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Open access 2026

A Bi-Stage Gaussian Process Framework for Modeling and Predicting Nonlinear Non-Autonomous Dynamics

Learning nonparametric systems of Ordinary Differential Equations (ODEs) from noisy data is challenging, especially when the system is input-dependent. Most current nonparametric approaches focus on autonomous systems, making them unable to capture the influence of external inputs. In this paper, we introduce a Bi-stage Gaussian Process (GP) framework for non-autonomous ODEs, capable of estimating system states and their derivatives directly from noisy measurements. The proposed method adopts a purely data-driven and nonparametric formulation, relying on Gaussian process regression and numerical integration without assuming explicit parametric system models or theoretical performance guarantees. The method is demonstrated on a scalar forced ODE with amplitudes <inline-formula> <tex-math notation="LaTeX">$A \in [{0.05, 2.5}]\pi $ </tex-math></inline-formula> and frequencies <inline-formula> <tex-math notation="LaTeX">$\omega \in [{0.1, 31.6}]$ </tex-math></inline-formula>, achieving state prediction errors below 2% for high signal-to-noise ratios (SNR = 1000) and derivative errors below 5% even for noisy measurements (SNR = 30). Furthermore, the approach is applied to a continuous stirred tank reactor (CSTR) system with inlet concentrations <inline-formula> <tex-math notation="LaTeX">$C_{A0}=1.0 2.0$ </tex-math></inline-formula> mol/m3 and flow rates <inline-formula> <tex-math notation="LaTeX">$F=0.01$ </tex-math></inline-formula> m3/s, successfully estimating reaction rates with relative errors below 4% across varying noise levels (SNR <inline-formula> <tex-math notation="LaTeX">$=100~30$ </tex-math></inline-formula>). Comparative results with non-parametric ODE (npODE), Gaussian Process ODE (GPODE) and continuous-time state-space neural network (CSNN) models demonstrate that the proposed Bi-stage GP achieves superior generalization performance under varying input conditions. The results demonstrate that the proposed method is robust, accurate, and capable of generalizing to unobserved inputs, providing a reliable alternative to classical ODE modeling in noisy and complex systems.

R. Fezai, Byanne Malluhi, N. Basha et al. · 0 citations
Conference Jul 2026

Low-Order Continuous-Time Koopman Operator Learning via Physics-Informed Neural Networks

One of the main objectives in control theory is to obtain a linear representation of inherently nonlinear systems in order to leverage the analytical and theoretical tools developed for linear systems. In this context, the Koopman operator has attracted increasing interest in recent years.Koopman operator theory provides a framework in which nonlinear dynamical systems are represented by a linear operator acting on an infinite-dimensional Hilbert space. Since such an infinite-dimensional representation is not numerically tractable, numerous finite-dimensional approximation methods have been proposed. These approaches typically rely on time-series data and include extended dynamic mode decomposition as well as deep learning–based variants. In this paper, we propose an original machine-learning-based approach for the synthesis of a fixed-dimensional Koopman approximant (lifting) of continuous-time nonlinear systems. A differential state-space representation of the system (as opposed to a recurrent state model) is assumed to be available through its vector field (f). The proposed encoder departs from conventional approaches in that it does not directly output the current latent state, but instead generates samples of the latent trajectory evaluated at user-defined time instants (temporal discretization). This formulation enables the integration into the learning process of Physical & Latent Continuous Losses, enforcing consistency between the physical dynamics and the Koopman dynamics, as well as Physical & Latent Boundary Losses, ensuring consistency with the prescribed initial conditions. In parallel, we introduce a structural stability constraint on the Koopman operator. The effectiveness of the proposed methodology is demonstrated through the analysis and simulation of two polynomial dynamical systems.

