Scalable physics-informed deep generative model for solving forward and inverse stochastic differential equations

TL;DR

A scalable physics-informed deep generative model, which is capable of solving SDE problems with both high-dimensional stochastic and spatial space and 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).

Abstract

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.

Figures (14)

• Figure 1: Schematic of scalable physics-informed deep generative models (sPI-GeM) for solving SDE problems, which consists of two different deep learning models: (a) Physics-informed basis networks (PI-BasisNet), and (b) Physics-informed deep generative model (PI-GeM).
• Figure 2: sGeM for approximating stochastic processes: Eigenvalues of the covariance matrix. (a) GP with $l = 0.2$, (b) GP with $l = 0.05$, and (c) a non-Gaussian process with $l = 0.1$.
• Figure 3: sGeM for approximating stochastic process: Loss history for (a) GP with $l = 0.2$, (b) GP with $l = 0.05$, and (c) a non-Gaussian process with $l = 0.1$. $\mathcal{L}_D$: Loss for the discriminator; $\mathcal{L}_G$: Loss for the generator; $\bm{W}$: Estimation of the Wasserstein-1 distance.
• Figure 4: sGeM for approximating GP with $l = 0.05$: (a) loss histories for sGeM and PI-GANs yang2020physics; (b) eigenvalues of the covariance matrix from sGeM and PI-GANs yang2020physics.
• Figure 5: sPI-GeM for forward stochastic Helmholtz problem ($D_{\bm{\zeta}} = 52$ and $D_{\bm{x}} = 2$): Predicted (a) mean, and (b) standard deviation for $u$. Colored background with white solid line: reference solution; Black dashed line: Predictions from sPI-GeM.