Capturing unsteady nonlinear fluid dynamics using physics-informed convolutional leaky-integrator recurrent network
Accurately capturing unsteady flow dynamics remains challenging for physics-informed neural networks because nonlinear convection, multiscale structures, boundary effects, and accumulated temporal errors can degrade long-time prediction accuracy. This work proposes a physics-informed convolutional leaky-integrator recurrent network for capturing time-dependent flow fields. By combining local convolutions with learnable leaky memory, the proposed method provides a compact recurrent time-marching model with fewer parameters than multi-gate recurrent architectures. To support long-time prediction, the temporal domain is decomposed into overlapping windows coupled by consistency constraints, governing-equation residuals are evaluated using finite-difference differentiation matrices, and prescribed boundary conditions are enforced through hard constraints. The framework is evaluated on incompressible Navier–Stokes flows, including forced two-dimensional flow and the pre-merger interaction of two co-rotating vortices in a no-slip square cavity. Numerical results show that the proposed method reconstructs velocity and vorticity fields with low errors and provides temporally consistent solutions over the tested intervals. For the co-rotating-vortex case, vortex-core kinematics, vorticity statistics, circulation, restricted enstrophy, and the second moment of vorticity are evaluated. The proposed method captures the mutual rotation and gradual approach of the vortex cores and reproduces the principal trends in vorticity redistribution and restricted-enstrophy decay. Compared with the Convolutional Long Short-Term Memory variant, the proposed method achieves errors of the same order of magnitude while using fewer trainable parameters and requiring shorter runtimes in the main recurrent comparisons. Additional tests in Appendix B illustrate its applicability to selected nonlinear evolutionary partial differential equations.