Optimally weighted loss functions for solving PDEs with Neural Networks: theory, practice and improvements

Abstract:  Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural. We introduce a generalization for these methods that manifests as a scaling parameter which balances the relative importance of the different constraints imposed by partial differential equations. We discuss the theoretical motivation of this method and show its benefits in practice on a number of PDEs. We also discuss the limitations of the methodology and present ideas for improvement and future works.