Goals and achievements

This methodologically oriented project was devoted to the development and numerical analysis of learning algorithms for (typically large) data sets exploiting a priori knowledge in terms of an auxiliary physical model formulated as partial differential equation. We developed a rigorous error analysis of supervised learning with given data augmented by pre-knowledge on the underlying physical process in terms of an (inexact) PDE model. The resulting a priori error estimates are the first results of this kind and are based on the variational structure of the PDE together with a novel nonlinear Céa-Lemma. The nonlinear Céa-Lemma also gives first theoretical insight why Galerkin discretizations with neural network ansatz functions work so well.

Publications

Carsten Gräser and Prem Alathur Srinivasan (2020) Error bounds for {PDE}-regularized learning.  Preprint arXiv:2003.06524 (submitted).

Hanna Wulkow (2020) Regularization of Elliptic Partial Differential Equations Using Neural Networks. Master thesis, FU Berlin

Hanna Wulkow and Tim Conrad and Natasa Djurdjevac Conrad and Sebastian A. Müller and Kai Nagel and Christof Schütte (2021) Prediction of Covid-19 spreading and optimal coordination of counter-measures: From microscopic to macroscopic models to Pareto fronts. PLOS One 16 (4),   DOI 10.1371/journal.pone.0249676

Margarita Kostre and Christof Schütte and Frank Noé and Mauricio del Razo Sarmina (2021)
Coupling Particle-Based Reaction-Diffusion Simulations with Reservoirs Mediated by Reaction-Diffusion PDEs. Multiscale Modeling \& Simulation 19 (4), pp. 1659 -- 1683, DOI 10.1137/20M1352739.