Pilot Project 4

Deep Learning Methods for Parametric PDEs

Parametric partial differential equations (PDEs) are ubiquitous and used e.g. in control, optimal design, and modeling of complex systems
Efficient methods for solving them and estimating parameters are based on model reduction (grasping the most important parameters)
Reduced models are learned either from analysis or data
Deep Learning is expected to be effective in model reduction
Objective: provide rigorous mathematics and extensive numerics to understand how to best utilize Deep Learning for parametric PDEs

NSF-DMS: Comparative Study of Finite Element and Neural Network Discretizations for Partial Differential Equations, 2021-24 (local co-PI: Siegel)
ONR MURI: Theoretical Foundations of Deep Learning, 2020-23 (local PI: DeVore, local coPIs: Foucart, Petrova)

P. Binev, A. Bonito, A. Cohen, W. Dahmen, R. DeVore, G. Petrova. Solving PDEs with incomplete information. SIAM Journal on Numerical Analysis, 62/3, 1278-1312, 2024. (doi)
P. Binev, A. Bonito, R. DeVore, G. Petrova. Optimal Learning. Calcolo 61/15, 2024. (doi)
J. W. Siegel, Q. Hong, X. Jin, W. Hao, J. Xu. Greedy training algorithms for neural networks and applications to PDEs. Journal of Computational Physics, 484, 112084, 2023. (doi)
E.J.R. Coutinho, M. Dall'Aqua, L. McClenny, M. Zhong, U. Braga-Neto, E. Gildin. Physics-informed neural networks with adaptive localized artificial viscosity. Journal of Computational Physics, 489, 112265, 2023. (doi)
A. Bonito, A. Cohen, R. DeVore, D. Guignard, P. Jantsch, G. Petrova. Nonlinear methods for model reduction. ESAIM: M2AN, 55/2, 507-531, 2021. (doi)
A. Bonito, R. DeVore, D. Guignard, P. Jantsch, G. Petrova. Polynomial approximation of anisotropic analytic functions of several variables. Constructive Approximation, 53, 319-348, 2021. (doi)

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