2D 3T PINNs for solving 2-D 3-T heat conduction equations based on physics-informed neural networks

Abstract

The 2-D 3-T heat conduction equations can approximate the propagation of energy in the material, as well as the energy exchange process of electrons, ions and photons. In many cases, the computational time to solve these equations accounts for a large proportion (more than 80%) of that of the entire simulation of radiation hydrodynamics. PINNs is a promising way to solve partial differential equations (PDEs). Although numerical methods have been successful in solving 2-D 3-T heat conduction equations, the PINNs method also has some advantages, such as mesh-free, suitable to high dimension and complex domain problems. But the original PINNs cannot solve the 2-D 3-T heat conduction equations to a reasonable precision. This work aims to explore which techniques need to be added to PINNs and to what extent it can address the challenges posed by strong nonlinearity and multi-scale phenomena present in 2-D 3-T heat conduction equations. Hence, we adopt guaranteed positive constraint to the outputs so that the network can be trained, give a relatively large weight to the initial loss and Dirichlet boundary loss, take the logarithm of the initial loss, use transfer learning and Fourier feature embedding to improve accuracy. We call our improved approach 2D 3T PINNs. Numerical experiments show that the relative $L_2$ and the absolute $L_{\infty }$ error between the 2D 3T PINNs prediction and the reference solution is of the order of ${10}^{-2}-{10}^{-3}$.