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

TL;DR

This work extends the immersed boundary-aware PINN framework to address long-time integration challenges in unsteady flow with moving boundaries by introducing sequential learning strategies. It contrasts a baseline MB-PINN with two classes of sequential methods: time marching (Class I) and time-domain decomposition with transfer learning (Class II), showing that decomposition-based approaches (especially M2_TD+TL and M2_TD+TLLR) better handle temporal sparsity and long horizons while reducing network size. For periodic and quasi-periodic flapping-foil flows, preferential spatiotemporal sampling and relaxation of boundary losses further improve near-field pressure recovery and aerodynamic load reconstruction, achieving data-efficient performance under fixed training budgets. The results highlight the limitations of standard PINNs for long-time moving-boundary problems and demonstrate practical, data-efficient strategies to stabilize training and reduce computational cost, with implications for pressure recovery, load prediction, and potential extensions to 3D and parametric scenarios.

Abstract

For a data-driven and physics combined modelling of unsteady flow systems with moving immersed boundaries, Sundar {\it et al.} 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, however, this approach struggles with error accumulation over long time domains; and - a time decomposition which divides the temporal domain into smaller segments, combined with transfer learning it effectively reduces error propagation and computational complexity. The key findings for modelling 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.

Figures (22)

• Figure 1: Schematics of the problem setup: computational domain $\Omega$ is chosen for the IBM solver, and the truncated domain $\Omega^r$ is chosen for the surrogate model. $\Omega_f$ and $\Omega_f^r$ are the fluid regions excluding the solid boundary $\Gamma_{IB}$ and solid region $\Omega_s$ at any given time instant. $\Gamma_{inlet}^r,$$\Gamma_{outlet}^r,$$\Gamma_{upper}^r$ and $\Gamma_{lower}^r$ are the inlet, outlet, upper and lower boundaries of the truncated domain $\Omega_r$, respectively.
• Figure 2: A schematic depicting the MB-PINN formulation under the immersed boundary aware (IBA) framework for pressure recovery from velocity data.
• Figure 3: Schematic depicting different computational and physical sources of temporal domain complexities with application to unsteady flow past a sinusoidally plunging airfoil.
• Figure 4: Schematics of different types of sequential learning variant of MB-PINN considered. The entire time domain is partitioned into subdomains (five, as an example) of equal length. The green, red, and blue outlined boxes refer to broad classification in terms of the time domain size the model sees during training; ClassI: $\mathbf{M1}_{TM}$ and $\mathbf{M1}_{TM+BC}$ fall under this for which a single network sees a gradual increase in the time domain by appending the subdomains one by one during training; Class II: $\mathbf{M2}_{TD}$ and $\mathbf{M2}_{TD+TL}$ fall under this class for which multiple networks form the model, each network inheriting the data of a subdomain.
• Figure 5: Dynamics and data analysis of the periodic case: (a-b) coefficients of lift ($C_L$) and drag ($C_D$) with time, (c) $C_D$ Fourier spectrum, (d) $C_L-C_D$ phase portrait, (e) streamwise velocity ($u$) at probe location $x/c = 2.5,$ and, (f) corresponding streamwise velocity correlation ($\rho$) as a function of time ($t/T$).