A Theoretical Analysis of Deep Neural Networks and Parametric PDEs
This work provides theoretical foundations for efficient neural network approximations in computational science, addressing a bottleneck in solving parametric PDEs, though it is incremental as it builds on existing reduced basis methods.
The paper tackles the problem of approximating solution maps of parametric partial differential equations (PDEs) using ReLU neural networks, deriving upper bounds on network complexity that are significantly better than classical results by leveraging the low-dimensionality of solution manifolds and reduced bases.
We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent low-dimensionality of the solution manifold to obtain approximation rates which are significantly superior to those provided by classical neural network approximation results. Concretely, we use the existence of a small reduced basis to construct, for a large variety of parametric partial differential equations, neural networks that yield approximations of the parametric solution maps in such a way that the sizes of these networks essentially only depend on the size of the reduced basis.