR-RKL achieves numerically stable and accurate online modeling, and when embedded in model predictive control, it maintains reliable tracking performance under uncertain, time-varying dynamics.

Weibin Gu, Chen Yang, Lu Shi +1

1Tsinghua University, PR China · 2China University of Petroleum-Beijing at Karamay, PR China · 3Xinchen Qihang Inc., PR China

Online System Identification Recursive Least Squares Koopman Operator Learning Nonlinear System Identification

Jun 11, 2026cs.LG

Extracting Governing Equations from Latent Dynamics via Multi-View Contrastive Learning

Identifying latent dynamical systems from noisy, high-dimensional measurements is a central problem at the intersection of representation learning, system identification, and scientific discovery. We present DYSCO, a multi-view temporal contrastive learning algorithm that jointly recovers latent trajectories and the governing dynamics from such observations, by leveraging multiple independent noisy views of the same underlying process to disentangle signal from noise. By parameterizing the dynamics in a structured functional basis, our framework further enables symbolic recovery of the governing equations within an affine gauge. We offer theoretical guarantees for strong identification up to an affine indeterminacy, extending prior identifiability results to the realistic setting of noisy nonlinear observations. Empirically, we demonstrate accurate recovery of both latent trajectories and flow fields across a diverse set of dynamical regimes (e.g., chaotic, oscillatory, and metastable) under both Gaussian and Poisson observation noise, the latter being particularly relevant for neural recordings.

Paolo Muratore, Mackenzie Weygandt Mathis

EPFL

Contrastive Learning Parameter Identifiability Latent Dynamics Modeling Nonlinear System Identification Autonomous Scientific Discovery

Jun 10, 2026cs.LG

How Low Can You Go? Active Learning for Sparse Model Discovery in the Ultra-Low-Data Limit

Identifying the governing equations of complex dynamical systems remains a fundamental challenge across science and engineering. While early approaches relied on empirical data and heuristics, modern data-driven methods offer greater flexibility and fewer assumptions. However, data acquisition in real-world settings is often expensive. This work addresses this challenge by introducing an active learning strategy for dynamics discovery in the ultra-low data limit. Rather than sampling randomly, our method iteratively prioritizes regions that are most informative for model identification. This approach builds on Sparse Identification of Nonlinear Dynamics (SINDy), and utilizes an ensemble extension, E-SINDy, to estimate epistemic uncertainty and guide the sampling for both ordinary and partial differential equations (ODEs/PDEs). For ODEs, an exhaustive analysis is conducted on the Lorenz system across varying data budgets and noise levels. For PDEs, two systems with contrasting dynamical characteristics are examined: the Burgers' equation, where a sharp shock front creates a distinction between informative and uninformative regions, and the Kuramoto-Sivashinsky equation, which presents a more spatially complex sampling landscape. Across all scenarios, the proposed method accurately identifies the governing dynamics with significantly fewer data samples than random sampling.

Ana Larrañaga, Urban Fasel, Steven L. Brunton

Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, United States · NSF AI Institute in Dynamic Systems, University of Washington, Seattle, WA 98195, United States · Department of Aeronautics, Imperial College London, SW7 2AZ, United Kingdom

Sparse Identification of Nonlinear Dynamics PDE Discovery Nonlinear System Identification Active Learning System Identification Data-Efficient Learning

Jun 10, 2026math.DS

Integral Formulation of QENDy for Robust Nonlinear System Identification

This manuscript proposes an integral formulation of the newly defined quadratic embedding method for identifying nonlinear systems (QENDy). In the original algorithm, trajectory data points along with their time derivatives are used. Methods for calculating time derivatives make the algorithm sensitive to noise. Our integral formulation does not use the time derivatives. This results in a more robust method to learn the dynamics.

