Physics-informed neural network based on control volumes for solving time-independent problems

Physics-informed neural networks (PINNs) have been employed as a new type of solver of partial differential equations (PDEs). However, PINNs suffer from two limitations that impede their further development. First, PINNs exhibit weak physical constraints that may result in unsatisfactory results for complex physical problems. Second, the differential operation using automatic differentiation (AD) in the loss function may contaminate backpropagated gradients hindering the convergence of neural networks. To address these issues and improve the ability of PINNs, this paper introduces a novel PINN, referred to as CV-PINN, based on control volumes with the collocation points as their geometric centers. In CV-PINN, the physical laws are incorporated in a reformulated loss function in the form of discretized algebraic equations derived by integrating the PDEs over the control volumes by means of the finite volume method (FVM). In this way, the physical constraints are transformed from a single local collocation point to a control volume. Furthermore, the use of algebraic discretized equations in the loss function eliminates the derivative terms and, thereby, avoids the differential operation using AD. To validate the performance of CV-PINN, several benchmark problems are solved. CV-PINN is first used to solve Poisson's equation and the Helmholtz equation in square and irregular domains, respectively. CV-PINN is then used to simulate the lid-driven cavity flow problem. The results demonstrate that CV-PINN can precisely predict the velocity distributions and the primary vortex. The numerical experiments demonstrate that enhanced physical constraints of CV-PINN improve its prediction performance in solving different PDEs.