Team: A. Bonito, R. DeVore (lead), G. Petrova, J. Siegel

Synopsizing points

• 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

Related grants

• NSF-DMS: Comparative Study of Finite Element and Neural Network Discretizations for Partial Differential Equations, 2021-2024 (local co-PI: Siegel)

• ONR MURI: Theoretical Foundations of Deep Learning, 2020-23 (local PI: DeVore, local coPIs: Foucart, Petrova)

Recent relevant papers

• P. Binev, A. Bonito, R. DeVore, G. Petrova. Optimal Learning. (arXiv)

• J. W. Siegel, Q. Hong, X. Jin, W. Hao, J. Xu. Greedy Training Algorithms for Neural Networks and Applications to PDEs. (arXiv)

• 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. (arXiv)

• 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)