Adaptive Multilevel Neural Networks for Parametric PDEs with Error Estimation

arXiv:2403.12650v1 Announce Type: cross
Abstract: To solve high-dimensional parameter-dependent partial differential equations (pPDEs), a neural network architecture is presented. It is constructed to map parameters of the model data to corresponding finite element solutions. To improve training efficiency and to enable control of the approximation error, the network mimics an adaptive finite element method (AFEM). It outputs a coarse grid solution and a series of corrections as produced in an AFEM, allowing a tracking of the error decay over successive …

abstract approximation architecture arxiv control cs.lg cs.na data differential efficiency element error map math.na network network architecture networks neural network neural networks parameters parametric solutions solve training type