EFiGP: Eigen-Fourier Physics-Informed Gaussian Process for Inference of Dynamic Systems

TL;DR

EFiGP is a novel Gaussian Process-based method that efficiently estimates parameters and reconstructs trajectories of nonlinear dynamical systems by integrating Fourier transforms and eigen-decomposition, improving accuracy and computational speed.

Contribution

The paper introduces EFiGP, a physics-informed Gaussian Process framework that incorporates Fourier transformation and eigen-decomposition to enhance inference of dynamical systems without numerical integration.

Findings

EFiGP achieves accurate trajectory and parameter recovery on benchmark problems.

The method significantly reduces computational cost compared to traditional approaches.

EFiGP effectively denoises data by truncating high-frequency Fourier components.

Abstract

Parameter estimation and trajectory reconstruction for data-driven dynamical systems governed by ordinary differential equations (ODEs) are essential tasks in fields such as biology, engineering, and physics. These inverse problems -- estimating ODE parameters from observational data -- are particularly challenging when the data are noisy, sparse, and the dynamics are nonlinear. We propose the Eigen-Fourier Physics-Informed Gaussian Process (EFiGP), an algorithm that integrates Fourier transformation and eigen-decomposition into a physics-informed Gaussian Process framework. This approach eliminates the need for numerical integration, significantly enhancing computational efficiency and accuracy. Built on a principled Bayesian framework, EFiGP incorporates the ODE system through probabilistic conditioning, enforcing governing equations in the Fourier domain while truncating high-frequency terms to achieve denoising and computational savings. The use of eigen-decomposition further simplifies Gaussian Process covariance operations, enabling efficient recovery of trajectories and parameters even in dense-grid settings. We validate the practical effectiveness of EFiGP on three benchmark examples, demonstrating its potential for reliable and interpretable modeling of complex dynamical systems while addressing key challenges in trajectory recovery and computational cost.