Fractional conservative variational physics-informed neural networks: A method for solving Stokes equations with time Caputo derivative

Abstract In this paper, we propose a fractional conservative variational physics-informed neural network (f-cVPINNs) for solving the time-fractional Stokes equations, where the time-fractional derivative is in the Caputo sense with order α ∈ (0, 1). This method combines the ideas of conservative PINNs and variational PINNs from machine learning. Since the time derivative of the proposed neural network is fractional, the chain rule cannot be directly used for differentiation. Instead, it needs to be obtained by numerical approximation. Therefore, we use the commonly employed L1 formula for time direction discretization. In the spatial direction, the computational domain is divided into several sub-domains. At the interfaces between adjacent sub-domains, additional interface conditions are imposed using the concept of ‘flux’, which facilitates natural communication between the sub-domains. Computational experiments show that f-cVPINNs has significant advantages over fractional PINNs in terms of convergence speed and flexibility. Additionally, we applied this algorithm to solve the time-fractional Navier–Stokes equations and found it to be effective as well.