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（GPs）的偏微分方程求解方法，通过桥接传统数值算法的理论严谨性与机器学习求解器的灵活设计，展现出巨大潜力。高斯过程方法的主要瓶颈在于协方差矩阵的求逆，其计算成本随样本数量呈三次方增长。受神经网络启发，我们提出一种融合高斯过程的小型批处理算法来求解非线性偏微分方程。该算法在每一步中仅使用一小批样本更新高斯过程模型，从而将计算成本分配到每次迭代中。通过稳定性分析和凸性论证，我们证明该小型批处理方法能以O(1/K + 1/M)的速率将误差的固有度量稳定降至零，其中K为迭代次数，M为批处理大小。数值结果表明，光滑问题可从小批处理规模中获益，而欠正则问题则需要谨慎的样本选择才能达到最优精度。