Physics informed neural networks for solving highly anisotropic diffusion equations

In recent years, Physics Informed Neural Networks (PINNs) have generated considerable interest in the scientific computing community as an alternative, or potential contender, to traditional numerical discretization methods such as finite- difference, volume, and element methods, for solving partial differential equations (PDE). In PINNs, the governing equations are incorporated into a loss function as a regularization term to guide the neural network so that the solution respects the underlying physics, hence the name ‘physics informed’. In this work, we investigate the performance of PINNs in solving the highly anisotropic diffusion equation that models fluid flow in subsurface porous media. Several levels of permeability anisotropy are tested. The results show that PINNs have excellent performance when the solution is smooth regardless of the strength of permeability anisotropy. However, PINNs struggle to give adequate results when the solution has large gradients, for example, when the solution is induced by a concentrated source term. The problem is exacerbated by higher levels of permeability anisotropy. Our results highlight a few limitations in the current implementation of physics-informed neural networks for fluid flow in porous media and show that we still have ways to go before it can compete with traditional numerical methods.