VarNet: Variational Neural Networks for the Solution of Partial Differential Equations

Reza Khodayi-Mehr, Michael Zavlanos
Proceedings of the 2nd Conference on Learning for Dynamics and Control, PMLR 120:298-307, 2020.

Abstract

We propose a new model-based unsupervised learning method, called VarNet, for the solution of partial differential equations (PDEs) using deep neural networks. Particularly, we propose a novel loss function that relies on the variational (integral) form of PDEs as apposed to their differential form which is commonly used in the literature. Our loss function is discretization-free, highly parallelizable, and more effective in capturing the solution of PDEs since it employs lower-order derivatives and trains over measure non-zero regions of space-time. The models obtained using VarNet are smooth and do not require interpolation. They are also easily differentiable and can directly be used for control and optimization of PDEs. Finally, VarNet can straight-forwardly incorporate parametric PDE models making it a natural tool for model order reduction of PDEs.

Cite this Paper

BibTeX
@InProceedings{pmlr-v120-khodayi-mehr20a, title = {VarNet: Variational Neural Networks for the Solution of Partial Differential Equations}, author = {Khodayi-Mehr, Reza and Zavlanos, Michael}, booktitle = {Proceedings of the 2nd Conference on Learning for Dynamics and Control}, pages = {298--307}, year = {2020}, editor = {Bayen, Alexandre M. and Jadbabaie, Ali and Pappas, George and Parrilo, Pablo A. and Recht, Benjamin and Tomlin, Claire and Zeilinger, Melanie}, volume = {120}, series = {Proceedings of Machine Learning Research}, month = {10--11 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v120/khodayi-mehr20a/khodayi-mehr20a.pdf}, url = {https://proceedings.mlr.press/v120/khodayi-mehr20a.html}, abstract = {We propose a new model-based unsupervised learning method, called VarNet, for the solution of partial differential equations (PDEs) using deep neural networks. Particularly, we propose a novel loss function that relies on the variational (integral) form of PDEs as apposed to their differential form which is commonly used in the literature. Our loss function is discretization-free, highly parallelizable, and more effective in capturing the solution of PDEs since it employs lower-order derivatives and trains over measure non-zero regions of space-time. The models obtained using VarNet are smooth and do not require interpolation. They are also easily differentiable and can directly be used for control and optimization of PDEs. Finally, VarNet can straight-forwardly incorporate parametric PDE models making it a natural tool for model order reduction of PDEs.} }
Endnote
%0 Conference Paper %T VarNet: Variational Neural Networks for the Solution of Partial Differential Equations %A Reza Khodayi-Mehr %A Michael Zavlanos %B Proceedings of the 2nd Conference on Learning for Dynamics and Control %C Proceedings of Machine Learning Research %D 2020 %E Alexandre M. Bayen %E Ali Jadbabaie %E George Pappas %E Pablo A. Parrilo %E Benjamin Recht %E Claire Tomlin %E Melanie Zeilinger %F pmlr-v120-khodayi-mehr20a %I PMLR %P 298--307 %U https://proceedings.mlr.press/v120/khodayi-mehr20a.html %V 120 %X We propose a new model-based unsupervised learning method, called VarNet, for the solution of partial differential equations (PDEs) using deep neural networks. Particularly, we propose a novel loss function that relies on the variational (integral) form of PDEs as apposed to their differential form which is commonly used in the literature. Our loss function is discretization-free, highly parallelizable, and more effective in capturing the solution of PDEs since it employs lower-order derivatives and trains over measure non-zero regions of space-time. The models obtained using VarNet are smooth and do not require interpolation. They are also easily differentiable and can directly be used for control and optimization of PDEs. Finally, VarNet can straight-forwardly incorporate parametric PDE models making it a natural tool for model order reduction of PDEs.
APA
Khodayi-Mehr, R. & Zavlanos, M.. (2020). VarNet: Variational Neural Networks for the Solution of Partial Differential Equations. Proceedings of the 2nd Conference on Learning for Dynamics and Control, in Proceedings of Machine Learning Research 120:298-307 Available from https://proceedings.mlr.press/v120/khodayi-mehr20a.html.

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