Amortized Finite Element Analysis for Fast PDE-Constrained Optimization

Tianju Xue, Alex Beatson, Sigrid Adriaenssens, Ryan Adams
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:10638-10647, 2020.

Abstract

Optimizing the parameters of partial differential equations (PDEs), i.e., PDE-constrained optimization (PDE-CO), allows us to model natural systems from observations or perform rational design of structures with complicated mechanical, thermal, or electromagnetic properties. However, PDE-CO is often computationally prohibitive due to the need to solve the PDE—typically via finite element analysis (FEA)—at each step of the optimization procedure. In this paper we propose amortized finite element analysis (AmorFEA), in which a neural network learns to produce accurate PDE solutions, while preserving many of the advantages of traditional finite element methods. This network is trained to directly minimize the potential energy from which the PDE and finite element method are derived, avoiding the need to generate costly supervised training data by solving PDEs with traditional FEA. As FEA is a variational procedure, AmorFEA is a direct analogue to popular amortized inference approaches in latent variable models, with the finite element basis acting as the variational family. AmorFEA can perform PDE-CO without the need to repeatedly solve the associated PDE, accelerating optimization when compared to a traditional workflow using FEA and the adjoint method.

Cite this Paper


BibTeX
@InProceedings{pmlr-v119-xue20a, title = {Amortized Finite Element Analysis for Fast {PDE}-Constrained Optimization}, author = {Xue, Tianju and Beatson, Alex and Adriaenssens, Sigrid and Adams, Ryan}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {10638--10647}, year = {2020}, editor = {Hal Daumé III and Aarti Singh}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/xue20a/xue20a.pdf}, url = { http://proceedings.mlr.press/v119/xue20a.html }, abstract = {Optimizing the parameters of partial differential equations (PDEs), i.e., PDE-constrained optimization (PDE-CO), allows us to model natural systems from observations or perform rational design of structures with complicated mechanical, thermal, or electromagnetic properties. However, PDE-CO is often computationally prohibitive due to the need to solve the PDE—typically via finite element analysis (FEA)—at each step of the optimization procedure. In this paper we propose amortized finite element analysis (AmorFEA), in which a neural network learns to produce accurate PDE solutions, while preserving many of the advantages of traditional finite element methods. This network is trained to directly minimize the potential energy from which the PDE and finite element method are derived, avoiding the need to generate costly supervised training data by solving PDEs with traditional FEA. As FEA is a variational procedure, AmorFEA is a direct analogue to popular amortized inference approaches in latent variable models, with the finite element basis acting as the variational family. AmorFEA can perform PDE-CO without the need to repeatedly solve the associated PDE, accelerating optimization when compared to a traditional workflow using FEA and the adjoint method.} }
Endnote
%0 Conference Paper %T Amortized Finite Element Analysis for Fast PDE-Constrained Optimization %A Tianju Xue %A Alex Beatson %A Sigrid Adriaenssens %A Ryan Adams %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-xue20a %I PMLR %P 10638--10647 %U http://proceedings.mlr.press/v119/xue20a.html %V 119 %X Optimizing the parameters of partial differential equations (PDEs), i.e., PDE-constrained optimization (PDE-CO), allows us to model natural systems from observations or perform rational design of structures with complicated mechanical, thermal, or electromagnetic properties. However, PDE-CO is often computationally prohibitive due to the need to solve the PDE—typically via finite element analysis (FEA)—at each step of the optimization procedure. In this paper we propose amortized finite element analysis (AmorFEA), in which a neural network learns to produce accurate PDE solutions, while preserving many of the advantages of traditional finite element methods. This network is trained to directly minimize the potential energy from which the PDE and finite element method are derived, avoiding the need to generate costly supervised training data by solving PDEs with traditional FEA. As FEA is a variational procedure, AmorFEA is a direct analogue to popular amortized inference approaches in latent variable models, with the finite element basis acting as the variational family. AmorFEA can perform PDE-CO without the need to repeatedly solve the associated PDE, accelerating optimization when compared to a traditional workflow using FEA and the adjoint method.
APA
Xue, T., Beatson, A., Adriaenssens, S. & Adams, R.. (2020). Amortized Finite Element Analysis for Fast PDE-Constrained Optimization. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:10638-10647 Available from http://proceedings.mlr.press/v119/xue20a.html .

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