A Conservative and Positivity-Preserving Method for Solving Anisotropic Diffusion Equations with Deep Learning

In this paper, we propose a conservative and positivity-preserving method to 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 proven that the learned solution with this loss formulation only has the $\mathcal{O}(\varepsilon)$ flux conservation error theoretically, where the parameter $\varepsilon$ is small and user-defined, while the loss formulation with the original PDE with/without flux conservation constraints may have $\mathcal{O}(1)$ 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 the unknown discontinuous coefficients. Compared with the usual PINN even with the direct flux conservation constraints, it is shown that our method can significantly improve the solution accuracy due to the better flux conservation property, and indeed preserve the positivity strictly for the forward problems. It can predict the discontinuous diffusion coefficients accurately in the inverse problems setting.