A Mini-Batch Method for Solving Nonlinear PDEs with Gaussian Processes

TL;DR

This paper introduces a mini-batch algorithm for Gaussian process-based solutions to nonlinear PDEs, reducing computational costs and ensuring convergence through stability and convexity analysis.

Contribution

It proposes a novel mini-batch method for GPs to efficiently solve nonlinear PDEs, addressing the covariance matrix inversion bottleneck.

Findings

The method reduces error at a rate of O(1/K+1/M).

It demonstrates stable convergence for the proposed algorithm.

Computational cost per iteration is manageable with mini-batches.

Abstract

Gaussian processes (GPs) based methods for solving partial differential equations (PDEs) demonstrate great promise by bridging the gap between the theoretical rigor of traditional numerical algorithms and the flexible design of machine learning solvers. The main bottleneck of GP methods lies in the inversion of a covariance matrix, whose cost grows cubically concerning the size of samples. Drawing inspiration from neural networks, we propose a mini-batch algorithm combined with GPs to solve nonlinear PDEs. A naive deployment of a stochastic gradient descent method for solving PDEs with GPs is challenging, as the objective function in the requisite minimization problem cannot be depicted as the expectation of a finite-dimensional random function. To address this issue, we employ a mini-batch method to the corresponding infinite-dimensional minimization problem over function spaces. The algorithm takes a mini-batch of samples at each step to update the GP model. Thus, the computational cost is allotted to each iteration. Using stability analysis and convexity arguments, we show that the mini-batch method steadily reduces a natural measure of errors towards zero at the rate of $O(1/K+1/M)$, where $K$ is the number of iterations and $M$ is the batch size.