Math @ Duke
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Publications [#361668] of Jianfeng Lu
Papers Published
- Chen, Z; Lu, J; Lu, Y, On the Representation of Solutions to Elliptic PDEs in Barron Spaces
(June, 2021)
(last updated on 2024/04/23)
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 $\epsilon$-close with respect to the $H^1$ 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 $H^1$ norm with a
dimension-explicit convergence rate.
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