About optimal loss function for training physics-informed neural networks under respecting causality
This work addresses a specific optimization issue in PINNs for computational physics, offering an incremental improvement in loss function design.
The authors tackled the challenge of tuning scaling coefficients in physics-informed neural networks (PINNs) by reducing problems with differential equations and initial/boundary conditions to ones with only differential equations, enabling a single-term loss function and demonstrating accuracy in numerical experiments.
A method is presented that allows to reduce a problem described by differential equations with initial and boundary conditions to the problem described only by differential equations. The advantage of using the modified problem for physics-informed neural networks (PINNs) methodology is that it becomes possible to represent the loss function in the form of a single term associated with differential equations, thus eliminating the need to tune the scaling coefficients for the terms related to boundary and initial conditions. The weighted loss functions respecting causality were modified and new weighted loss functions based on generalized functions are derived. Numerical experiments have been carried out for a number of problems, demonstrating the accuracy of the proposed methods.