Nikhil Saran, Sushant Pokhriyal, Stefan Klus +2

Department of Mathematics and Statistics at the University of South Florida · Institute of Engineering and Technology, JK Lakshmipat University, Jaipur, India · School of Mathematical & Computer Sciences at the Heriot–Watt University, Edinburgh +1

Sparse Identification of Nonlinear Dynamics Nonlinear System Identification

Jun 8, 2026cs.LG

Nonlinear Bayesian Estimator for Parameter Learning: A Fixed-Point Characterization

This paper presents a nonlinear parameter estimator for Wiener-type state-space models obtained as a fixed-point architecture that couples two affine minimum mean-squared error (MMSE) estimators: one for the unknown parameters and one for latent variables. The architecture retains the functional structure of the optimal affine MMSE parameter estimator while incorporating Dynamic Basis Statistics (DBS) estimates that summarize nonlinear basis-function evaluations. Two DBS construction strategies are developed, leading to two nonlinear estimator frameworks. The dual basis-parameter estimator combines an affine basis estimator with the affine parameter estimator, whereas the dual state-parameter estimator first computes affine state estimates and their covariances, then maps these state-estimate statistics through a Gaussian DBS operator to obtain DBS estimates. Both dual estimators admit fixed-point characterizations that alternate between estimating each component using the updated prior of the other, obtained from that component's plug-in estimate statistics from the previous iteration. The efficacy of the proposed methods is examined via extensive Monte Carlo experiments, showing that the dual basis-parameter estimator attains parameter mean-squared errors comparable to those of the purely affine parameter estimator, while the dual state-parameter estimator achieves the lowest parameter mean-squared error, outperforming both the dual basis-parameter and purely affine parameter estimators, as well as sequential Monte Carlo variants of classical Particle Gibbs and Expectation-Maximization schemes.

Sasan Vakili, Daniël Woonings, Pradyumna Paruchuri +1

Delft University of Technology, The Netherlands · University of Toronto, Canada

Parameter Estimation Bayesian Inference Nonlinear System Identification State Space Models

Jun 8, 2026cs.RO

Physics-Aware Sparse Learning and Selective Online Adaptation for Euler-Lagrange Robot Dynamics

Accurate dynamics models are essential for model-based robotic control, yet nominal Euler--Lagrange models often become inaccurate in the presence of payload variation, unmodeled coupling, friction, aerodynamic effects, and changing operating conditions. Most learning-based correction methods improve prediction accuracy by introducing a single additive residual, but do not preserve the internal mechanical structure of Euler--Lagrange systems. This leads to models that do not preserve symmetry, positive-definiteness, or the coupling between inertia and velocity-dependent terms, which can result in physically inconsistent predictions and reduced reliability when embedded in model-based controllers. We propose a structure-preserving residual learning framework that decomposes model mismatch into an inertia correction, the corresponding induced Coriolis term, and a generalized-force residual. The mechanical component is learned under physical constraints, while the disturbance-sensitive component is represented through a sparse history-dependent latent interaction model and adapted online using Bayesian linear regression. This separation preserves key mechanical structure while restricting adaptation to the part of the dynamics most affected by changing conditions. Experiments across multiple robotic platforms, including mobile, aerial, and manipulator systems, show that the proposed method improves dynamics prediction and trajectory tracking under coupled and time-varying dynamics. These results highlight the value of combining structured residual modeling, compact latent interaction selection, and selective online adaptation for real-world model-based control.

Rishabh Dev Yadav, Samaksh Ujjawal, Sihao Sun +2

Department of Computer Science, The University of Manchaster, U.K. · Department of Cognitive Robotics, Delft University of Technology, Netherlands · International Institute of Information Technology Hyderabad. +1

Robotic Control Robot Dynamics Identification Online System Identification Residual Learning Residual Dynamics Learning Robot Dynamics Learning+2

Jun 8, 2026cs.LG

Mechanical Field Networks: Structured Neural Dynamics for Multivariate Systems

Many multivariate dynamical systems are observed only through trajectories, leaving the mechanisms governing their joint dynamics hidden. Existing approaches can impose interpretable dynamics or learn flexible state transitions, yet the resulting interaction structure is typically either specified in advance or left implicit within the learned dynamics. We introduce MF-Net, a recurrent dynamical model that represents all variables in a shared field state and updates this state through a learned relation law. Each variable carries a field component, and these components evolve jointly through a learnable mechanical transition. Here, mechanical refers to the relation-to-motion organization of the transition, where learned relations shape state-dependent flows, field responses, and motion tendencies that move the field state forward. The resulting structure is part of the rollout itself: learned relations influence how the field moves, and the same internal quantities support both forecasting and structural readout. Across known-law interaction systems, chaotic benchmarks, real neural recordings, and ecological time series, MF-Net achieves competitive short- and medium-horizon forecasting while retaining inspectable structural readout. On the 40-dimensional Lorenz--96 testbed, MF-Net achieves an eight-step

