Parameter Inference based on Gaussian Processes Informed by Nonlinear Partial Differential Equations

TL;DR

The paper tackles PDE parameter inference from sparse, noisy data by embedding the PDE structure into a Gaussian Process (GP) prior, yielding a solver-free inference framework. It introduces the PDE-Informed Gaussian Process (PIGP), which constrains GPs to lie on the manifold defined by the PDE and boundary/initial conditions, and extends to nonlinear PDEs via an augmentation that yields a linear system in derivatives. The approach delivers simultaneous inference for unknown parameters and PDE solutions, with uncertainty quantified via normal approximation or Hamiltonian Monte Carlo, and leverages KL-expansion for scalable dimension reduction and an efficient posterior. By incorporating initial/boundary conditions and employing tempering, PIGP achieves favorable accuracy and computational efficiency across linear and nonlinear, single- and multi-dimensional PDEs, including censored data scenarios. The method is demonstrated on diverse applications (LIDAR, Burgers’, diffusion-Brusselator) and shows substantial speedups over traditional solvers while maintaining reliable uncertainty quantification.

Abstract

Partial differential equations (PDEs) are widely used for the description of physical and engineering phenomena. Some key parameters involved in PDEs, which represent certain physical properties with important scientific interpretations, are difficult or even impossible to measure directly. Estimating these parameters from noisy and sparse experimental data of related physical quantities is an important task. Many methods for PDE parameter inference involve a large number of evaluations for numerical solutions to PDE through algorithms such as the finite element method, which can be time-consuming, especially for nonlinear PDEs. In this paper, we propose a novel method for the inference of unknown parameters in PDEs, called the PDE-Informed Gaussian Process (PIGP) based parameter inference method. Through modeling the PDE solution as a Gaussian process (GP), we derive the manifold constraints induced by the (linear) PDE structure such that, under the constraints, the GP satisfies the PDE. For nonlinear PDEs, we propose an augmentation method that transforms the nonlinear PDE into an equivalent PDE system linear in all derivatives, which our PIGP-based method can handle. The proposed method can be applied to a broad spectrum of nonlinear PDEs. The PIGP-based method can be applied to multi-dimensional PDE systems and PDE systems with unobserved components. Like conventional Bayesian approaches, the method can provide uncertainty quantification for both the unknown parameters and the PDE solution. The PIGP-based method also completely bypasses the numerical solver for PDEs. The proposed method is demonstrated through several application examples from different areas.

Figures (9)

• Figure 1: The computation time of MAP optimization using PIGP with and without augmented PDE, top left zoom-in figure plots the computation time of MAP optimization using PIGP with augmented PDE for better illustration of its nearly-linear computational time with respect to $n_{\bm{I}}$. (Online version of this figure is colored)
• Figure 2: LIDAR equation. The posterior sample obtained by HMC, posterior density estimate obtained by normal approximation.
• Figure 3: LIDAR equation. Posterior sample, posterior mean, and the true value of PDE solutions. The posterior sample and posterior mean are produced by HMC using PIGP, $n=800$.
• Figure 4: LIDAR equation. Posterior sample, posterior mean, and the true value of PDE solutions. The posterior sample and posterior mean are produced by HMC using PIGPP, $n=100$.
• Figure 5: The RMSEs of PDE solution estimation

Theorems & Definitions (13)

• Example 2.1: LIDAR Equation
• Definition 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Example 2.6: Burger's Equation
• Definition 2.7: Augmented PDE
• Remark 2.8
• Example 2.9: Counterexample: Eikonal Equation
• Remark 2.10