  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 networks. 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. A mathematical
motivation of these generalized methods is provided, which shows that for
linear and well-posed partial differential equations, the functional form is
convex. We then derive a choice for the scaling parameter that is optimal with
respect to a measure of relative error. Because this optimal choice relies on
having full knowledge of analytical solutions, we also propose a heuristic
method to approximate this optimal choice. The proposed methods are compared
numerically to the original methods on a variety of model partial differential
equations, with the number of data points being updated adaptively. For several
problems, including high-dimensional PDEs the proposed methods are shown to
significantly enhance accuracy.