R^2

of

0.798\pm0.018

; across five seeds, its learned relation matrix recovers the local coupling support with a local/nonlocal strength ratio of

19.80\pm1.00

and Precision@

K

of

1.000\pm0.000

. MF-Net provides a structure-readable dynamical modeling framework in which learned relations are trained through forward evolution and, on real data, interpreted as functional predictive couplings under appropriate observational limits.

Xingji Cui

Xi’an Jiaotong University, China

Multivariate Time Series Forecasting Dynamical Systems Latent Dynamics Modeling Nonlinear System Identification Recurrent Neural Networks

Jun 7, 2026cs.LG

Inferring hidden forcing in a biological oscillator using Kolmogorov-Arnold networks

Inferring the forces that drive a dynamical system from partial observations is a fundamental challenge across physics, particularly when distinct underlying mechanisms produce similar observable dynamics. Here we show that the effective muscular forcing underlying avian respiratory dynamics can be reconstructed from measurements of air-sac pressure alone. Using an interpretable learning framework based on Kolmogorov-Arnold networks, we infer the governing equations of the system directly from data and uncover a nontrivial structure in the underlying forcing that is not apparent from the pressure signal, which instead suggests a relaxation-like oscillation. The reconstructed dynamics predict a two-phase activation pattern within each respiratory cycle, which we independently validate through electromyographic recordings of expiratory muscles. These results demonstrate that data-driven reconstruction of dynamical laws can reveal hidden physical structure and provide access to unobserved driving variables, establishing a general route to infer latent forces in partially observed dynamical systems.

Julian Szereszewski, Facundo Fainstein, Leandro E. Fernandez +1

Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Física, Ciudad Universitaria, 1428 Buenos Aires, Argentina. · CONICET - Universidad de Buenos Aires, Instituto de Física Interdisciplinaria y Aplicada (INFINA), Ciudad Universitaria, 1428 Buenos Aires, Argentina.

Dynamical Systems Nonlinear System Identification

Jun 7, 2026cs.LG

When Are Neural Interaction Discoveries Real? Identifiability, Recoverability, and a Pre-Fit Diagnostic

When a neural time-series model reports that one variable modulates another's effect on a target, is the discovered interaction a property of the data or an artifact of model flexibility? We argue that this is fundamentally a question of identifiability, governed by the geometry of the observed input support rather than by the specific neural architecture. We study the problem in a multiplicative-gating extension of neural additive vector autoregression (GNAVAR), in which source contributions are modulated by other lagged variables. We show that representational capacity is not identifiability: dependent inputs induce leakage between edge-specific interaction terms, and low-dimensional support permits distinct interaction decompositions that agree on the observed data while differing elsewhere. We then prove a population identifiability theorem for normalized minimal GNAVAR decompositions under explicit support conditions, including settings with shared modulators. The theory yields a simple practitioner-facing diagnostic: the effective rank of the joint lag-block covariance predicts, before fitting, whether interaction recovery is feasible for a given candidate set. When the candidate set is unknown, a two-seed stability check provides a practical operational test. The same support condition organizes empirical outcomes into the three states predicted by the theory. Our results show that interaction recoverability depends on support geometry, that effective rank provides a practical pre-fit diagnostic, and that instability across independent fits is a characteristic signature of non-identifiable interaction discovery. The identifiability phenomenon, the support condition, and the instability signature are model-agnostic; GNAVAR is the vehicle that makes them provable.

