Physics-Aware Neural Implicit Solvers for multiscale, parametric PDEs with applications in heterogeneous media

TL;DR

This paper introduces Physics-Aware Neural Implicit Solvers (PANIS), a data-driven approach for efficiently approximating solutions to complex, multiscale, parametric PDEs in heterogeneous media without solving the PDE directly.

Contribution

The paper presents a novel framework combining probabilistic residual-based learning with physics-aware implicit solvers, enabling generalization and uncertainty quantification in high-dimensional PDE problems.

Findings

Successfully models multiscale heterogeneous media

Learns effective homogenized solutions without solving reference PDEs

Provides probabilistic surrogates with uncertainty quantification

Abstract

We propose Physics-Aware Neural Implicit Solvers (PANIS), a novel, data-driven framework for learning surrogates for parametrized Partial Differential Equations (PDEs). It consists of a probabilistic, learning objective in which weighted residuals are used to probe the PDE and provide a source of {\em virtual} data i.e. the actual PDE never needs to be solved. This is combined with a physics-aware implicit solver that consists of a much coarser, discretized version of the original PDE, which provides the requisite information bottleneck for high-dimensional problems and enables generalization in out-of-distribution settings (e.g. different boundary conditions). We demonstrate its capability in the context of random heterogeneous materials where the input parameters represent the material microstructure. We extend the framework to multiscale problems and show that a surrogate can be learned for the effective (homogenized) solution without ever solving the reference problem. We further demonstrate how the proposed framework can accommodate and generalize several existing learning objectives and architectures while yielding probabilistic surrogates that can quantify predictive uncertainty.