Numerical simulation of viscoelastic flows using physics-informed neural networks: R-PINN and adaptive IR-PINN

Purpose This paper aims to investigate viscoelastic flow problems governed by the coupled Navier–Stokes equations and Oldroyd-B constitutive model. To address the difficulties of residual physics-informed neural networks (R-PINNs) in such problems, including unstable training, large stress prediction errors and nonphysical solutions caused by strong coupling, strong nonlinearity and local stress concentration, an iterative R-PINN (IR-PINN) is proposed to enhance the solution capability for complex viscoelastic flows. An Adaptive IR-PINN with exponential moving average (EMA)-based weighting is further introduced to balance the coupled loss terms. Design/methodology/approach The R-PINN is first applied to the unsteady lid-driven cavity flow and to steady and unsteady viscoelastic plane Couette flows to assess its capability in predicting velocity, pressure and stress fields. For the more complex unsteady viscoelastic lid-driven cavity flow, the IR-PINN uses pretrained velocity-pressure and stress fields as baseline solutions. The coupled problem is decomposed into stress-evolution and velocity-pressure correction subproblems, which are solved alternately within a time-marching framework using Picard-type iteration, relaxation updates and anchor constraints. The adaptive IR-PINN further incorporates an EMA-based adaptive weighting strategy to balance different residual terms during coupled training. Findings The results demonstrate that the R-PINN can accurately solve the unsteady lid-driven cavity flow as well as the steady and unsteady viscoelastic plane Couette flows, with good agreement achieved between the predicted results and the analytical or benchmark solutions. For the unsteady viscoelastic lid-driven cavity flow, the proposed IR-PINN shows good consistency with the benchmark results in terms of the centerline velocity profiles, lid stress distributions and primary vortex location, while the adaptive IR-PINN further improves the Picard convergence behavior and improves the representation of localized stress peaks without altering the main velocity-field structure. Originality/value By integrating the R-PINN with an iterative splitting strategy under a time-marching framework, this study proposes an IR-PINN method for complex viscoelastic flows. The proposed method transforms the direct solution of the strongly coupled system into a stepwise process of local residual correction, thereby improving training stability and solution controllability. By further incorporating EMA-based adaptive weighting, the adaptive IR-PINN improves the balance among loss terms during coupled training, providing an effective PINN-based approach for more complex non-Newtonian flow problems.