A scalable framework for learning the geometry-dependent solution operators of partial differential equations

Abstract

Solving partial differential equations (PDEs) using numerical methods is a ubiquitous task in engineering and medicine. However, the computational costs can be prohibitively high when many-query evaluations of PDE solutions on multiple geometries are needed. Here we aim to address this challenge by introducing Diffeomorphic Mapping Operator Learning (DIMON), a generic artificial intelligence framework that learns geometry-dependent solution operators of different types of PDE on a variety of geometries. We present several examples to demonstrate the performance, efficiency and scalability of the framework in learning both static and time-dependent PDEs on parameterized and non-parameterized domains; these include solving the Laplace equations, reaction–diffusion equations and a system of multiscale PDEs that characterize the electrical propagation on thousands of personalized heart digital twins. DIMON can reduce the computational costs of solution approximations on multiple geometries from hours to seconds with substantially less computational resources. This work presents an artificial intelligence framework to learn geometry-dependent solution operators of partial differential equations (PDEs). The framework enables scalable and fast approximations of PDE solutions on a variety of 3D geometries.