Stochastic Deep-Ritz for Parametric Uncertainty Quantification

Abstract

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.

Modeling Information Diffusion in Online Social Networks with Partial Differential Equations

“…Since the above loss only involves the input a and not the solution (output) u, the physics-informed approach, in principle, does not require the knowledge of the data set D and hence has the key advantage of being "data-free" by design [15,18,29,31,34]. Nevertheless, the optimization of a physics-informed loss remains challenging and consequently limits its applicability to relatively simple problems [26,28,30].…”