Solving Partial Differential Equations with Deep Neural Networks

Speaker: Martin Magill (MCSC)

Abstract: Partial differential equations (PDEs) are powerful mathematical models with applications throughout science and engineering. Because they often cannot be solved exactly, numerical methods are commonly used in practice to derive approximate solutions. In this talk, I will be discussing the neural network method of solving PDEs. Interest in this technique has grown rapidly in recent years, in light of the remarkable success of deep neural networks on applications such as machine vision and natural language processing.
A growing body of empirical and theoretical evidence suggests that deep neural networks break the so-called curse of dimensionality. In particular, whereas the computational costs of most traditional numerical methods grow exponentially in the dimensionality of the PDE, the neural network method has been successfully applied to solve PDEs in tens, hundreds, and even thousands of dimensions. Despite these enticing demonstrations, however, the neural network method currently lacks the comprehensive theoretical framework of more mature techniques such as the finite difference, finite volume, or finite element techniques. Furthermore, most of the published applications of the neural network method are restricted to relatively simple PDEs. In this talk, I will share what we are learning by applying this technique to challenging problems from computational nanobiophysics, and by comparing it to traditional numerical methods.