Valentina Kuskova, Dmitry Zaytsev, Michael Coppedge

Lucy Family Institute for Data & Society · University of Notre Dame · Notre Dame, IN, USA +1

Feature Interaction Modeling Parameter Identifiability Nonlinear System Identification Neural Network Interpretability

Jun 3, 2026math.DS

Learning Control-Affine Reduced-Order Models via Autoencoders

We present in this paper a framework for the identification of control-affine reduced-order models (ROMs). The proposed method utilizes autoencoders (AEs) to transform the high-dimensional states, and potentially the high-dimensional inputs, into reduced latent ones suitable for control-affine state-space dynamics. This is achieved by simultaneous training of the AE and the state-space model. In addition, we extend the discrete ROM formulation to a sequence-based model, which processes state and input histories to improve prediction accuracy while preserving the control-affine structure. We motivate our framework by applying feedback linearization to the derived models, and we present guidelines for its efficient use. The proposed framework is assessed on two numerical examples and its performance is compared to a baseline model, where the AE identifies a latent space with linear state-space dynamics. The assessment involves evaluating the prediction accuracy of the ROM on test data and its effectiveness in controlling the system to a desired state or trajectory.

Ali Mjalled, Martin Mönnigmann

Automatic Control and Systems Theory Ruhr-Universität Bochum Universitätstraße 150, 44801, Bochum

Nonlinear System Identification System Identification Reduced-Order Modeling State Space Models

Jun 3, 2026cs.LG

AdaKoop: Efficient Modeling of Nonlinear Dynamics from Nonstationary Data Streams with Koopman Operator Regression

Real-time data analysis requires the ability to accurately and adaptively address nonlinear dynamics in a nonstationary data stream while preserving computational efficiency. However, nonlinear dynamics are so complex that capturing dynamically changing nonlinear patterns and utilizing them for downstream tasks under strict time constraints is nontrivial. To bridge the gap between nonlinear complexity and computational tractability, this study applies Koopman operator theory, which states that nonlinear dynamics can be represented as linear transitions in an infinite-dimensional space. Building upon finite-dimensional approximations of this operator, we present AdaKoop, an efficient streaming algorithm for modeling nonlinear dynamics over nonstationary data streams. Our approach utilizes a probabilistic framework grounded in Koopman operator theory, treating both raw observations and reproducing kernel Hilbert space (RKHS) features as emissions from latent vectors. This dual-view formulation allows nonlinear dynamics to be expressed as a tractable linear system. Therefore, AdaKoop enables the efficient and stable modeling of nonlinear dynamics in a streaming fashion, avoiding the prohibitive computational costs of iterative nonlinear optimization. Furthermore, to address nonstationarity in data streams, AdaKoop adaptively detects the switching of patterns via statistical hypothesis testing for abrupt pattern shifts and incrementally updates model parameters to handle continuous changes. Extensive experiments on a total of 71 practical benchmark datasets across various domains demonstrate that AdaKoop outperforms state-of-the-art methods in terms of real-time forecasting accuracy and computational efficiency.

Naoki Chihara, Ren Fujiwara, Yasuko Matsubara +1

SANKEN, The University of Osaka Osaka, Japan

Koopman Operator Learning Nonlinear System Identification Non-Stationary Time Series Forecasting Online Learning

Jun 2, 2026cs.LG

PE-MHL: Physics-Encoded Modular Hybrid Layers for Scalable Learning of Complex Systems

Hybrid models that combine physics-based and data-driven components have shown strong potential for achieving accuracy and interpretability in control applications. While recent methods have made progress in incorporating physical consistency, challenges remain in scalability, robustness to noise, and control of model complexity. This paper proposes a Physics-Encoded Modular Hybrid Layer (PE-MHL) framework, in which a baseline physics-based model is incrementally refined through the addition of new sub-models, where each new component adds complexity while preserving what previous components have already learned. We establish a theoretical guarantee for this construction: with a least-squares initialization of each new sub-model, the training error is monotonically non-increasing in the number of sub-models and provably converges. Empirical evaluations on a nonlinear NARX benchmark and the Quanser Aero 2 platform demonstrate that PE-MHL outperforms equivalently sized monolithic networks in both accuracy and generalization, while also providing more stable training dynamics and better preservation of underlying data structures.

