Physics Informed Neural Networks for heat conduction with phase change

TL;DR

Numerical algorithms to solve a specific Partial Differential Equation (PDE), namely the Stefan problem, using Physics Informed Neural Networks (PINNs), and compares with classical solvers for PDEs (finite differences).

Abstract

We study numerical algorithms to solve a specific Partial Differential Equation (PDE), namely the Stefan problem, using Physics Informed Neural Networks (PINNs). This problem describes the heat propagation in a liquid-solid phase change system. It implies a heat equation and a discontinuity at the interface where the phase change occurs. In the context of PINNs, this model leads to difficulties in the learning process, especially near the interface of phase change. We present different strategies that can be used in this context. We illustrate our results and compare with classical solvers for PDEs (finite differences).

Figures (14)

• Figure 1: Convergence of the finite differences scheme used to solve \ref{['eq:CN']}. The $L_2$ error is calculated relative to a reference solution obtained with a step size $h_{\rm min} = 3.9\times 10^{-5}$, i.e. Rel. $L_2$ error: $\|\theta_h - \theta_{h_{\rm min}}\|_2/\|\theta_{h_{\rm min}}\|_2$.
• Figure 2: Problem \ref{['eq:3.8']}. Left: relative $L_2$ error $\|\widehat{\theta} - \theta\|_2/\|\theta\|_2$ during training. Right: absolute error $|\widehat{\theta} - \theta|$ at the end of training (i.e. at epoch $=10^5$) .
• Figure 3: Solution of \ref{['eq:3.8']} at the end of training for $t=0.05$, $t=0.53$ and $t=1$.
• Figure 4: Absolute error $|\widehat{\theta} - \theta|$ for Eq. \ref{['eq:3.9']} at the end of training (i.e. at epoch $=10^5$) for three choices of loss weights.
• Figure 5: Solution of \ref{['eq:3.9']} at the end of training at $t=0.05$, $t=0.53$ and $t=1$.