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

TL;DR

It is shown that a surrogate can be learned for the effective (homogenized) solution without ever solving the reference problem and extended to multiscale problems and shown that a surrogate can be learned for the effective (homogenized) solution without ever solving the reference problem.

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.

Figures (10)

• Figure 1: Schematic illustration of the proposed, physics-aware architecture for parametrizing the mean solution $\boldsymbol{\mu}_{\boldsymbol{\psi}} (\boldsymbol{x})$ as described in subsection \ref{['sec:ApproximatingDensity']}.
• Figure 2: Convergence of the ELBO $\mathcal{F}$ for PANIS, when trained on microstructures with $VF=50\%$ and BCs $u_0=0$ (left). Illustration of 100 and 500 randomly selected RBF-based weight functions $w_j$ according to algorithm \ref{['alg:RandomResidual']} (middle-right). Each one of these $w_j$'s corresponds to a single weighted residual.
• Figure 3: Indicative pairs of the full permeability field $c(\boldsymbol{s}; \boldsymbol{x})$ (generated for $l=0.05$ as described in section \ref{['sub:2D_Darcy']}) and the learned coarse-grained input $\boldsymbol{X}$.
• Figure 4: Predictive accuracy of FNO and PANIS when trained and tested on microstructures with $VF=50 ~\%$. The right column shows a one-dimensional slice of the solution along the vertical line from (0.5,0) to (0.5, 1.0). The shaded blue area corresponds to $\pm 2$ posterior standard deviations computed as described in Algorithm \ref{['alg:MakingPredictions']}.
• Figure 5: Comparison between PANIS and PINO.