Ismail Hassaballa, Mircea Lazar

Eindhoven University of Technology Eindhoven, Netherlands

Nonlinear System Identification Physics-Informed ML Modular Neural Networks

Jun 2, 2026cs.RO

Hybrid Dynamics Modeling for a Flexible 2-DoF Robotic Arm

This paper examines three approaches for modeling the dynamics of a flexible-link 2-DoF robotic arm to address unmodeled dynamics not captured by rigid-body models. Two physics informed models combine rigid-body dynamics (RBD) formulations with a Gaussian Mixture Model (GMM) to capture residual model errors and linkage flexibility. A kinematics-based regression model serves as a purely data-driven baseline. Using an open-source dataset, torque predictions are first estimated using Ridge regression on kinematic features, while the physicsbased baseline is constructed from published specifications, and ordinary least-squares regression is subsequently used to estimate the same parameter set directly from data. Results show that the physics-based parameters yield the poorest accuracy, while regularized and least-squares estimators align more closely with measured torques. Residual analysis and error metrics highlight the limitations of purely parametric models for flexible-link systems and underscore the value of regularization and data-driven identification, supporting developments of semi-parametric residual learning methods.

Maciek Popik, Daniel Yang, Mahdis Bisheban

Dept. of Mechanical and Manufacturing Eng at the Schulich School of Engineering, University of Calgary, Alberta, Canada · Intelligent Dynamics and Control Lab, University of Calgary, Alberta, Canada

Robot Dynamics Identification Nonlinear System Identification Robotics Inverse Dynamics

Jun 1, 2026math.NA

Learning Chaotic Dynamics through Second-Order Geometric Supervision

Learning chaotic dynamical systems from data requires more than short-term predictive accuracy: the learned model must preserve the attractor geometry and its invariant statistics. Trajectory (zero-order) and Jacobian (first-order) matching supervise the values and tangent structure of the vector field, but neither constrains how the field bends away from its tangent plane. A model can thus match values and tangents at the supervised states yet curve differently from the truth, remaining locally accurate while drifting toward spurious attractors and distorting long-time statistics. We show that enforcing second-order consistency mitigates these failures, but forming the full Hessian is prohibitive in high dimensions. We propose model-constrained randomized Jacobian matching, which compares the Jacobians of the true and learned vector fields at randomly perturbed inputs. A Taylor expansion shows that the expected randomized Jacobian loss decomposes into the nominal Jacobian mismatch plus a Hessian mismatch scaled by the noise variance, implicitly enforcing second-order consistency at

\mathcal{O}(d^2)

cost without forming the

\mathcal{O}(d^3)

Hessian tensor. Using only Jacobian evaluations, the method scales to high dimensions where explicit Hessian matching does not. Numerical experiments confirm that second-order methods are robust. For Lorenz63, first-order methods produce catastrophic Lyapunov-exponent outliers under minimal temporal supervision, which second-order methods eliminate while recovering the correct attractor. For coupled Lorenz96, an out-of-distribution forcing sweep separates the methods: all agree up to

F=16

, but beyond

F=18

only second-order methods preserve the invariant measure and Lyapunov spectrum. On both systems, randomized Jacobian matching performs comparably to explicit Hessian matching at much lower cost.

Shinhoo Kang, Hai V. Nguyen, Tan Bui-Thanh

Department of Computer Science and Software Engineering, Korea University, Sejong 30019, KR · Department of Aerospace Engineering and Engineering Mechanics, The Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, 2026 TX, USA

Nonlinear System Identification Chaotic Dynamical Systems Chaotic Time-Series Prediction

May 29, 2026cs.LG

Flow map learning in nonlinear vector autoregressive models: influence of the feature-library structure on the training error

Time series forecasting often requires learning nonlinear and time-delayed dependencies. A paradigmatic class of forecasting models are nonlinear vector autoregressive processes (NVAR), also known as next-generation reservoir computers (NG-RCs). These models approximate the Koopman operator on the space spanned by their explicit feature library. We consider the identifiability problem for learning Markovian nonlinear dynamical systems and show that the training error as a function of time resolution follows characteristic (pre-)asymptotic scaling laws. These laws depend on whether the feature library can represent the early Lie-series coefficients of the flow map (propagator) exactly or merely approximately. For dynamical systems governed by polynomial vector fields, we demonstrate the mechanism for NVAR/NG-RC models with monomial and Fourier feature libraries. We determine the dependence of the training error on the temporal resolution, the involved nonlinear degree, and the number of delay terms. While delay terms reduce the optimal one-step training error, they improve long-horizon forecasts only when the library provides sufficient nonlinearity. Thus, small training error coexists with weak generalization as the model class is mismatched to the true data-generating process. Numerical experiments on various chaotic dynamical systems confirm the theoretical predictions.

