Learning Fluid-Structure Interaction with Physics-Informed Machine Learning and Immersed Boundary Methods

TL;DR

The paper tackles the challenge of modeling fluid-structure interaction with moving boundaries using physics-informed neural networks. It introduces a decoupled Eulerian-Lagrangian PINN with domain-specific fluid and structure networks, augmented by learnable B-spline SiLU activations and interface-informed coupling inspired by the immersed boundary method. Across a 2D lid-driven cavity with a moving disc, the domain-decomposed EL-L architecture achieves substantial improvements, notably reducing structural pressure error from 12.9% to 2.39% and delivering near-CFD accuracy in both velocity and pressure. The results support a broader principle that aligning neural architecture with the physics of distinct domains, together with adaptive activations, yields the most effective PINN-based FSI models.

Abstract

Physics-informed neural networks (PINNs) have emerged as a promising approach for solving complex fluid dynamics problems, yet their application to fluid-structure interaction (FSI) problems with moving boundaries remains largely unexplored. This work addresses the critical challenge of modeling FSI systems with moving interfaces, where traditional unified PINN architectures struggle to capture the distinct physics governing fluid and structural domains simultaneously. We present an innovative Eulerian-Lagrangian PINN architecture that integrates immersed boundary method (IBM) principles to solve FSI problems with moving boundary conditions. Our approach fundamentally departs from conventional unified architectures by introducing domain-specific neural networks: an Eulerian network for fluid dynamics and a Lagrangian network for structural interfaces, coupled through physics-based constraints. Additionally, we incorporate learnable B-spline activation functions with SiLU to capture both localized high-gradient features near interfaces and global flow patterns. Empirical studies on a 2D cavity flow problem involving a moving solid structure show that while baseline unified PINNs achieve reasonable velocity predictions, they suffer from substantial pressure errors (12.9%) in structural regions. Our Eulerian-Lagrangian architecture with learnable activations (EL-L) achieves better performance across all metrics, improving accuracy by 24.1-91.4% and particularly reducing pressure errors from 12.9% to 2.39%. These results demonstrate that domain decomposition aligned with physical principles, combined with locality-aware activation functions, is essential for accurate FSI modeling within the PINN framework.

Figures (13)

• Figure 1: Illustration of the computational domain of the FSI problem considered in the present work, showing the movement of a soft disc in a lid-driven cavity flow at different time steps.
• Figure 2: Baseline (B): Standard PINN with fixed Tanh activation. $\{t, x, y\}$ as inputs and output $\{\hat{u}, \hat{v}, \hat{p}\}$ for velocity components and pressure in both Eulerian and Lagrangian domains. The loss function $\mathcal{L}(\theta)$ is defined in Eq. \ref{['equ:m1_main_loss']}.
• Figure 3: Baseline enhanced with learnable activation (B-L) PINN, which includes trainable B-spline+SiLU activation functions. Both models take $\{t, x, y\}$ as inputs and output $\{\hat{u}, \hat{v}, \hat{p}\}$ for velocity components and pressure in both Eulerian and Lagrangian domains. The loss function $\mathcal{L}(\theta)$ is defined in Eq. \ref{['equ:m1_main_loss']}.
• Figure 4: Eulerian-Lagrangian network: the inputs of the Eulerian Network ($\theta_1$) are $\{t_e, x_e, y_e\}$ from the Eulerian fluid domain, and the output variables are $\{\hat{u}, \hat{v},\hat{p}\}$ representing the Eulerian velocity and pressure fields respectively. For the Lagrangian Network ($\theta_2$), the input variables are $\{t_l, x_l, y_l\}$ representing the Lagrangian structure and the output variables are $\{\Tilde{u}, \Tilde{v}, \Tilde{p}\}$ representing the Lagrangian velocity and pressure values respectively. The inputs are also shared between the networks for the velocity prediction at the interface. See, Eq.\ref{['eq:m2_main_loss']} for details of the loss function $\mathcal{L}(\theta)$.
• Figure 5: