Learning computable models from data

Publikation: Beitrag zu KonferenzenPaperBeigetragenBegutachtung

Beitragende

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

OriginalspracheEnglisch
Seiten1-6
Seitenumfang6
PublikationsstatusVeröffentlicht - 2021
Peer-Review-StatusJa

Konferenz

Titel14th World Congress of Computational Mechanics and ECCOMAS Congress, WCCM-ECCOMAS 2020
Dauer11 - 15 Januar 2021
StadtVirtual, Online

Externe IDs

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

Schlagworte

ASJC Scopus Sachgebiete

Schlagwörter

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