Sequential learning based PINNs to overcome temporal domain complexities in unsteady flow past flapping wings

For a data-driven and physics combined modeling of unsteady flow systems with moving immersed boundaries, Sundar et al. (Sundar et al. 2024) introduced an immersed boundary-aware (IBA) framework, combining Physics-Informed Neural Networks (PINNs) and the immersed boundary method (IBM). This approach was beneficial because it avoided case-specific transformations to a body-attached reference frame. Building on this, we now address the challenges of long time integration in velocity reconstruction and pressure recovery by extending this IBA framework with sequential learning strategies. Key difficulties for PINNs in long time integration include temporal sparsity, long temporal domains and rich spectral content. To tackle these, a moving boundary-enabled PINN is developed, proposing two sequential learning strategies: - a time marching with gradual increase in time domain size, training a monolithic PINN and - a time decomposition which divides the temporal domain into smaller segments, training a PINN over each subdomains and combining them together. While the former approach may struggle with error accumulation over long time domains, the latter one, eventually combined with transfer learning, effectively reduces error propagation and computational complexity. The key findings for modeling of incompressible unsteady flows past a flapping airfoil include: - for quasi-periodic flows, the time decomposition approach with preferential spatio-temporal sampling improves accuracy and efficiency for pressure recovery and aerodynamic load reconstruction, and, - for long time domains, decomposing it into smaller temporal segments and employing multiple sub-networks, simplifies the problem ensuring stability and reduced network sizes. This study highlights the limitations of traditional PINNs for long time integration of flow-structure interaction problems and demonstrates the benefits of decomposition-based strategies for addressing error accumulation, computational cost, and complex dynamics.