2026· IEEE Access· Vol 14, pp. 118816-118831· 0 citations
Computer Science
TL;DR
A Bi-stage Gaussian Process (GP) framework for non-autonomous ODEs, capable of estimating system states and their derivatives directly from noisy measurements, providing a reliable alternative to classical ODE modeling in noisy and complex systems.
Abstract
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.
A stochastic behavioral modeling framework, termed Gaussian behaviors, which augments a deterministic linear time-invariant behavior with a Gaussian noise component is proposed, which enables simple and tractable stochastic data-driven control methods.
András Sasfi, A. Padoan, I. Markovsky et al.· arXiv.org· 0 citations
A hybrid neural--physics framework in which the known components of the ODE are kept explicit and the missing components are represented by a neural network to improve latent-state reconstruction and long-horizon prediction.
Ahmet Demirkaya, Georgios Stratis, Tales Imbiriba et al.· arXiv.org· 0 citations
Many physical, biological, and epidemiological processes are governed by ordinary differential equations (ODEs) that are nonlinear in the state variable, including logistic population growth, chemical reaction kinetics, and epidemiological compartment models. We develop a differential equation-constrained local polynomial regression (DE-constrained LPR) framework for the general first-order ODE constraint g'(x) = F(x, g(x)), where F may be any Lipschitz continuous function, extending prior work restricted to exponential and linear ODE structures. Because F is generally nonlinear in g, the Taylor coefficients of the DE1-k estimator cannot be written in closed form; instead they are obtained by successive symbolic differentiation of F, and the estimator is computed by nonlinear least squares, requiring only a single local parameter at each evaluation point regardless of polynomial degree k. We derive the asymptotic conditional bias and variance of the DE1-k estimator, propose an AIMSE-optimal bandwidth that exploits the ODE structure to avoid direct estimation of high-order derivatives, and evaluate the method in a simulation study based on logistic growth, benchmarking against the parameter cascading method of Ramsay et al. (2007) (PCODE) and classical local linear regression. The DE-constrained estimator consistently outperforms local linear regression and is competitive with PCODE even though it estimates no structural parameter of the ODE; a sensitivity analysis across growth rates shows DE-constrained estimation becomes more accurate and more robust than PCODE as the curve steepens and PCODE's parameter estimation grows less stable. These results position DE-constrained LPR as a practical nonparametric alternative to parametric ODE-fitting methods when structural parameters are difficult to identify reliably.
Data-driven modelling of nonlinear dynamical systems based on machine learning is reviewed in this paper. Nonlinear dynamics refer to all sorts of complicated and irregular phenomena in mathematics, physics, engineering and life that are not well explained by linear models. Based on the accumulation of observation data from sensors, experiments and high-fidelity simulations, machine learning has recently begun to be applied in system identification, state estimation and the discovery of governing equations. This paper systematically presents the basic ideas of dynamical systems and then introduces several data-driven methods, such as Sparse Identification of Nonlinear Dynamics (SINDy), Neural Ordinary Differential Equations (Neural ODEs) and Koopman operator theory. Based on the above research, a number of methods have been proposed to solve the problems of noise, irregular sampling, etc., in high-dimensional chaos, and their respective advantages are introduced below. At present, the problems of poor data quality, partial observability and lack of interpretability in deep learning models are well-known. In the future, Physics-Informed Machine Learning, uncertainty quantification and robust model methods will be applied in engineering. Data science and applied mathematics will be combined in this paper to offer a comprehensive introduction to the problems and models of complex systems.
Ming-Yang Wang· Theoretical and Natural Scie...· 0 citations
In this paper, we study the extension of Neural Jump ODEs to infinite-dimensional function spaces. In particular, the underlying process $X$ now takes values in $L^2(\Xi, \mathbb{R}^{d_X})$ instead of $\mathbb{R}^{d_X}$ and the Operator NJ-ODE approximates the optimal predictor of this process by producing a representative of the conditional expectation. The NJ-ODE model is a framework for online learning the optimal prediction of continuous-time stochastic processes, given discrete, possibly irregular and incomplete past observations. In a series of works, this model has been extended to deal with generic path-dependent processes, with observation noise and dependent observations, with long-term predictions, and with input-output systems. However, throughout all of these works, the underlying processes were restricted to be finite-dimensional. In particular, function-valued problems, like yield curve or volatility surface predictions, could only be handled through discretization, which inherently leads to a loss of information. In this work, we build on ideas from Neural Operator methods that allow us to extend the NJ-ODE framework to an infinite-dimensional output process. To prove convergence of the NJ-ODE to the optimal prediction process, we develop a new approximation strategy that also generalizes previous works in the finite-dimensional setting by considerably weakening the assumptions.
F. Krach, Oliver Löthgren, Josef Teichmann· arXiv.org· 0 citations
This work introduces a unified hybrid analytical–numerical–ML framework for deterministic and stochastic differential equations that provides a principled pathway toward physically consistent, scalable, and uncertainty-aware solvers for complex dynamical systems.
Suresh Kumar Sahani· Journal of Intelligent Decis...· 2 citations
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