Physics-Informed Neural Networks for the Heat Equation with Source Term under Various Boundary Conditions

Abstract

Modeling of physical processes as partial differential equations (PDEs) is often carried out with computationally expensive numerical solvers. A common, and important, process to model is that of laser interaction with biological tissues. Physics-informed neural networks (PINNs) have been used to model many physical processes, though none have demonstrated an approximation involving a source term in a PDE, which modeling laser-tissue interactions requires. In this work, a numerical solver for simulating tissue interactions with lasers was surrogated using PINNs while testing various boundary conditions, one with a radiative source term involved. Models were tested using differing activation function combinations in their architectures for comparison. The best combinations of activation functions were different for cases with and without a source term, and R2 scores and average relative errors for the predictions of the best PINN models indicate that it is an accurate surrogate model for corresponding solvers. PINNs appear to be valid replacements for numerical solvers for one-dimensional tissue interactions with electromagnetic radiation.