Markus Gross

Institute for AI Safety and Security, German Aerospace Center (DLR), Sankt Augustin and Ulm, Germany

Koopman Operator Learning Nonlinear System Identification Reservoir Computing Flow Map Learning Chaotic Time-Series Prediction Autoregressive Modeling

May 28, 2026physics.ao-ph

Learning effective Sargassum transport dynamics from limited drifter observations

Floating-material transport is influenced by unresolved processes that are often absent from available circulation products. We develop a data-driven transport-learning framework for learning effective transport corrections from limited Lagrangian observations using physically motivated ocean--atmosphere diagnostics and finite-memory representations motivated in part by inertial-particle memory effects. The diagnostic representation is analyzed through predictive and sparse symbolic-discovery approaches under leave-one-trajectory-out validation. Applications to Sargassum-following drifters in the Puerto Rico region and the Gulf Stream show that the diagnostics contain transport-relevant information beyond the baseline circulation products. Multilayer perceptron (MLP) ensembles provide flexible predictive trajectory corrections, while Sparse Identification of Nonlinear Dynamics (SINDy) tests whether instantaneous or delayed sparse symbolic transport structure can be extracted from the diagnostics. The results differ across flow regimes: (i) in Puerto Rico, delayed sparse symbolic corrections provide modest but systematic improvement; (ii) in the Gulf Stream application, dynamically useful sparse symbolic corrections remain primarily instantaneous even though delayed predictive information persists. These results support finite-memory transport effects in coarse-grained floating-material transport while also illustrating the difficulty of obtaining stable delayed sparse symbolic closures.

F. J. Beron-VEra, M. J. Olascoaga, J. Morell +1

Department of Atmospheric Sciences Rosenstiel School of Marine, Atmospheric, and Earth Science University of Miami Miami, Florida, USA · Department of Ocean Sciences Rosenstiel School of Marine, Atmospheric, and Earth Science University of Miami Miami, Florida, USA · Department of Marine Sciences University of Puerto Rico Mayagüez, Puerto Rico

Sparse Identification of Nonlinear Dynamics Nonlinear System Identification Trajectory Prediction

May 28, 2026math.DS

Learning effective models from network dynamics data with multiple initial conditions using weak form SINDy

Social systems consist of networks of individuals who influence one another through social interactions. Studying how processes evolve on these networks can help us better understand patterns of social behavior. We study a system that couples online and offline social activity and investigate how to learn effective models directly from data using Weak Form Sparse Identification of Nonlinear Dynamics (WSINDy), a method for discovering governing equations. We assess learning performance using data generated by a mean-field approximation model of a stochastic interaction process on networks and test how accurately the system can be recovered under different noise levels. Our results show that using more trajectories improves accuracy when noise is high, but only a small number of additional trajectories is needed to gain most of the benefit, with little improvement beyond that. We also learn effective ODE models from averaged stochastic data on networks. When traditional mean-field approximations fail, identifying continuum ODEs directly from stochastic processes yields efficient models that better match the data and provide deeper insight into the underlying dynamics.

Moyi Tian, Daniel A. Messenger, Vanja Dukic +2

Department of Applied Mathematics, University of Colorado, Boulder, CO 80309 United States · Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545 United States

Sparse Identification of Nonlinear Dynamics Nonlinear System Identification Computational Social Science

May 27, 2026cs.RO

Identifying Explicit Parsimonious Piece-wise Polynomial Relationships in Industrial time-series: Application to manipulator robots

This paper addresses the problem of identifying parsimonious explicit piece-wise polynomial relationships that might involve a relatively large number of raw features. The algorithm leverages a recently proposed identification algorithm that yields parsimonious implicit relationships enabling to derive normality characterization in the context of anomaly detection and localization. The algorithm proposed in this paper goes a step further by deriving explicit piece-wise representations that are built using the set of polynomials involved in the implicit representations. The framework is illustrated on the problem of identifying parsimonious explicit representations of the inverse model of a 6-axis manipulator robot. Moreover, further experiments on a 4-axis robot are also shown which are designed to investigate the generalization capability of parsimonious models compared to state-of-the-art DNNs structures, when models face unseen contexts of use.

