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.

翻译：基于高斯过程的偏微分方程求解方法兼具传统数值算法理论严谨性与机器学习求解器灵活设计优势，展现出巨大应用潜力。但高斯过程方法的主要瓶颈在于协方差矩阵求逆，其计算成本随样本量呈立方增长。受神经网络启发，我们提出将迷你批次算法与高斯过程相结合以求解非线性偏微分方程。该算法每步取迷你批次样本更新高斯过程模型，从而将计算成本分配至各迭代步骤。通过稳定性分析与凸性论证，我们证明迷你批次方法能以O(1/K + 1/M)的速率将误差自然度量稳步降至零（K为迭代次数，M为批次大小）。数值实验表明，光滑问题采用小批次即可奏效，而非正则问题需谨慎选择样本以实现最优精度。