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Mathematics Colloquium

Marios Mattheakis
Harvard University

Title: Artificial neural networks for solving differential equations
Date: Friday, March 5, 2021
Place and Time: Zoom, 3:05-3:55 pm

Abstract. There has been a wave of interest in applying machine learning to study differential equations. The universal approximation theorem states that a neural network can approximate any continuous function with arbitrary accuracy. Moreover, the obtained predictions are analytical and differentiable, making neural networks a suitable approach to solving complicated problems governed by differential equations. In contrast to conventional data-driven machine learning methods, neural network solvers are equation-driven models that construct analytical functions that satisfy a particular differential equations system. Subsequently, the optimization is a fully data-free process resulting in an unsupervised learning method. The resulting network solvers are mesh-free and can predict solutions that identically satisfy any boundary or initial conditions, and have an analytical form of the general solution, namely a solution as a function of the variables, initial and boundary conditions. A vital consequence of this is that they are differentiable, and they can be inverted. This presentation will review the formulation of neural network solvers and discuss recent advances in solving ordinary, partial, and eigenvalue differential equation problems.