含随机参数的偏微分方程的自适应高斯过程求解器

An adaptive Gaussian process emulator for partial differential equations withstochastic parameters

摘要:

对于数值求解含随机参数的偏微分方程的问题，本文基于以高斯过程为核心的求解器，提出了一种自适应挑选训练数据的求解模型.该模型从极少的初始训练数据集出发训练高斯过程求解器，将参数池中预测方差指示变量最大的参数及其对应的偏微分方程的高精度解加入训练数据集中，然后重复上述过程，直到所训练出来的高斯过程求解器在测试数据集上达到要求的精度.此外，本文还将该自适应模型在带有二维随机参数的扩散方程上进行测试，结果表明所提出的自适应选点策略有效，模型的预测准确度随着训练数据的增加迅速提高，最终只需要40个训练数据即可在测试数据集上达到要求的精度.

Abstract:

This paper aims at numerical solution of the partial differential equations (PDEs) with stochastic parameters. We propose a Gaussian-process-based emulator which is capable of choosing the training data adaptively. This model begins with limited training data, trains the Gaussian process emulator, adds the parameters with the highest prediction variance indicator from the parameter pool, along with the corresponding high-fidelity PDE output, into the training data set, until the model achieves a desired accuracy. A 2D parametric diffusion equation is used to test the model. Numerical results demonstrate the efficiency of the model. The accuracy of the model increases rapidly with the growth of training data. Only 40 training data allow us to obtain the desired accuracy.