Model Reduction and Neural Networks for Parametric PDEs
This work addresses the challenge of efficiently approximating complex operators in parametric PDEs, which is incremental as it builds on existing neural network and model reduction techniques.
The authors tackled the problem of approximating input-output maps between infinite-dimensional spaces, such as PDE solution operators, by combining neural networks with model reduction, resulting in a method that is robust to discretization size and proven to converge for certain classes of maps.
We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with ideas from model reduction. This combination results in a neural network approximation which, in principle, is defined on infinite-dimensional spaces and, in practice, is robust to the dimension of finite-dimensional approximations of these spaces required for computation. For a class of input-output maps, and suitably chosen probability measures on the inputs, we prove convergence of the proposed approximation methodology. We also include numerical experiments which demonstrate the effectiveness of the method, showing convergence and robustness of the approximation scheme with respect to the size of the discretization, and compare it with existing algorithms from the literature; our examples include the mapping from coefficient to solution in a divergence form elliptic partial differential equation (PDE) problem, and the solution operator for viscous Burgers' equation.