Variable coefficient-informed neural network for PDE inverse problem in fluid dynamics

Abstract

Variable-coefficient equations are crucial in the field of fluid dynamics as they accurately capture the spatial and temporal properties of fluid. In many cases, there exist some constraints among the coefficients and embedding these constraints into neural networks poses a challenge. In this paper, we design a variable coefficient-informed neural network (VCINN) to address the inverse problem of variable-coefficient partial differential equation in fluid dynamics. The VCINN framework integrates the physics-informed neural network (PINN) with the constraints among multiple coefficients, encoding both constraints and physics information into the neural networks. Compared to classical PINN, VCINN enjoys such advantages as parallelization capacity, embedding constraint information and efficient hyperparameter adjustment. Through a series of examples, the capability of the approach to recover coefficients from observations has been validated. Numerical results indicate that the present method achieves higher accuracy and lower training error compared to classical PINN.