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

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. 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. Numerical results show that smooth problems benefit from a small batch size, while less regular problems require careful sample selection for optimal accuracy.