Stochastic Deep-Ritz for Parametric Uncertainty Quantification

Scientific machine learning has become an increasingly popular tool for solving high dimensional differential equations and constructing surrogates of complex physical models. In this work, we propose a deep learning based numerical method for solving elliptic partial differential equations (PDE) with random coefficients. We elucidate the stochastic variational formulation for the problem by recourse to the direct method of calculus of variations. The formulation allows us to reformulate the random coefficient PDE into a stochastic optimization problem, subsequently solved by a combination of Monte Carlo sampling and deep-learning approximation. The resulting method is simple yet powerful. We carry out numerical experiments to demonstrate the efficiency and accuracy of the proposed method.