PDE-DKL: PDE-constrained deep kernel learning in high dimensionality

TL;DR

The paper introduces PDE-DKL, a novel framework combining deep learning and Gaussian processes under PDE constraints to efficiently solve high-dimensional PDE problems with limited data and provide uncertainty quantification.

Contribution

It proposes a PDE-constrained deep kernel learning approach that leverages low-dimensional latent representations to unify neural networks and Gaussian processes for high-dimensional PDEs.

Findings

Achieves high accuracy with limited data

Provides robust uncertainty quantification

Demonstrates scalability to high-dimensional problems

Abstract

Many physics-informed machine learning methods for PDE-based problems rely on Gaussian processes (GPs) or neural networks (NNs). However, both face limitations when data are scarce and the dimensionality is high. Although GPs are known for their robust uncertainty quantification in low-dimensional settings, their computational complexity becomes prohibitive as the dimensionality increases. In contrast, while conventional NNs can accommodate high-dimensional input, they often require extensive training data and do not offer uncertainty quantification. To address these challenges, we propose a PDE-constrained Deep Kernel Learning (PDE-DKL) framework that combines DL and GPs under explicit PDE constraints. Specifically, NNs learn a low-dimensional latent representation of the high-dimensional PDE problem, reducing the complexity of the problem. GPs then perform kernel regression subject to the governing PDEs, ensuring accurate solutions and principled uncertainty quantification, even when available data are limited. This synergy unifies the strengths of both NNs and GPs, yielding high accuracy, robust uncertainty estimates, and computational efficiency for high-dimensional PDEs. Numerical experiments demonstrate that PDE-DKL achieves high accuracy with reduced data requirements. They highlight its potential as a practical, reliable, and scalable solver for complex PDE-based applications in science and engineering.