Learning computable models from data

Research output: Contribution to conferencesPaperContributedpeer-review

Contributors

Abstract

Numerical methods for approximately solving partial differential equations (PDE) are at the core of scientific computing. Often, this requires high-resolution or adaptive discretization grids to capture relevant spatio-temporal features in the PDE solution, e.g., in applications like turbulence, combustion, and shock propagation. Numerical approximation also requires knowing the PDE in order to construct problem-specific discretizations. Systematically deriving solution-adaptive discrete operators, however, is a current challenge. Here we present an artificial neural network architecture for data-driven learning of problem-and resolution-specific local discretizations of nonlinear PDEs. Our proposed method achieves numerically stable discretization of the operators in an unknown nonlinear PDE by spatially and temporally adaptive parametric pooling on regular Cartesian grids, and by incorporating knowledge about discrete time integration. Knowing the actual PDE is not necessary, as solution data is sufficient to train the network to learn the discrete operators. A once-trained network can be used to predict solutions of the PDE on larger spatial domains and for longer times than it was trained for, addressing the problem of PDE-constrained extrapolation from data. We present examples on long-term forecasting of hard numerical problems including equation-free forecasting of the nonlinear dynamics of the forced Burgers problem on coarse spatio-temporal grids.

Details

Original languageEnglish
Pages1-6
Number of pages6
Publication statusPublished - 2021
Peer-reviewedYes

Conference

Title14th World Congress of Computational Mechanics and ECCOMAS Congress, WCCM-ECCOMAS 2020
Duration11 - 15 January 2021
CityVirtual, Online

External IDs

ORCID /0000-0003-4414-4340/work/142252155

Keywords

ASJC Scopus subject areas

Keywords

  • Differential operators, ENO-WENO, Neural networks, Surrogate modeling