Optimally weighted loss functions for solving PDEs with Neural Networks

Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise the framework of physics informed (Raissi et al., 2007). We introduce a generalization for these methods manifests as scaling parameter which balances relative importance different constraints imposed by equations. A mathematical motivation generalized is provided, shows linear and well-posed functional form convex. then derive choice optimal with respect measure error. Because this relies on having full knowledge analytical solutions, we also propose heuristic method approximate choice. The proposed are compared numerically original variety model number data points being updated adaptively. For several problems, including high-dimensional PDEs significantly enhance accuracy.