Neural Operator induced Gaussian Process framework for probabilistic solution of parametric partial differential equations

TL;DR

The paper addresses the lack of uncertainty quantification in neural operators for solving PDEs by proposing NOGaP, which uses a Wavelet Neural Operator as the mean function within a Gaussian Process framework. The model employs a separable Kronecker-structured covariance with Matérn kernels, enabling scalable training via NLML optimization and closed-form predictive inference. Across four nonlinear PDEs (1D Burger, 1D wave advection, 2D Darcy with a notch, and 2D Poisson), NOGaP achieves higher accuracy than standalone WNO, zero-mean GP, and Bayesian WNO, while providing reliable uncertainty estimates through 95% credible intervals. This uncertainty-informed, scalable surrogate framework is particularly valuable for data-limited, safety-critical PDE applications and can be extended by exploring alternative kernels and deeper operator architectures.

Abstract

The study of neural operators has paved the way for the development of efficient approaches for solving partial differential equations (PDEs) compared with traditional methods. However, most of the existing neural operators lack the capability to provide uncertainty measures for their predictions, a crucial aspect, especially in data-driven scenarios with limited available data. In this work, we propose a novel Neural Operator-induced Gaussian Process (NOGaP), which exploits the probabilistic characteristics of Gaussian Processes (GPs) while leveraging the learning prowess of operator learning. The proposed framework leads to improved prediction accuracy and offers a quantifiable measure of uncertainty. The proposed framework is extensively evaluated through experiments on various PDE examples, including Burger's equation, Darcy flow, non-homogeneous Poisson, and wave-advection equations. Furthermore, a comparative study with state-of-the-art operator learning algorithms is presented to highlight the advantages of NOGaP. The results demonstrate superior accuracy and expected uncertainty characteristics, suggesting the promising potential of the proposed framework.

Figures (7)

• Figure 1: Schematic of NOGaP: The illustration depicts the NOGaP framework, featuring a Mean and Covariance block. The computation of the mean function is facilitated by the Wavelet Neural Operator (WNO) block. NOGaP operates by utilizing N-training samples comprising input-output pairs alongside the grid. The corresponding N outputs encompass both the mean of the predicted samples and the associated standard deviation.
• Figure 2: 1D Burger equation with periodic boundary condition. This figure shows the ground truth and predicted solution for different initial conditions along with the $95\%$ confidence interval (grey region) associated with each prediction. The results clearly show that NOGaP can make excellent predictions for this example.
• Figure 3: NOGaP predictions for 1D Burgers' equation for different sizes of training data samples (TDS). This figure shows the ground truth and the predicted solution along with the $95\%$ confidence interval (CI). It can be observed that the predictive uncertainty decreases with an increase in training sample sizes, thus reducing the CI. It is the expected behavior in GPR.
• Figure 4: 1D Wave advection equation with periodic boundary condition. The plot illustrates the predictions of the NOGaP model, which effectively learns the mapping between input and output function spaces. The predicted solutions for three different initial conditions obtained by the proposed NOGaP framework for $u(x,t)$ at $t=0.5$ are plotted, along with a 95 % confidence interval.
• Figure 5: NOGaP predictions for 1D Wave advection equation for different training sample sizes. The plot shows the ground truth and predicted solution along with the $95\%$ CI. It can be observed that the predictive uncertainty decreases with an increasing number of training sample sizes.