Learning Physical Models that Can Respect Conservation Laws

Derek Hansen, Danielle C. Maddix, Shima Alizadeh, Gaurav Gupta, Michael W. Mahoney
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:12469-12510, 2023.

Abstract

Recent work in scientific machine learning (SciML) has focused on incorporating partial differential equation (PDE) information into the learning process. Much of this work has focused on relatively "easy” PDE operators (e.g., elliptic and parabolic), with less emphasis on relatively “hard” PDE operators (e.g., hyperbolic). Within numerical PDEs, the latter problem class requires control of a type of volume element or conservation constraint, which is known to be challenging. Delivering on the promise of SciML requires seamlessly incorporating both types of problems into the learning process. To address this issue, we propose ProbConserv, a framework for incorporating constraints into a generic SciML architecture. To do so, ProbConserv combines the integral form of a conservation law with a Bayesian update. We provide a detailed analysis of ProbConserv on learning with the Generalized Porous Medium Equation (GPME), a widely-applicable parameterized family of PDEs that illustrates the qualitative properties of both easier and harder PDEs. ProbConserv is effective for easy GPME variants, performing well with state-of-the-art competitors; and for harder GPME variants it outperforms other approaches that do not guarantee volume conservation. ProbConserv seamlessly enforces physical conservation constraints, maintains probabilistic uncertainty quantification (UQ), and deals well with shocks and heteroscedasticity. In each case, it achieves superior predictive performance on downstream tasks.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-hansen23b, title = {Learning Physical Models that Can Respect Conservation Laws}, author = {Hansen, Derek and Maddix, Danielle C. and Alizadeh, Shima and Gupta, Gaurav and Mahoney, Michael W.}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {12469--12510}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/hansen23b/hansen23b.pdf}, url = {https://proceedings.mlr.press/v202/hansen23b.html}, abstract = {Recent work in scientific machine learning (SciML) has focused on incorporating partial differential equation (PDE) information into the learning process. Much of this work has focused on relatively "easy” PDE operators (e.g., elliptic and parabolic), with less emphasis on relatively “hard” PDE operators (e.g., hyperbolic). Within numerical PDEs, the latter problem class requires control of a type of volume element or conservation constraint, which is known to be challenging. Delivering on the promise of SciML requires seamlessly incorporating both types of problems into the learning process. To address this issue, we propose ProbConserv, a framework for incorporating constraints into a generic SciML architecture. To do so, ProbConserv combines the integral form of a conservation law with a Bayesian update. We provide a detailed analysis of ProbConserv on learning with the Generalized Porous Medium Equation (GPME), a widely-applicable parameterized family of PDEs that illustrates the qualitative properties of both easier and harder PDEs. ProbConserv is effective for easy GPME variants, performing well with state-of-the-art competitors; and for harder GPME variants it outperforms other approaches that do not guarantee volume conservation. ProbConserv seamlessly enforces physical conservation constraints, maintains probabilistic uncertainty quantification (UQ), and deals well with shocks and heteroscedasticity. In each case, it achieves superior predictive performance on downstream tasks.} }
Endnote
%0 Conference Paper %T Learning Physical Models that Can Respect Conservation Laws %A Derek Hansen %A Danielle C. Maddix %A Shima Alizadeh %A Gaurav Gupta %A Michael W. Mahoney %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-hansen23b %I PMLR %P 12469--12510 %U https://proceedings.mlr.press/v202/hansen23b.html %V 202 %X Recent work in scientific machine learning (SciML) has focused on incorporating partial differential equation (PDE) information into the learning process. Much of this work has focused on relatively "easy” PDE operators (e.g., elliptic and parabolic), with less emphasis on relatively “hard” PDE operators (e.g., hyperbolic). Within numerical PDEs, the latter problem class requires control of a type of volume element or conservation constraint, which is known to be challenging. Delivering on the promise of SciML requires seamlessly incorporating both types of problems into the learning process. To address this issue, we propose ProbConserv, a framework for incorporating constraints into a generic SciML architecture. To do so, ProbConserv combines the integral form of a conservation law with a Bayesian update. We provide a detailed analysis of ProbConserv on learning with the Generalized Porous Medium Equation (GPME), a widely-applicable parameterized family of PDEs that illustrates the qualitative properties of both easier and harder PDEs. ProbConserv is effective for easy GPME variants, performing well with state-of-the-art competitors; and for harder GPME variants it outperforms other approaches that do not guarantee volume conservation. ProbConserv seamlessly enforces physical conservation constraints, maintains probabilistic uncertainty quantification (UQ), and deals well with shocks and heteroscedasticity. In each case, it achieves superior predictive performance on downstream tasks.
APA
Hansen, D., Maddix, D.C., Alizadeh, S., Gupta, G. & Mahoney, M.W.. (2023). Learning Physical Models that Can Respect Conservation Laws. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:12469-12510 Available from https://proceedings.mlr.press/v202/hansen23b.html.

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