Composing Partial Differential Equations with Physics-Aware Neural Networks

Matthias Karlbauer, Timothy Praditia, Sebastian Otte, Sergey Oladyshkin, Wolfgang Nowak, Martin V. Butz
Proceedings of the 39th International Conference on Machine Learning, PMLR 162:10773-10801, 2022.

Abstract

We introduce a compositional physics-aware FInite volume Neural Network (FINN) for learning spatiotemporal advection-diffusion processes. FINN implements a new way of combining the learning abilities of artificial neural networks with physical and structural knowledge from numerical simulation by modeling the constituents of partial differential equations (PDEs) in a compositional manner. Results on both one- and two-dimensional PDEs (Burgers’, diffusion-sorption, diffusion-reaction, Allen{–}Cahn) demonstrate FINN’s superior modeling accuracy and excellent out-of-distribution generalization ability beyond initial and boundary conditions. With only one tenth of the number of parameters on average, FINN outperforms pure machine learning and other state-of-the-art physics-aware models in all cases{—}often even by multiple orders of magnitude. Moreover, FINN outperforms a calibrated physical model when approximating sparse real-world data in a diffusion-sorption scenario, confirming its generalization abilities and showing explanatory potential by revealing the unknown retardation factor of the observed process.

Cite this Paper

BibTeX
@InProceedings{pmlr-v162-karlbauer22a, title = {Composing Partial Differential Equations with Physics-Aware Neural Networks}, author = {Karlbauer, Matthias and Praditia, Timothy and Otte, Sebastian and Oladyshkin, Sergey and Nowak, Wolfgang and Butz, Martin V.}, booktitle = {Proceedings of the 39th International Conference on Machine Learning}, pages = {10773--10801}, year = {2022}, editor = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan}, volume = {162}, series = {Proceedings of Machine Learning Research}, month = {17--23 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v162/karlbauer22a/karlbauer22a.pdf}, url = {https://proceedings.mlr.press/v162/karlbauer22a.html}, abstract = {We introduce a compositional physics-aware FInite volume Neural Network (FINN) for learning spatiotemporal advection-diffusion processes. FINN implements a new way of combining the learning abilities of artificial neural networks with physical and structural knowledge from numerical simulation by modeling the constituents of partial differential equations (PDEs) in a compositional manner. Results on both one- and two-dimensional PDEs (Burgers’, diffusion-sorption, diffusion-reaction, Allen{–}Cahn) demonstrate FINN’s superior modeling accuracy and excellent out-of-distribution generalization ability beyond initial and boundary conditions. With only one tenth of the number of parameters on average, FINN outperforms pure machine learning and other state-of-the-art physics-aware models in all cases{—}often even by multiple orders of magnitude. Moreover, FINN outperforms a calibrated physical model when approximating sparse real-world data in a diffusion-sorption scenario, confirming its generalization abilities and showing explanatory potential by revealing the unknown retardation factor of the observed process.} }
Endnote
%0 Conference Paper %T Composing Partial Differential Equations with Physics-Aware Neural Networks %A Matthias Karlbauer %A Timothy Praditia %A Sebastian Otte %A Sergey Oladyshkin %A Wolfgang Nowak %A Martin V. Butz %B Proceedings of the 39th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2022 %E Kamalika Chaudhuri %E Stefanie Jegelka %E Le Song %E Csaba Szepesvari %E Gang Niu %E Sivan Sabato %F pmlr-v162-karlbauer22a %I PMLR %P 10773--10801 %U https://proceedings.mlr.press/v162/karlbauer22a.html %V 162 %X We introduce a compositional physics-aware FInite volume Neural Network (FINN) for learning spatiotemporal advection-diffusion processes. FINN implements a new way of combining the learning abilities of artificial neural networks with physical and structural knowledge from numerical simulation by modeling the constituents of partial differential equations (PDEs) in a compositional manner. Results on both one- and two-dimensional PDEs (Burgers’, diffusion-sorption, diffusion-reaction, Allen{–}Cahn) demonstrate FINN’s superior modeling accuracy and excellent out-of-distribution generalization ability beyond initial and boundary conditions. With only one tenth of the number of parameters on average, FINN outperforms pure machine learning and other state-of-the-art physics-aware models in all cases{—}often even by multiple orders of magnitude. Moreover, FINN outperforms a calibrated physical model when approximating sparse real-world data in a diffusion-sorption scenario, confirming its generalization abilities and showing explanatory potential by revealing the unknown retardation factor of the observed process.
APA
Karlbauer, M., Praditia, T., Otte, S., Oladyshkin, S., Nowak, W. & Butz, M.V.. (2022). Composing Partial Differential Equations with Physics-Aware Neural Networks. Proceedings of the 39th International Conference on Machine Learning, in Proceedings of Machine Learning Research 162:10773-10801 Available from https://proceedings.mlr.press/v162/karlbauer22a.html.

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