Deep Learning in Numerical Analysis

Abstract: The development of new classification and regression algorithms based on deep neural networks coined Deep Learning have had a dramatic impact in the areas of artificial intelligence, machine learning, and data analysis. More recently, these methods have been applied successfully to the numerical solution of partial differential equations (PDEs). However, a rigorous analysis of their potential and limitations is still largely open. In this talk we will survey recent results contributing to such an analysis. In particular I will present recent empirical and theoretical results supporting the capability of Deep Learning based methods to break the curse of dimensionality for several high dimensional PDEs, including nonlinear Black Scholes equations used in computational finance, Hamilton Jacobi Bellman equations used in optimal control, and stationary Schrödinger equations used in quantum chemistry. Despite these encouraging results, it is still largely unclear for which problem classes a Deep Learning based ansatz can be beneficial. To this end I will, in a second part, present recent work establishing fundamental limitations on the computational efficiency of Deep Learning based numerical algorithms that, in particular, confirm a previously empirically observed "theory-to-practice gap".

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysis

Audience: researchers in the topic

"Partial Differential Equations and Applications" Webinar