Solving inverse-PDE problems with physics-aware neural networks

Abstract

We propose a novel composite framework that enables finding unknown fields in the context of inverse problems for partial differential equations (PDEs). We blend the high expressibility of deep neural networks as universal function estimators with the accuracy and reliability of existing numerical algorithms for partial differential equations. Our design brings together techniques of computational mathematics, machine learning and pattern recognition under one umbrella to seamlessly incorporate any domain-specific knowledge and insights through modeling. The network is explicitly aware of the governing physics through a hard-coded PDE solver layer in contrast to existing methods that incorporate the governing equations in the loss function; this subsequently focuses the computational load to only the discovery of the hidden fields and enables incorporating more sophisticated neural network architectures in the scientific computing domain. In addition, techniques of pattern recognition and surface reconstruction are used to further represent the unknown fields in a straightforward fashion. Most importantly, our inverse-PDE solver enables effortless integration of domain-specific knowledge about the physics of the underlying fields, such as symmetries and proper basis functions. We call this architecture Blended inverse-PDE networks (hereby dubbed BiPDE networks) and demonstrate its applicability on recovering the variable diffusion coefficient in Poisson problems in one and two spatial dimensions, as well as the diffusion coefficient in the time-dependent and nonlinear Burgers equation in one dimension. We also show that this approach is robust to noise.