A Conservative and Positivity-Preserving Method for Solving Anisotropic Diffusion Equations with Deep Learning (Communications in Computational Physics, 2024)
In this paper, we propose a conservative and positivity-preserving methodto solve the anisotropic diffusion equations with the physics-informed neural network(PINN). Due to the possible complicated discontinuity of diffusion coefficients, without employing multiple neural networks, we approximate the solution and its gradients by one single neural network with a novel first-order loss formulation. It is proventhat the learned solution with this loss formulation only has the $\mathcal{O}(ε)$ flux conservation error theoretically, where the parameter $ε$ is small and user-defined, while the lossformulation with the original PDE with/without flux conservation constraints mayhave $\mathcal{O}(1)$ flux conservation error. To keep positivity with the neural network approximation, some positive functions are applied to the primal neural network solution.This loss formulation with some observation data can also be employed to identify theunknown discontinuous coefficients. Compared with the usual PINN even with thedirect flux conservation constraints, it is shown that our method can significantly improve the solution accuracy due to the better flux conservation property, and indeedpreserve the positivity strictly for the forward problems. It can predict the discontinuous diffusion coefficients accurately in the inverse problems setting.
Keywords: Anisotropic diffusion; physics-informed neural network; first-order loss; flux conservation; positivity.