Physics-informed deep learning approaches have been developed to solve forward and inverse stochastic differential equation (SDE) problems with high-dimensional stochastic space. However, the existing deep learning models have difficulties solving SDEs with high-dimensional spatial space. In the present study, we propose a scalable physics-informed deep generative model (sPI-GeM), which is capable of solving SDE problems with both high-dimensional stochastic and spatial space. The sPI-GeM consists of two deep learning models, i.e., (1) physics-informed basis networks (PI-BasisNet), which are used to learn the basis functions as well as the coefficients given data on a certain stochastic process or random field, and (2) physics-informed deep generative model (PI-GeM), which learns the distribution over the coefficients obtained from the PI-BasisNet. The new samples for the learned stochastic process can then be obtained using the inner product between the output of the generator and the basis functions from the trained PI-BasisNet. The sPI-GeM addresses the scalability in the spatial space in a similar way as in the widely used dimensionality reduction technique, i.e., principal component analysis (PCA). A series of numerical experiments, including approximation of Gaussian and non-Gaussian stochastic processes, forward and inverse SDE problems, are performed to demonstrate the accuracy of the proposed model. Furthermore, we also show the scalability of the sPI-GeM in both the stochastic and spatial space using an example of a forward SDE problem with 38- and 20-dimension stochastic and spatial space, respectively.

翻译：物理信息深度学习方法已被开发用于求解具有高维随机空间的正演与反演随机微分方程问题。然而，现有深度学习模型难以求解具有高维空间域的随机微分方程。本研究提出一种可扩展的物理信息深度生成模型，该模型能够同时处理高维随机空间与高维空间域的随机微分方程问题。该模型由两个深度学习模块构成：（1）物理信息基函数网络——用于根据特定随机过程或随机场的数据学习基函数及其系数；（2）物理信息深度生成模型——用于学习从物理信息基函数网络所获系数的概率分布。通过将生成器的输出与训练后的物理信息基函数网络所得的基函数进行内积运算，即可获得已学习随机过程的新样本。该模型通过类似主成分分析这一广泛使用的降维技术的方式，实现了空间域的可扩展性。通过一系列数值实验（包括高斯与非高斯随机过程的逼近、正演与反演随机微分方程问题），验证了所提模型的准确性。此外，我们通过一个随机维度为38维、空间维度为20维的正演随机微分方程算例，展示了该模型在随机空间与物理空间的双重可扩展性。