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Maddu, S.* ; Sturm, D.* ; Cheeseman, B.L.* ; Müller, C.L.

Learning computable models from data.

In: (14th World Congress in Computational Mechanics (WCCM), ECCOMAS Congress 2020, 11-15 January 2021, Barcelona). 2021. 1-6 ( ; 1700)
DOI
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.
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Publikationstyp Artikel: Konferenzbeitrag
Schlagwörter Differential Operators ; Eno-weno ; Neural Networks ; Surrogate Modeling
Konferenztitel 14th World Congress in Computational Mechanics (WCCM), ECCOMAS Congress 2020
Konferzenzdatum 11-15 January 2021
Konferenzort Barcelona
Quellenangaben Band: 1700, Heft: , Seiten: 1-6 Artikelnummer: , Supplement: ,