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

Parametric PDEs appear in a wide variety of problems of engineering relevance, and their repeated solution under different parametric values in the context of many-query applications represents a major computational roadblock in achieving analysis and design objectives. Perhaps one of the most challenging applications, which lies at the core of this investigation, is encountered in the context of (random) heterogeneous media in which microstructural details determine their macroscopic properties [1]. These are found in a multitude of engineering applications, such as aligned and chopped fiber composites, porous membranes, particulate composites, cellular solids, colloids, microemulsions, concrete [1]. Their microstructural properties can vary, most often randomly, at multiple length-scales [2]. Capturing this variability requires, in general, very high-dimensional representations and very fine discretizations, which in turn imply a significant cost for each solution of the governing PDEs in order to predict their response [3]. Being able to efficiently obtain accurate solutions under varying microstructures represents a core challenge that can enable the solution of various forward analysis problems such as uncertainty quantification [4, 5]. More importantly, however, it is of relevance in the context of inverse design where one attempts to identify (families of) microstructures that achieve extremal or target properties [6]. While several different tools come into play, data-driven strategies, to which our contribution belongs, have risen into prominence in recent years [7, 8] as in many cases they have produced high-throughput, forward-model surrogates which are essential for inverting the microstructure-to-property link [9].