Scaling up Deep Learning for PDE-based Models

Philipp Haehnel, Julien Monteil, Jakub Marecek, Fearghal O'Donncha

Abstract:

Across numerous applications, forecasting relies on numerical solvers for partial differential equations (PDEs). Although the use of deep-learning techniques has been proposed, the uses have been restricted by the fact the training data are obtained using PDE solvers. Thereby, the uses were limited to domains, where the PDE solver was applicable, but no further.

We present methods for training on small domains, while applying the trained models on larger domains, with consistency constraints ensuring the solutions are physically meaningful even at the boundary of the small domains. We demonstrate the results on an air-pollution forecasting model for Dublin, Ireland.

Keywords: recurrent neural networks, partial differential equation, domain decomposition, consistency constraints, advection, diffusion

TL;DR: We present RNNs for training surrogate models of PDEs, wherein consistency constraints ensure the solutions are physically meaningful, even when the training uses much smaller domains than the trained model is applied to.

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:1810.09425/code)