Unified finite-volume physics informed neural networks to solve the heterogeneous partial differential equations

Abstract

The emerging physics informed neural network (PINN) has been recently applied to a wide range of mathematical problems. It is promising to precisely solve the partial differential equations (PDEs) in a fast, flexible manner. Whereas, PINN struggles with poor accuracy and costly computation in case of heterogeneous PDE coefficients. To mitigate these issues, a new PINN, which is known as the unified finite volume PINN (UFV-PINN), is proposed to unify the sub-domain decomposition, finite volume discretization, and conventional numerical solvers. The output by neural network (NN) over the boundaries of agglomerated sub-domains functions as boundary conditions (BCs) for UFV-PINN training. In this connection, the customized differentiable conventional numerical solver further solves the PDEs. The discrepancy between NN prediction and the conventional numerical solution within the sub-domains is taken as the novel training loss, enforcing the conservation law of PDE. For illustration, the Poisson and advection–diffusion equations (ADE) are solved, which are classical but still challenging to PINN in the presence of heterogeneity. Numerical experiments are conducted to compare the performance of the proposed UFV-PINN and the standard PINN, both in terms of accuracy and efficiency. Results indicate that UFV-PINN attains remarkable accuracy improvement with less computation time. Hessian spectrum analysis indicates that the loss Hessian matrix of UFV-PINN is more inclined to be positive definite, highlighting its superior performance over the standard PINN. It is the first time that the FV numerical solver is seamlessly embedded into the PINN training for performance improvement.