Physics-informed neural network based on the finite volume method for solving forward and inverse problems

The physics-informed neural network (PINN) is a promising approach in scientific computing. However, PINNs still face significant challenges, including high training costs, weak physical constraints, and difficulties in handling multi-scale problems. To address these challenges, this paper presents a PINN based on the finite volume method (FVM), referred to as FVM-PINN. FVM-PINN replaces the partial differential equations in the loss function with discretized equations derived using the FVM. Since the discretized equations exclude derivative terms and enforce conservation principles at more collocation points, FVM-PINN avoids the computationally intensive automatic differentiation operations in the complex computational graph and also imposes more rigorous physical constraints in the loss function. To demonstrate the performance of FVM-PINN, the forward and inverse problems of lid-driven square cavity flow are studied. The results show that FVM-PINN achieves a higher accuracy than PINN while requiring only one-tenth of the training time. Notably, it can predict the subtle, multi-scale behavior of lid-driven square cavity flow at higher Reynolds numbers, which outperforms PINN. This study also investigates the influence of different discretization schemes on the performance of FVM-PINN. It is shown that the schemes based on more physically rational profile assumptions and involving more surrounding points enhance the model's compliance with physical constraints, thereby improving the model's convergence performance and prediction accuracy. Additionally, the accuracy of the discretization schemes is also important for the accuracy of the model. These findings provide the guidelines for selecting appropriate discretization schemes when constructing the loss function of FVM-PINN for different flow problems.