Error Estimates for the Deep Ritz Method with Boundary Penalty

Johannes Müller, Marius Zeinhofer
Proceedings of Mathematical and Scientific Machine Learning, PMLR 190:215-230, 2022.

Abstract

We estimate the error of the Deep Ritz Method for linear elliptic equations. For Dirichlet boundary conditions, we estimate the error when the boundary values are imposed through the boundary penalty method. Our results apply to arbitrary sets of ansatz functions and estimate the error in dependence of the optimization accuracy, the approximation capabilities of the ansatz class and – in the case of Dirichlet boundary values – the penalization strength $\lambda$. To the best of our knowledge, our results are presently the only ones in the literature that treat the case of Dirichlet boundary conditions in full generality, i.e., without a lower order term that leads to coercivity on all of $H^1(\Omega)$. Further, we discuss the implications of our results for ansatz classes which are given through ReLU networks and the relation to existing estimates for finite element functions. For high dimensional problems our results show that the favourable approximation capabilities of neural networks for smooth functions are inherited by the Deep Ritz Method.

Cite this Paper

BibTeX
@InProceedings{pmlr-v190-muller22a, title = {Error Estimates for the Deep Ritz Method with Boundary Penalty}, author = {M\"{u}ller, Johannes and Zeinhofer, Marius}, booktitle = {Proceedings of Mathematical and Scientific Machine Learning}, pages = {215--230}, 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/muller22a/muller22a.pdf}, url = {https://proceedings.mlr.press/v190/muller22a.html}, abstract = {We estimate the error of the Deep Ritz Method for linear elliptic equations. For Dirichlet boundary conditions, we estimate the error when the boundary values are imposed through the boundary penalty method. Our results apply to arbitrary sets of ansatz functions and estimate the error in dependence of the optimization accuracy, the approximation capabilities of the ansatz class and – in the case of Dirichlet boundary values – the penalization strength $\lambda$. To the best of our knowledge, our results are presently the only ones in the literature that treat the case of Dirichlet boundary conditions in full generality, i.e., without a lower order term that leads to coercivity on all of $H^1(\Omega)$. Further, we discuss the implications of our results for ansatz classes which are given through ReLU networks and the relation to existing estimates for finite element functions. For high dimensional problems our results show that the favourable approximation capabilities of neural networks for smooth functions are inherited by the Deep Ritz Method.} }
Endnote
%0 Conference Paper %T Error Estimates for the Deep Ritz Method with Boundary Penalty %A Johannes Müller %A Marius Zeinhofer %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-muller22a %I PMLR %P 215--230 %U https://proceedings.mlr.press/v190/muller22a.html %V 190 %X We estimate the error of the Deep Ritz Method for linear elliptic equations. For Dirichlet boundary conditions, we estimate the error when the boundary values are imposed through the boundary penalty method. Our results apply to arbitrary sets of ansatz functions and estimate the error in dependence of the optimization accuracy, the approximation capabilities of the ansatz class and – in the case of Dirichlet boundary values – the penalization strength $\lambda$. To the best of our knowledge, our results are presently the only ones in the literature that treat the case of Dirichlet boundary conditions in full generality, i.e., without a lower order term that leads to coercivity on all of $H^1(\Omega)$. Further, we discuss the implications of our results for ansatz classes which are given through ReLU networks and the relation to existing estimates for finite element functions. For high dimensional problems our results show that the favourable approximation capabilities of neural networks for smooth functions are inherited by the Deep Ritz Method.
APA
Müller, J. & Zeinhofer, M.. (2022). Error Estimates for the Deep Ritz Method with Boundary Penalty. Proceedings of Mathematical and Scientific Machine Learning, in Proceedings of Machine Learning Research 190:215-230 Available from https://proceedings.mlr.press/v190/muller22a.html.

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