Active Learning with Selective Time-Step Acquisition for PDEs

Yegon Kim, Hyunsu Kim, Gyeonghoon Ko, Juho Lee
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:30199-30223, 2025.

Abstract

Accurately solving partial differential equations (PDEs) is critical to understanding complex scientific and engineering phenomena, yet traditional numerical solvers are computationally expensive. Surrogate models offer a more efficient alternative, but their development is hindered by the cost of generating sufficient training data from numerical solvers. In this paper, we present a novel framework for active learning (AL) in PDE surrogate modeling that reduces this cost. Unlike the existing AL methods for PDEs that always acquire entire PDE trajectories, our approach strategically generates only the most important time steps with the numerical solver, while employing the surrogate model to approximate the remaining steps. This dramatically reduces the cost incurred by each trajectory and thus allows the active learning algorithm to try out a more diverse set of trajectories given the same budget. To accommodate this novel framework, we develop an acquisition function that estimates the utility of a set of time steps by approximating its resulting variance reduction. We demonstrate the effectiveness of our method on several benchmark PDEs, including the Burgers’ equation, Korteweg–De Vries equation, Kuramoto–Sivashinsky equation, the incompressible Navier-Stokes equation, and the compressible Navier-Stokes equation. Experiments show that our approach improves performance by large margins over the best existing method. Our method not only reduces average error but also the 99%, 95%, and 50% quantiles of error, which is rare for an AL algorithm. All in all, our approach offers a data-efficient solution to surrogate modeling for PDEs.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-kim25m, title = {Active Learning with Selective Time-Step Acquisition for {PDE}s}, author = {Kim, Yegon and Kim, Hyunsu and Ko, Gyeonghoon and Lee, Juho}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {30199--30223}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/kim25m/kim25m.pdf}, url = {https://proceedings.mlr.press/v267/kim25m.html}, abstract = {Accurately solving partial differential equations (PDEs) is critical to understanding complex scientific and engineering phenomena, yet traditional numerical solvers are computationally expensive. Surrogate models offer a more efficient alternative, but their development is hindered by the cost of generating sufficient training data from numerical solvers. In this paper, we present a novel framework for active learning (AL) in PDE surrogate modeling that reduces this cost. Unlike the existing AL methods for PDEs that always acquire entire PDE trajectories, our approach strategically generates only the most important time steps with the numerical solver, while employing the surrogate model to approximate the remaining steps. This dramatically reduces the cost incurred by each trajectory and thus allows the active learning algorithm to try out a more diverse set of trajectories given the same budget. To accommodate this novel framework, we develop an acquisition function that estimates the utility of a set of time steps by approximating its resulting variance reduction. We demonstrate the effectiveness of our method on several benchmark PDEs, including the Burgers’ equation, Korteweg–De Vries equation, Kuramoto–Sivashinsky equation, the incompressible Navier-Stokes equation, and the compressible Navier-Stokes equation. Experiments show that our approach improves performance by large margins over the best existing method. Our method not only reduces average error but also the 99%, 95%, and 50% quantiles of error, which is rare for an AL algorithm. All in all, our approach offers a data-efficient solution to surrogate modeling for PDEs.} }
Endnote
%0 Conference Paper %T Active Learning with Selective Time-Step Acquisition for PDEs %A Yegon Kim %A Hyunsu Kim %A Gyeonghoon Ko %A Juho Lee %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-kim25m %I PMLR %P 30199--30223 %U https://proceedings.mlr.press/v267/kim25m.html %V 267 %X Accurately solving partial differential equations (PDEs) is critical to understanding complex scientific and engineering phenomena, yet traditional numerical solvers are computationally expensive. Surrogate models offer a more efficient alternative, but their development is hindered by the cost of generating sufficient training data from numerical solvers. In this paper, we present a novel framework for active learning (AL) in PDE surrogate modeling that reduces this cost. Unlike the existing AL methods for PDEs that always acquire entire PDE trajectories, our approach strategically generates only the most important time steps with the numerical solver, while employing the surrogate model to approximate the remaining steps. This dramatically reduces the cost incurred by each trajectory and thus allows the active learning algorithm to try out a more diverse set of trajectories given the same budget. To accommodate this novel framework, we develop an acquisition function that estimates the utility of a set of time steps by approximating its resulting variance reduction. We demonstrate the effectiveness of our method on several benchmark PDEs, including the Burgers’ equation, Korteweg–De Vries equation, Kuramoto–Sivashinsky equation, the incompressible Navier-Stokes equation, and the compressible Navier-Stokes equation. Experiments show that our approach improves performance by large margins over the best existing method. Our method not only reduces average error but also the 99%, 95%, and 50% quantiles of error, which is rare for an AL algorithm. All in all, our approach offers a data-efficient solution to surrogate modeling for PDEs.
APA
Kim, Y., Kim, H., Ko, G. & Lee, J.. (2025). Active Learning with Selective Time-Step Acquisition for PDEs. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:30199-30223 Available from https://proceedings.mlr.press/v267/kim25m.html.

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