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On the Representation of Solutions to Elliptic PDEs in Barron Spaces
Ziang Chen · Jianfeng Lu · Yulong Lu
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Numerical solutions to highdimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of the solutions of $d$dimensional secondorder 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 $\epsilon$close with respect to the $H^1$ norm to a Barron function. Moreover, we prove dimensionexplicit 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 twolayer neural network with respect to the $H^1$ norm with a dimensionexplicit convergence rate.
Author Information
Ziang Chen (Duke University)
Jianfeng Lu (Duke University)
Yulong Lu (University of Massachusetts Amherst)
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2021 Poster: On the Representation of Solutions to Elliptic PDEs in Barron Spaces »
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