Numerical simulation is a critical tool in analysis, design, and control of complex systems, which are usually described by partial differential equations (PDE). The traditional approach to solving these PDEs has been via their numerical approximations (for example, finite difference, finite element). Recently, advances in Scientific machine learning (SciML) has opened up the possibility of training deep networks to solve complex PDEs. Several promising SciML approaches –for example, PINNs, FNO – have recently been proposed to solve PDEs under various amounts of data availability. 

In this talk, I will discuss some of my group’s contribution to this effort in training neural PDE solvers. This discussion will cover (a) formulating and training a mesh-based neural network approach that solves for a large parametric family of PDEs, (b) how we accelerate training a large models (for mega voxel PDE predictions) via a method analogous to the multigrid technique used in numerical linear algebra, (c) approaches to solve PDEs over domains with irregularly shaped (non-rectilinear) geometric boundaries, and (d) applying these approaches for solving design problems in energy technology. Whenever possible, we perform analysis which reveals theoretical insights into several sources of error incurred in the model-building process.