Operator Learning with Gaussian Processes

TL;DR

This paper introduces a hybrid Gaussian Process and Neural Network framework for operator learning, improving accuracy and enabling zero-shot predictions in solving parametric PDEs by leveraging bilinear forms and physics-informed training.

Contribution

It proposes a novel GP/NN hybrid approach that approximates bilinear forms instead of operators directly, enhancing interpretability, performance, and zero-shot capabilities in operator learning.

Findings

Improves neural operator performance using GP mean functions.

Enables zero-shot data-driven predictions without prior training.

Handles multi-output operators efficiently with product kernels.

Abstract

Operator learning focuses on approximating mappings $\mathcal{G}^\dagger:\mathcal{U} \rightarrow\mathcal{V}$ between infinite-dimensional spaces of functions, such as $u: \Omega_u\rightarrow\mathbb{R}$ and $v: \Omega_v\rightarrow\mathbb{R}$. This makes it particularly suitable for solving parametric nonlinear partial differential equations (PDEs). While most machine learning methods for operator learning rely on variants of deep neural networks (NNs), recent studies have shown that Gaussian Processes (GPs) are also competitive while offering interpretability and theoretical guarantees. In this paper, we introduce a hybrid GP/NN-based framework for operator learning that leverages the strengths of both methods. Instead of approximating the function-valued operator $\mathcal{G}^\dagger$, we use a GP to approximate its associated real-valued bilinear form $\widetilde{\mathcal{G}}^\dagger: \mathcal{U}\times\mathcal{V}^*\rightarrow\mathbb{R}.$ This bilinear form is defined by $\widetilde{\mathcal{G}}^\dagger(u,\varphi) := [\varphi,\mathcal{G}^\dagger(u)],$ which allows us to recover the operator $\mathcal{G}^\dagger$ through $\mathcal{G}^\dagger(u)(y)=\widetilde{\mathcal{G}}^\dagger(u,\delta_y).$ The GP mean function can be zero or parameterized by a neural operator and for each setting we develop a robust training mechanism based on maximum likelihood estimation (MLE) that can optionally leverage the physics involved. Numerical benchmarks show that (1) it improves the performance of a base neural operator by using it as the mean function of a GP, and (2) it enables zero-shot data-driven models for accurate predictions without prior training. Our framework also handles multi-output operators where $\mathcal{G}^\dagger:\mathcal{U} \rightarrow\prod_{s=1}^S\mathcal{V}^s$, and benefits from computational speed-ups via product kernel structures and Kronecker product matrix representations.