Online Weak-form Sparse Identification of Partial Differential Equations

Daniel A.Messenger, Emiliano Dall’Anese, David Bortz
Proceedings of Mathematical and Scientific Machine Learning, PMLR 190:241-256, 2022.

Abstract

This paper presents an online algorithm for identification of partial differential equations (PDEs) based on the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy). The algorithm is online in a sense that if performs the identification task by processing solution snapshots that arrive sequentially. The core of the method combines a weak-form discretization of candidate PDEs with an online proximal gradient descent approach to the sparse regression problem. In particular, we do not regularize the $\ell_0$-pseudo-norm, instead finding that directly applying its proximal operator (which corresponds to a hard thresholding) leads to efficient online system identification from noisy data. We demonstrate the success of the method on the Kuramoto-Sivashinsky equation, the nonlinear wave equation with time-varying wavespeed, and the linear wave equation, in one, two, and three spatial dimensions, respectively. In particular, our examples show that the method is capable of identifying and tracking systems with coefficients that vary abruptly in time, and offers a streaming alternative to problems in higher dimensions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v190-a-messenger22a, title = {Online Weak-form Sparse Identification of Partial Differential Equations}, author = {A.Messenger, Daniel and Dall'Anese, Emiliano and Bortz, David}, booktitle = {Proceedings of Mathematical and Scientific Machine Learning}, pages = {241--256}, year = {2022}, editor = {Dong, Bin and Li, Qianxiao and Wang, Lei and Xu, Zhi-Qin John}, volume = {190}, series = {Proceedings of Machine Learning Research}, month = {15--17 Aug}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v190/a-messenger22a/a-messenger22a.pdf}, url = {https://proceedings.mlr.press/v190/a-messenger22a.html}, abstract = {This paper presents an online algorithm for identification of partial differential equations (PDEs) based on the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy). The algorithm is online in a sense that if performs the identification task by processing solution snapshots that arrive sequentially. The core of the method combines a weak-form discretization of candidate PDEs with an online proximal gradient descent approach to the sparse regression problem. In particular, we do not regularize the $\ell_0$-pseudo-norm, instead finding that directly applying its proximal operator (which corresponds to a hard thresholding) leads to efficient online system identification from noisy data. We demonstrate the success of the method on the Kuramoto-Sivashinsky equation, the nonlinear wave equation with time-varying wavespeed, and the linear wave equation, in one, two, and three spatial dimensions, respectively. In particular, our examples show that the method is capable of identifying and tracking systems with coefficients that vary abruptly in time, and offers a streaming alternative to problems in higher dimensions.} }
Endnote
%0 Conference Paper %T Online Weak-form Sparse Identification of Partial Differential Equations %A Daniel A.Messenger %A Emiliano Dall’Anese %A David Bortz %B Proceedings of Mathematical and Scientific Machine Learning %C Proceedings of Machine Learning Research %D 2022 %E Bin Dong %E Qianxiao Li %E Lei Wang %E Zhi-Qin John Xu %F pmlr-v190-a-messenger22a %I PMLR %P 241--256 %U https://proceedings.mlr.press/v190/a-messenger22a.html %V 190 %X This paper presents an online algorithm for identification of partial differential equations (PDEs) based on the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy). The algorithm is online in a sense that if performs the identification task by processing solution snapshots that arrive sequentially. The core of the method combines a weak-form discretization of candidate PDEs with an online proximal gradient descent approach to the sparse regression problem. In particular, we do not regularize the $\ell_0$-pseudo-norm, instead finding that directly applying its proximal operator (which corresponds to a hard thresholding) leads to efficient online system identification from noisy data. We demonstrate the success of the method on the Kuramoto-Sivashinsky equation, the nonlinear wave equation with time-varying wavespeed, and the linear wave equation, in one, two, and three spatial dimensions, respectively. In particular, our examples show that the method is capable of identifying and tracking systems with coefficients that vary abruptly in time, and offers a streaming alternative to problems in higher dimensions.
APA
A.Messenger, D., Dall’Anese, E. & Bortz, D.. (2022). Online Weak-form Sparse Identification of Partial Differential Equations. Proceedings of Mathematical and Scientific Machine Learning, in Proceedings of Machine Learning Research 190:241-256 Available from https://proceedings.mlr.press/v190/a-messenger22a.html.

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