ERROR ESTIMATES OF PHYSICS-INFORMED NEURAL NETWORKS FOR INITIAL VALUE PROBLEMS

Abstract

This paper reviews basic concepts for Physics-Informed Neural Networks (PINN) applied to the initial value problems for ordinary differential equations. In particular, using only basic calculus, we derive the error estimates where the error functions (the differences between the true solution and the approximations expressed by neural networks) are dominated by train- ing loss functions. Numerical experiments are conducted to validate our error estimates, visual- izing the relationship between the error and the training loss for various first-order differential equations and a second-order linear equation.