In my current position as postdoc at the Seminar for Applied Mathematics at ETH Zurich, Switzerland, I am interested in the approximation of functions by deep neural networks (DNNs) in the context of partial differential equations (PDEs). This includes the approximation of PDE solutions as a function of spatial coordinates, taken from a bounded, Lipschitz, polyhedral domain, as well as uncertainty quantification, where we consider a PDE containing parameters and approximate the parameter-dependent solution and other quantities of interest. As a general theme, we prove for a range of applications that DNNs can perform almost as well as the best currently known (piecewise) polynomial approximation methods.
Preprints
arxiv, preprint, Joost A.A. Opschoor, Christoph Schwab, Exponential expressivity of ReLU^k neural networks on Gevrey classes with point singularities, 2024
arxiv, preprint, Joost A.A. Opschoor, Christoph Schwab, Christos Xenophontos, Neural networks for singular perturbations, 2024
arxiv, preprint, Joost A.A. Opschoor, Christoph Schwab, Deep ReLU networks and high-order finite element methods II: Chebyshev emulation, 2023
Publications
doi, Marcello Longo, Joost A.A. Opschoor, Nico Disch, Christoph Schwab, Jakob Zech, De Rham compatible Deep Neural Network FEM, Neural Networks 165, 721--739, 2023
doi, Joost A.A. Opschoor, Constructive deep neural network approximations of weighted analytic solutions to partial differential equations in polygons, Dissertation 29278, ETH Zurich, Switzerland, 2023
doi, Carlo Marcati, Joost A.A. Opschoor, Philipp C. Petersen, Christoph Schwab, Exponential ReLU neural network approximation rates for point and edge singularities, Foundations of Computational Mathematics 23(3), 1043--1127, 2023
doi, preprint, Joost A.A. Opschoor, Christoph Schwab, Jakob Zech, Deep learning in high dimension: ReLU neural network expression for Bayesian PDE inversion, In: Optimization and Control for Partial Differential Equations: Uncertainty quantification, open and closed-loop control, and shape optimization. Eds. Roland Herzog, Matthias Heinkenschloss, Dante Kalise, Georg Stadler, Emmanuel Trélat. De Gruyter, 2022
doi, Lukas Herrmann, Joost A.A. Opschoor, Christoph Schwab, Constructive deep ReLU neural network approximation, Journal of Scientific Computing 90(2), 75, 2022
doi, Joost A.A. Opschoor, Christoph Schwab, Jakob Zech, Exponential ReLU DNN expression of holomorphic maps in high dimension, Constructive Approximation 55(1), 537--582, 2022
doi, Joost A.A. Opschoor, Philipp C. Petersen, Christoph Schwab, Deep ReLU networks and high-order finite element methods, Analysis and Applications 18(5), 715--770, 2020