A physics-informed kernel approach to learning the operator for parametric PDEs

Abstract

Operator networks are neural networks designed to learn operators with special emphasis on solution operators for parameterized families of partial differential equations (PDEs). Once trained, operator networks can provide a solution to a PDE more quickly than current numerical PDE solvers by several orders of magnitude. Fourier neural operators (FNOs) and deep operator networks (DeepONets) are the two primary operator networks in existence for learning the solution operator to PDEs and have mostly only been applied to two-dimensional or three-dimensional problems, due to the computational expense of training networks in higher dimensional settings. The sole exception is a model-parallel FNO, which decomposes the function input domain space. We demonstrate a neural operator network with a physics-informed integral kernel that, once trained, is able to predict skin and ocular media’s time-dependent thermal response to incident laser radiation much more rapidly than existing numerical algorithms.