Mazen Alamir, Sacha Clavel

Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, 38000 Grenoble, France · Staubli company. Auvergne-Rhône-Alpes, 74210, Faverges, France.

Nonlinear System Identification Inverse Dynamics

May 26, 2026cs.CV

Model discovery for dynamical systems with complex-valued product units

Discovering the governing equations of a dynamical system from observed trajectories provides deeper insight into its structure than mere prediction of future states. We present a data-driven approach to model discovery based on complex-valued product-unit networks, in which each unit represents a complex monomial and the network output is a sparse linear combination of such monomials. In contrast to established library-based methods such as SINDy, our approach does not require a predefined set of candidate functions: the relevant monomials, including those with fractional or negative exponents, are learned directly from data. Across four chaotic benchmark systems (Lorenz63, Lorenz84, the Four-Wing attractor, and a fractional variant of Lorenz63), we recover the exact governing equations in 90% of trials for the first three systems, and in 70-90% of trials for the fractional case, using at least 3000 training points. Applied to real-world human-gait accelerometer signals, the model produced stable trajectories with bounded prediction errors, corresponding to an RMSE of approximately 12-14% of the signal amplitude range over a test horizon three times longer than the training interval, demonstrating its potential for high-dimensional systems in which analytic equations are unavailable.

Martin Brückmann, Babette Dellen, Uwe Jaekel

Department of Mathematics, Informatics, and Technology, RheinAhrCampus Remagen University of Applied Sciences Koblenz Joseph-Rovan-Allee 2, 53424 Remagen, Germany

Nonlinear System Identification Complex-Valued Neural Networks Chaotic Dynamical Systems

May 26, 2026cs.LG

PIDM-DP: Physics-Informed Diffusion with Dormand-Prince Integration for Chaotic System Identification and State Reconstruction across Multiple Dynamical Regimes

Reconstructing continuous state trajectories of chaotic dynamical systems from sparse, noisy observations remains a fundamental open problem in nonlinear science. We introduce the Physics-Informed Diffusion Model with Dormand-Prince Integration (PIDM-DP), which embeds a fully differentiable 5th-order Dormand-Prince (DP-RK45) ODE integrator directly into the reverse sampling loop of a Denoising Diffusion Probabilistic Model (DDPM). At each denoising step, physics residuals are back-propagated via automatic differentiation, constraining every generated trajectory to satisfy the system's governing equations to 5th-order accuracy. A linear-scheduled guidance mechanism that ramps the physics weight from zero at high noise levels to its full value near the clean-data limit prevents the gradient explosions that cause naive physics-informed approaches to fail on stiff systems with Jacobian eigenvalues of order

O(10^3)

. Evaluated across five benchmark systems of increasing complexity 3D Lorenz, 3D Rössler, 5D Hyperchaotic, 20D Lorenz-96, and the stiff 3D Rabinovich-Fabrikant at 10% observation density with additive Gaussian noise (

σ=0.05

), PIDM-DP achieves reconstruction RMSE improvements of up to

15.4\times

over an unconstrained diffusion baseline and decisively outperforms the Ensemble Kalman Filter on stiff systems where ensemble covariance collapses. On the Rabinovich-Fabrikant out-of-distribution benchmark, PIDM-DP attains RMSE

0.1097 \pm 0.0269

versus

0.9443 \pm 0.5288

(unconstrained diffusion,

8.6\times

worse) and

0.3561 \pm 0.3040

(EnKF,

3.2\times

worse), with

p<0.001

in paired Wilcoxon tests (

N = 30

). Topological validation via the Rosenstein Lyapunov estimator confirms that PIDM-DP preserves the chaotic invariant measure.

Shailendra Dabral

Indian Institute of Technology Indore

Physics-Informed Diffusion Models Nonlinear System Identification Chaotic Dynamical Systems Physics-Informed ML Diffusion Models