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

Jianhong Chen, Shihao Yang

TL;DR

EFiGP addresses the challenge of inferring parameters and trajectories for nonlinear ODEs from noisy observations by integrating physics-informed Gaussian processes with Fourier-domain constraints and eigen-decomposition. By enforcing the ODE structure in the Fourier domain and reparameterizing via a truncated spectral basis, it bypasses costly numerical solvers and achieves substantial speedups while improving trajectory reconstruction across benchmark systems. The approach offers MAP estimates with uncertainty quantification via posterior means and intervals, and demonstrates robustness to dense discretization where prior methods like MAGI struggle. This yields a scalable, interpretable framework for data-driven dynamics with broad applicability in biology, engineering, and physics.

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.

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

TL;DR

EFiGP addresses the challenge of inferring parameters and trajectories for nonlinear ODEs from noisy observations by integrating physics-informed Gaussian processes with Fourier-domain constraints and eigen-decomposition. By enforcing the ODE structure in the Fourier domain and reparameterizing via a truncated spectral basis, it bypasses costly numerical solvers and achieves substantial speedups while improving trajectory reconstruction across benchmark systems. The approach offers MAP estimates with uncertainty quantification via posterior means and intervals, and demonstrates robustness to dense discretization where prior methods like MAGI struggle. This yields a scalable, interpretable framework for data-driven dynamics with broad applicability in biology, engineering, and physics.

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.
Paper Structure (18 sections, 2 theorems, 39 equations, 4 figures, 8 tables)

This paper contains 18 sections, 2 theorems, 39 equations, 4 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. ODE Inverse Problem
  4. MAGI: Manifold-Constrained Gaussian Process Inference
  5. Spectral Decomposition
  6. Fourier Transformation
  7. EFiGP: Eigen-Fourier Physics-Informed Gaussian Process
  8. Simulation results
  9. FN system
  10. Hes1 system
  11. LV system
  12. Discussion and Conclusion
  13. supplementary materials
  14. GP Smoothing
  15. Full $d$ notation
  16. ...and 3 more sections

Key Result

Lemma 2.1

Let $\Sigma$ be a covariance matrix with eigendecomposition $\Sigma = V \Lambda V^{\top}$, where $V$ is the matrix of eigenvectors, $\Lambda$ is the diagonal matrix of eigenvalues, and $J$ is the number of non-zero eigenvalues. Consider a random vector $\bm{Z} \sim \mathcal{N}(0, I_J)$, where $I_J$ where $\lambda_i$ are the eigenvalues in $\Lambda$, $\bm{v}_i$ are the corresponding eigenvectors i

Figures (4)

  • Figure 1: Predicted trajectory from EFiGP (red solid and dashed line) and from MAGI (blue solid and dashed line) for a 1281 discretization size on the FN system with ground-truth trajectory (black solid and dashed line) and 41 observed data points.
  • Figure 2: Predicted trajectory from EFiGP (red solid, dashed and dotted line) and from MAGI (blue solid, dashed and dotted line) for a 1281 discretization size on the log-transformed Hes1 system with ground-truth trajectory (black solid, dashed and dotted line) and 41 observed data points. At 1281 discretization, MAGI failed to converge while EFiGP still produce meaningful results.
  • Figure 3: Predicted trajectory from EFiGP (red solid and dashed line) and from MAGI (blue solid and dashed line) for a 1281 discretization size on the LV system with ground-truth trajectory (black solid and dashed line) and 41 observed data points.
  • Figure 4: Single-dataset comparison between the predicted trajectory (red solid and dashed lines) and the ground-truth trajectory (black solid and dashed lines) at a discretization level of 161 (LHS) and 1,281 (RHS). Also shown is a different trajectory (blue solid and dashed lines) inferred using the estimated initial values and parameters. This comparison is conducted on a single dataset for illustrative purposes. The true parameter values are $a = 1.5$, $b = 1$, $c = 1$, and $d = 3$, with initial conditions $x_1(0) = 5$ and $x_2(0) = 0.2$. As evident from the figure, at a discretization level of 161 (LHS), the inferred parameters are more accurate. However, at a finer discretization level of 1,281 (RHS), the trajectory RMSE is lower despite greater parameter estimation errors. This highlights the phenomenon of weakly identifiable parameters, where a parameter set with higher error can still yield trajectories with improved accuracy.

Theorems & Definitions (2)

  • Lemma 2.1
  • Lemma 2.2