# Simulator-free Solution of High-Dimensional Stochastic Elliptic Partial Differential Equations using Deep Neural Networks

## TL;DR

This paper introduces a novel deep neural network method that solves high-dimensional stochastic elliptic PDEs without relying on traditional forward solvers, effectively addressing the curse of dimensionality in uncertainty propagation.

## Contribution

The authors develop a physics-informed deep residual network approach that bypasses the need for deterministic solvers, enabling efficient high-dimensional uncertainty analysis of elliptic SPDEs.

## Key findings

- Successfully applied to various high-dimensional uncertainty scenarios
- Achieved accurate uncertainty propagation without traditional forward model evaluations
- Demonstrated effectiveness in inverse problems involving high-dimensional inputs

## Abstract

Stochastic partial differential equations (SPDEs) are ubiquitous in engineering and computational sciences. The stochasticity arises as a consequence of uncertainty in input parameters, constitutive relations, initial/boundary conditions, etc. Because of these functional uncertainties, the stochastic parameter space is often high-dimensional, requiring hundreds, or even thousands, of parameters to describe it. This poses an insurmountable challenge to response surface modeling since the number of forward model evaluations needed to construct an accurate surrogate grows exponentially with the dimension of the uncertain parameter space; a phenomenon referred to as the \textit{curse of dimensionality}. State-of-the-art methods for high-dimensional uncertainty propagation seek to alleviate the curse of dimensionality by performing dimensionality reduction in the uncertain parameter space. However, one still needs to perform forward model evaluations that potentially carry a very high computational burden. We propose a novel methodology for high-dimensional uncertainty propagation of elliptic SPDEs which lifts the requirement for a deterministic forward solver. Our approach is as follows. We parameterize the solution of the elliptic SPDE using a deep residual network (ResNet). In a departure from the traditional squared residual (SR) based loss function for training the ResNet, we introduce a novel physics-informed loss function derived from variational principles. Specifically, our loss function is the expectation of the energy functional of the PDE over the stochastic variables. We demonstrate our solver-free approach through various examples where the elliptic SPDE is subjected to different types of high-dimensional input uncertainties. Also, we solve high-dimensional uncertainty propagation and inverse problems.