Recent empirical results show that deep networks can approximate solutions to high dimensional PDEs, seemingly escaping the curse of dimensionality. However many open questions remain regarding the theoretical basis for such approximations, including the number of parameters required . In this paper, we investigate the representational power of neural networks for approximating solutions to linear elliptic PDEs with Dirichlet Boundary conditions. We prove that when a PDE's coefficients are representable by small neural networks, the parameters required to approximate its solution scale polynomially with the input dimension d and are proportional to the parameter counts of the coefficient neural networks. Our proof is based on constructing a neural network which simulates gradient descent  in an appropriate Hilbert space which converges to the solution of the PDE. Moreover, we bound the size of the neural network needed to represent each iterate in terms of the neural network representing the previous iterate, resulting in a final network whose parameters depend polynomially on d and does not depend on the volume of the domain.
Recent experiments have shown that deep networks can approximate solutions to high-dimensional PDEs, seemingly escaping the curse of dimensionality. However , questions regarding the theoretical basis for such approximations, including the required network size, remain open . In this paper, we investigate the representational power of neural networks for approximating solutions to linear elliptic PDEs with Dirichlet boundary conditions. We prove that when a PDE's coefficients are representable by small neural networks, the parameters required to approximate its solution scale polynomially with the input dimension d and proportionally to the parameter counts of the coefficient networks. To this we end, we develop a proof technique that simulates gradient descent ( in an appropriate Hilbert space ) by growing a neural network architecture whose iterates each participate as sub-networks in their (slightly larger) successors, and converge to the solution of the PDE. We bound the size of the solution, showing a polynomial dependence on d and no dependence on the volume of the domain.
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empirical results show
experiments have shown
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high dimensional
high-dimensional
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many open questions remain
, questions
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number of parameters required
required network size, remain open
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Boundary
boundary
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are proportional
proportionally
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neural networks. Our proof is based on constructing a neural network which
networks. To this we end, we develop a proof technique that
A
(
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which converges
) by growing a neural network architecture whose iterates each participate as sub-networks in their (slightly larger) successors, and converge
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Moreover, we
We
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neural network needed to represent each iterate in terms of the neural network representing the previous iterate, resulting in a final network whose parameters depend polynomially
solution, showing a polynomial dependence
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does not depend
no dependence