On the Representation of Solutions to Elliptic PDEs in Barron Spaces

Ziang Chen; Jianfeng Lu; Yulong Lu

On the Representation of Solutions to Elliptic PDEs in Barron Spaces

Ziang Chen
, Jianfeng Lu
, Yulong Lu

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Abstract

Numerical solutions to high-dimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of the solutions of d-dimensional second-order elliptic PDEs in the Barron space, that is a set of functions admitting the integral of certain parametric ridge function against a probability measure on the parameters. We prove under some appropriate assumptions that if the coefficients and the source term of the elliptic PDE lie in Barron spaces, then the solution of the PDE is ǫ-close with respect to the H¹ norm to a Barron function. Moreover, we prove dimension-explicit bounds for the Barron norm of this approximate solution, depending at most polynomially on the dimension d of the PDE. As a direct consequence of the complexity estimates, the solution of the PDE can be approximated on any bounded domain by a two-layer neural network with respect to the H¹ norm with a dimension-explicit convergence rate.

Original language	English (US)
Title of host publication	Advances in Neural Information Processing Systems 34 - 35th Conference on Neural Information Processing Systems, NeurIPS 2021
Editors	Marc'Aurelio Ranzato, Alina Beygelzimer, Yann Dauphin, Percy S. Liang, Jenn Wortman Vaughan
Publisher	Neural information processing systems foundation
Pages	6454-6465
Number of pages	12
ISBN (Electronic)	9781713845393
State	Published - 2021
Externally published	Yes
Event	35th Conference on Neural Information Processing Systems, NeurIPS 2021 - Virtual, Online Duration: Dec 6 2021 → Dec 14 2021

Publication series

Name	Advances in Neural Information Processing Systems
Volume	8
ISSN (Print)	1049-5258

Conference

Conference	35th Conference on Neural Information Processing Systems, NeurIPS 2021
City	Virtual, Online
Period	12/6/21 → 12/14/21

Bibliographical note

Funding Information:
The work of Z.C. and J.L. is supported in part by the National Science Foundation via grants DMS-2012286 and CCF-1934964. Y.L. thanks the National Science foundation for its support through the award DMS-2107934.

Publisher Copyright:
© 2021 Neural information processing systems foundation. All rights reserved.

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Chen, Z., Lu, J., & Lu, Y. (2021). On the Representation of Solutions to Elliptic PDEs in Barron Spaces. In MA. Ranzato, A. Beygelzimer, Y. Dauphin, P. S. Liang, & J. Wortman Vaughan (Eds.), Advances in Neural Information Processing Systems 34 - 35th Conference on Neural Information Processing Systems, NeurIPS 2021 (pp. 6454-6465). (Advances in Neural Information Processing Systems; Vol. 8). Neural information processing systems foundation.

On the Representation of Solutions to Elliptic PDEs in Barron Spaces. / Chen, Ziang; Lu, Jianfeng; Lu, Yulong.
Advances in Neural Information Processing Systems 34 - 35th Conference on Neural Information Processing Systems, NeurIPS 2021. ed. / Marc'Aurelio Ranzato; Alina Beygelzimer; Yann Dauphin; Percy S. Liang; Jenn Wortman Vaughan. Neural information processing systems foundation, 2021. p. 6454-6465 (Advances in Neural Information Processing Systems; Vol. 8).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Chen, Z, Lu, J & Lu, Y 2021, On the Representation of Solutions to Elliptic PDEs in Barron Spaces. in MA Ranzato, A Beygelzimer, Y Dauphin, PS Liang & J Wortman Vaughan (eds), Advances in Neural Information Processing Systems 34 - 35th Conference on Neural Information Processing Systems, NeurIPS 2021. Advances in Neural Information Processing Systems, vol. 8, Neural information processing systems foundation, pp. 6454-6465, 35th Conference on Neural Information Processing Systems, NeurIPS 2021, Virtual, Online, 12/6/21.

Chen Z, Lu J, Lu Y. On the Representation of Solutions to Elliptic PDEs in Barron Spaces. In Ranzato MA, Beygelzimer A, Dauphin Y, Liang PS, Wortman Vaughan J, editors, Advances in Neural Information Processing Systems 34 - 35th Conference on Neural Information Processing Systems, NeurIPS 2021. Neural information processing systems foundation. 2021. p. 6454-6465. (Advances in Neural Information Processing Systems).

Chen, Ziang ; Lu, Jianfeng ; Lu, Yulong. / On the Representation of Solutions to Elliptic PDEs in Barron Spaces. Advances in Neural Information Processing Systems 34 - 35th Conference on Neural Information Processing Systems, NeurIPS 2021. editor / Marc'Aurelio Ranzato ; Alina Beygelzimer ; Yann Dauphin ; Percy S. Liang ; Jenn Wortman Vaughan. Neural information processing systems foundation, 2021. pp. 6454-6465 (Advances in Neural Information Processing Systems).

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abstract = "Numerical solutions to high-dimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of the solutions of d-dimensional second-order elliptic PDEs in the Barron space, that is a set of functions admitting the integral of certain parametric ridge function against a probability measure on the parameters. We prove under some appropriate assumptions that if the coefficients and the source term of the elliptic PDE lie in Barron spaces, then the solution of the PDE is ǫ-close with respect to the H1 norm to a Barron function. Moreover, we prove dimension-explicit bounds for the Barron norm of this approximate solution, depending at most polynomially on the dimension d of the PDE. As a direct consequence of the complexity estimates, the solution of the PDE can be approximated on any bounded domain by a two-layer neural network with respect to the H1 norm with a dimension-explicit convergence rate.",

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Y1 - 2021

N2 - Numerical solutions to high-dimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of the solutions of d-dimensional second-order elliptic PDEs in the Barron space, that is a set of functions admitting the integral of certain parametric ridge function against a probability measure on the parameters. We prove under some appropriate assumptions that if the coefficients and the source term of the elliptic PDE lie in Barron spaces, then the solution of the PDE is ǫ-close with respect to the H1 norm to a Barron function. Moreover, we prove dimension-explicit bounds for the Barron norm of this approximate solution, depending at most polynomially on the dimension d of the PDE. As a direct consequence of the complexity estimates, the solution of the PDE can be approximated on any bounded domain by a two-layer neural network with respect to the H1 norm with a dimension-explicit convergence rate.

AB - Numerical solutions to high-dimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of the solutions of d-dimensional second-order elliptic PDEs in the Barron space, that is a set of functions admitting the integral of certain parametric ridge function against a probability measure on the parameters. We prove under some appropriate assumptions that if the coefficients and the source term of the elliptic PDE lie in Barron spaces, then the solution of the PDE is ǫ-close with respect to the H1 norm to a Barron function. Moreover, we prove dimension-explicit bounds for the Barron norm of this approximate solution, depending at most polynomially on the dimension d of the PDE. As a direct consequence of the complexity estimates, the solution of the PDE can be approximated on any bounded domain by a two-layer neural network with respect to the H1 norm with a dimension-explicit convergence rate.

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On the Representation of Solutions to Elliptic PDEs in Barron Spaces

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