Judicious Integration of Deep Learning for Solving PDEs

Abstract

Computing numerical solutions to partial differential equations (PDEs) has been an ongoing area of mathematical research for over one hundred years. Currently, with more powerful computers and algorithms, the world of numerical PDE solvers is more rich than ever before. The most recent developments involve incorporating machine learning, a very powerful computational technique that has found applications in nearly every science and engineering field. This dissertation explores how deep learning, in particular, can be carefully leveraged to develop and improve numerical PDE solvers. There are many ways in which one can navigate the intersection of traditional numerical solvers and deep learning, from addressing a computational bottleneck in an existing solver to completely replacing the solver with a trained network. The first area we consider is hyperbolic conservation laws, where shocks can develop in the solutions in finite time. We are interested in designing a high-order entropy stable numerical solver which requires a specialized reconstruction algorithm satisfying certain properties. As will be discussed, there are only a handful of existing reconstructions that can work, but they have their own issues. The key idea here is that the reconstruction can be replaced with a properly constrained deep-learned algorithm, while the bulk of the numerical solver and its properties, e.g. high-order accuracy, is retained. Moving on to more general PDEs, we consider a framework where we are interested in learning the underlying solution operator. Instead of using a traditional finite element-type numerical solver, we seek to develop a neural network that mimics the weak form of the PDE. In particular, we wish to emulate the Petrov-Galerkin weak formulation by having a network that learns the solution operator, which recovers the optimal solution with respect to a suitable norm. In summary, we are using mathematical theory to inform our approach of incorporating deep learning, rather than treating it as a purely data-driven black box. As we will discuss, this demonstrates interesting new prospects in the ever-expanding field at the intersection of deep learning and numerical PDEs.