Abstract: Solving high-dimensional partial differential equations (PDEs) is a major challenge in scientific computing. We develop a new numerical method for solving elliptic-type PDEs by adapting the Q-learning algorithm in reinforcement learning.

Using a neural tangent kernel (NTK) approach, we prove that the neural network approximator for the PDE solution, trained with the Q-PDE algorithm, converges to the trajectory of an infinite-dimensional ordinary differential equation (ODE) as the number of hidden units becomes infinite. For monotone PDE (i.e. those given by monotone operators), despite the lack of a spectral gap in the NTK, we then prove that the limit neural network, which satisfies the infinite-dimensional ODE, converges in $L^2$ to the PDE solution as the training time increases. The numerical performance of the Q-PDE algorithm is studied for several elliptic PDEs.

Based on joint work with Deqing Jiang and Justin Sirignano

Bio: Samuel is an Associate Professor in the Mathematical Institute at Oxford and the theme lead for Machine Learning in Finance at the Alan Turing Institute. He is also an associate member of the Oxford-Man Institute, a member of the Oxford-Nie Financial Big Data Lab and a Senior Research Fellow at New College.

Samuel's main research interests are in the areas of stochastic analysis and mathematical finance, in particular, in the interaction between statistical and machine learning, decision making and control and uncertainty aversion. Professor Cohen has published papers in very top journals including Annals of Applied probability, SIAM Journal on Control and Optimization, etc. and he is the co-author of the textbook Stochastic Calculus and Applications. He servers as the associate editor for the journals Stochastics, Journal of Stochastic Analysis and Applications, and Communications on Stochastic Analysis.

Meeting Recording: https://ucsb.zoom.us/rec/share/ZwydB_H0A4HMq4G585veXNgBBYyMKlbhxwsviADqCt9AQnSiJ2xV68F0q6Cxo6D7.eVgU_4zl7j3EZnml

Access Passcode: vC?n+e$7