Closure Discovery for Coarse-Grained Partial Differential Equations Using Grid-based Reinforcement Learning

Jan-Philipp von Bassewitz, Sebastian Kaltenbach, Petros Koumoutsakos
Conference on Parsimony and Learning, PMLR 280:960-984, 2025.

Abstract

Reliable predictions of critical phenomena, such as weather, wildfires and epidemics often rely on models described by Partial Differential Equations (PDEs). However, simulations that capture the full range of spatio-temporal scales described by such PDEs are often prohibitively expensive. Consequently, coarse-grained simulations are usually deployed that adopt various heuristics and empirical closure terms to account for the missing information. We propose a novel and systematic approach for identifying closures in under-resolved PDEs using grid-based Reinforcement Learning. This formulation incorporates inductive bias and exploits locality by deploying a central policy represented efficiently by a Fully Convolutional Network (FCN). We demonstrate the capabilities and limitations of our framework through numerical solutions of the advection equation and the Burgers’ equation. Our results show accurate predictions for in- and out-of-distribution test cases as well as a significant speedup compared to resolving all scales.

Cite this Paper

BibTeX
@InProceedings{pmlr-v280-bassewitz25a, title = {Closure Discovery for Coarse-Grained Partial Differential Equations Using Grid-based Reinforcement Learning}, author = {Bassewitz, Jan-Philipp von and Kaltenbach, Sebastian and Koumoutsakos, Petros}, booktitle = {Conference on Parsimony and Learning}, pages = {960--984}, year = {2025}, editor = {Chen, Beidi and Liu, Shijia and Pilanci, Mert and Su, Weijie and Sulam, Jeremias and Wang, Yuxiang and Zhu, Zhihui}, volume = {280}, series = {Proceedings of Machine Learning Research}, month = {24--27 Mar}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v280/main/assets/bassewitz25a/bassewitz25a.pdf}, url = {https://proceedings.mlr.press/v280/bassewitz25a.html}, abstract = {Reliable predictions of critical phenomena, such as weather, wildfires and epidemics often rely on models described by Partial Differential Equations (PDEs). However, simulations that capture the full range of spatio-temporal scales described by such PDEs are often prohibitively expensive. Consequently, coarse-grained simulations are usually deployed that adopt various heuristics and empirical closure terms to account for the missing information. We propose a novel and systematic approach for identifying closures in under-resolved PDEs using grid-based Reinforcement Learning. This formulation incorporates inductive bias and exploits locality by deploying a central policy represented efficiently by a Fully Convolutional Network (FCN). We demonstrate the capabilities and limitations of our framework through numerical solutions of the advection equation and the Burgers’ equation. Our results show accurate predictions for in- and out-of-distribution test cases as well as a significant speedup compared to resolving all scales.} }
Endnote
%0 Conference Paper %T Closure Discovery for Coarse-Grained Partial Differential Equations Using Grid-based Reinforcement Learning %A Jan-Philipp von Bassewitz %A Sebastian Kaltenbach %A Petros Koumoutsakos %B Conference on Parsimony and Learning %C Proceedings of Machine Learning Research %D 2025 %E Beidi Chen %E Shijia Liu %E Mert Pilanci %E Weijie Su %E Jeremias Sulam %E Yuxiang Wang %E Zhihui Zhu %F pmlr-v280-bassewitz25a %I PMLR %P 960--984 %U https://proceedings.mlr.press/v280/bassewitz25a.html %V 280 %X Reliable predictions of critical phenomena, such as weather, wildfires and epidemics often rely on models described by Partial Differential Equations (PDEs). However, simulations that capture the full range of spatio-temporal scales described by such PDEs are often prohibitively expensive. Consequently, coarse-grained simulations are usually deployed that adopt various heuristics and empirical closure terms to account for the missing information. We propose a novel and systematic approach for identifying closures in under-resolved PDEs using grid-based Reinforcement Learning. This formulation incorporates inductive bias and exploits locality by deploying a central policy represented efficiently by a Fully Convolutional Network (FCN). We demonstrate the capabilities and limitations of our framework through numerical solutions of the advection equation and the Burgers’ equation. Our results show accurate predictions for in- and out-of-distribution test cases as well as a significant speedup compared to resolving all scales.
APA
Bassewitz, J.v., Kaltenbach, S. & Koumoutsakos, P.. (2025). Closure Discovery for Coarse-Grained Partial Differential Equations Using Grid-based Reinforcement Learning. Conference on Parsimony and Learning, in Proceedings of Machine Learning Research 280:960-984 Available from https://proceedings.mlr.press/v280/bassewitz25a.html.

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