M. Zodros, A. Colotti, M. Yagoubi et al. · 0 citations
Review Open access May 2026

Data-Driven Identification of Stochastic Dynamical Systems

Identifying stochastic dynamical systems from observational data remains a major challenge in applied mathematics and engineering, particularly when complex systems are influenced by random perturbations and incomplete empirical information. This comprehensive review aims to examine state-of-the-art data-driven methods for discovering governing equations, estimating parameters, and predicting the behavior of stochastic dynamical systems. The review systematically analyzes key methodological approaches, including Sparse Identification of Nonlinear Dynamics (SINDy), Dynamic Mode Decomposition (DMD) and its extensions, Koopman operator theory, neural ordinary differential equations, and Bayesian inference. Each approach is evaluated in terms of its theoretical foundations, computational requirements, robustness to noise, and applicability to different classes of stochastic systems. Drawing on numerical experiments and real-world case studies, the findings show that no single method consistently outperforms others across all scenarios. Instead, hybrid approaches that integrate physics-informed constraints with machine learning demonstrate the strongest potential for advancing data-driven system identification. The review concludes that future research should address real-time identification, uncertainty quantification, and the integration of multi-fidelity data sources to improve the reliability and scalability of stochastic system modeling. This work contributes a comprehensive framework for guiding researchers and practitioners in selecting and implementing appropriate identification methods for stochastic dynamical systems.

Rishav Jha, Kameshwar Sahani, S. K. Sahani et al. · 0 citations
Preprint Jul 2026

Gaussian behaviors and stochastic data-driven control

We propose a stochastic behavioral modeling framework, termed Gaussian behaviors, which augments a deterministic linear time-invariant (LTI) behavior with a Gaussian noise component. We show that this notion is a tractable subclass of stochastic behaviors and encompasses classical parametric stochastic LTI state-space system models as special cases. Analogously to deterministic LTI behaviors, the framework enables simple and tractable stochastic data-driven control methods. To this end, we obtain a method for prediction by conditioning the Gaussian behavior on the known part of the trajectory, which is identified directly from the sample covariance of trajectory data. Building on this method, we develop predictive control formulations that optimize over feedforward or disturbance affine feedback policies. The resulting formulations are shown to be convex. We further derive a finite-sample confidence bound on the prediction accounting for both aleatoric and epistemic uncertainty, and incorporate it into a robust control method, for which a tractable convex upper bound is obtained. Within this framework, subspace predictive control is recovered when only the mean prediction is used, while data-enabled predictive control is shown to account for the prediction uncertainty in an optimistic fashion. Numerical case studies illustrate the benefits of the proposed methods.

András Sasfi, A. Padoan, I. Markovsky et al. · 0 citations
Preprint Aug 2026

Spectral Distillation: From Nonlinear Dynamics to Linear State-Space Models

This work first learns an implicit spectral predictor using Observation Spectral Filtering using Observation Spectral Filtering, a convex method that competes with the best linear observer for the system, and applies spectral-to-LDS distillation to convert this predictor into an explicit recurrent linear dynamical system.

Liane Galanti, Devan Shah, Shlomo Fortgang et al. · 0 citations
Open access Aug 2026

An interpretable data-driven identification of dynamical systems via universal neural ordinary differential equations

Interpretable data-driven techniques are of significant research value in dynamical system identification. This paper develops an interpretable identification framework based on universal neural ordinary differential equations (UNODEs), symbolic regression, and parameter refinement. In the proposed framework, UNODEs are used as a front-end model to learn state- and time-dependent vector fields, while symbolic regression is employed to transform the learned black-box vector field into an explicit governing equation. After the symbolic structure is identified, model parameters are further refined at the trajectory level to improve the numerical consistency of the mechanistic model. Theoretical justification for the uniqueness and identifiability of the recovered model is provided via the regularized Neural ODE framework. Comparative experiments against SINDy, PNODE, and ODENet demonstrate that the proposed method is competitive on autonomous polynomial systems and more effective in recovering compact symbolic structures for non-polynomial and explicitly time-varying dynamics.

Qing-Tong Dong · 0 citations