Multilevel CNNs for Parametric PDEs
This addresses the challenge of efficiently solving high-dimensional parametric PDEs in uncertainty quantification, offering a novel method that reduces training data and time, though it is incremental in combining existing concepts.
The authors tackled the problem of solving high-dimensional parametric PDEs by proposing a multilevel CNN architecture that approximates multigrid V-cycles, achieving substantial improvements over state-of-the-art deep learning-based solvers, with the number of weights depending logarithmically on mesh resolution.
We combine concepts from multilevel solvers for partial differential equations (PDEs) with neural network based deep learning and propose a new methodology for the efficient numerical solution of high-dimensional parametric PDEs. An in-depth theoretical analysis shows that the proposed architecture is able to approximate multigrid V-cycles to arbitrary precision with the number of weights only depending logarithmically on the resolution of the finest mesh. As a consequence, approximation bounds for the solution of parametric PDEs by neural networks that are independent on the (stochastic) parameter dimension can be derived. The performance of the proposed method is illustrated on high-dimensional parametric linear elliptic PDEs that are common benchmark problems in uncertainty quantification. We find substantial improvements over state-of-the-art deep learning-based solvers. As particularly challenging examples, random conductivity with high-dimensional non-affine Gaussian fields in 100 parameter dimensions and a random cookie problem are examined. Due to the multilevel structure of our method, the amount of training samples can be reduced on finer levels, hence significantly lowering the generation time for training data and the training time of our method.