About optimal loss function for training physics-informed neural networks under respecting causality

Understand

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.