NeuralFluid: Physics-Informed FSI Inference

• NeuralFluid is a framework that fuses physics-informed neural networks with low-dimensional modal surface representations to model fluid-structure interactions.
• It accurately reconstructs coupled unsteady flow and structural motion from minimal Lagrangian particle data without needing direct surface observations.
• The approach utilizes composite physics-based losses and Monte Carlo integration to ensure robust, regularization-free inference even with over-parameterized modal bases.

NeuralFluid encompasses a suite of physics-informed neural network methodologies and hybrid operator-learning frameworks for modeling, inference, and control of fluid-structure interaction (FSI) systems under extreme data sparsity. It achieves quantitatively accurate reconstructions of coupled unsteady flow and structural motion from minimal Lagrangian particle measurements, without requiring direct observations of solid boundaries or constitutive solid models. By fusing coordinate neural representations, low-dimensional modal surface descriptions, and composite physics-based losses using only off-body flow tracks and weak interface constraints, NeuralFluid enables robust, regularization-free quantitative analysis of canonical multiphase dynamics. This framework represents a substantial advance in the non-intrusive inference of FSI, bridging the experimental gap prevalent in laboratory and field scenarios with asynchronous or missing measurements of structural motion (Tang et al., 30 Jun 2025).

1. Governing Equations and Fluid-Structure Coupling

NeuralFluid imposes the full incompressible Navier–Stokes equations in an Eulerian domain Ω(t): $$\frac{\partial u}{\partial t} + (u\cdot\nabla)u = -\frac{1}{\rho}\nabla p + \nu \nabla^2 u,\qquad \nabla\cdot u = 0,$$ where $u(x,t)$ is velocity, $p(x,t)$ pressure, $\rho$ density, $\nu$ kinematic viscosity. The solid boundary is encoded as a moving domain boundary $\partial\Omega(t)$ subject to physically motivated interface constraints. The principal interface condition is velocity continuity (no-slip),

$$e_s(x,t) = u(x,t) - v_s(x,t),\qquad x\in\partial\Omega(t),$$

where $v_s(x,t)$ is the surface velocity of the solid, itself a function of low-dimensional modal coordinates.

Weak enforcement of stress continuity can be included if surface stress data are available, but the core approach requires only velocity matching.

2. Low-Dimensional Modal Surface Representation

The moving solid boundary is formulated as a modal expansion about a base configuration: $p(t) \approx \bar p + \Phi \beta(t),$ with $\bar p \in \mathbb{R}^{d n_s}$ the mean nodal positions, $\Phi \in \mathbb{R}^{(d n_s)\times n_m}$ the modal basis (from, e.g., eigenfrequency or POD analysis), and $\beta(t)\in\mathbb{R}^{n_m}$ the modal coefficients parameterizing structural deformation.

A feed-forward neural “structure PINN” S maps scalar time to the modal coefficients: $S : t \mapsto \beta(t).$ Automatic differentiation yields surface velocities $\dot{p}(t) = \Phi \frac{d\beta}{dt}$, which in turn define the moving interface and the fluid mesh Ω(t), dynamically deforming throughout training.

This compact representation eliminates the need for explicit surface tracking instrumentation and enables model generalization across different FSI scenarios.

3. Neural Network Architecture and Physical Constraints

The core neural representations consist of:

• Fluid velocity network $F_u$ and pressure network $F_p$: Each takes $x,t\in\mathbb{R}^d\times\mathbb{R}$ as input and outputs $u$ or $p$. They use a Fourier feature embedding on spatiotemporal coordinates to overcome spectral bias, followed by 6 hidden layers of 100 neurons (swish activation); Fourier embedding size is 256.
• Structure network $S$: A 4-layer MLP with width 150 (swish), Fourier time embedding, mapping $t\mapsto\beta(t)$.
• Per-track kinematic networks $P[k]$: Each off-body Lagrangian particle measured gets a physics-constrained kinematic model (implemented via the Theory of Functional Connections), enforcing exact fitting of the measured track and integrating the advection equation $dx_k/dt = v_k(t)$ as a hard constraint.

All neural fields are auto-differentiable, enabling calculation of time and spatial derivatives as required by the composite physics-informed loss.

4. Composite Loss Structure and Optimization

The overall objective combines three principal losses: $$J_{\mathrm{total}} = \gamma_1 J_{\mathrm{flow}} + \gamma_2 J_{\mathrm{surf}} + \gamma_3 J_{\mathrm{part}},$$

• Fluid physics loss ($J_{\mathrm{flow}}$) enforces Navier–Stokes incompressibility ($\nabla\cdot u=0$) and momentum balance over Ω(t), normalized by domain volume and variable dimensions.
• Interface continuity loss ($J_{\mathrm{surf}}$) penalizes velocity mismatch at the moving fluid–structure interface, quantified in a weak (least-squares) sense over $\partial\Omega(t)$.
• Particle (data) loss ($J_{\mathrm{part}}$) enforces consistency between the reconstructed field $u(x_k(t),t)$ and the “true” velocity $v_k(t)$ along each particle path inferred from the kinematic subsystem, again normalized by sample number and vector dimensions.

All loss integrals are approximated via Monte Carlo sampling in dynamic, deformation-aware meshes for bulk and interface points, with an adaptive batching strategy for particles to mitigate data sparsity and spatial non-uniformity. The optimization uses Adam with learning rate $1\times 10^{-3}$, optimal network sizes, and hyperparameter weighting searched via validation error minimization.

5. Handling of Sparse Data and Regularization-Free Over-Parameterization

NeuralFluid requires only off-body Lagrangian track measurements and the moving boundary condition imposed weakly via $J_{\mathrm{surf}}$. Critically, a key innovation is explicitly modeling the kinematic constraint for each particle via hard neural projections so that noisy, sparse tracks are exactly fit—this structurally prevents ill-posed boundary drift arising from lack of data near deforming interfaces.

A significant robustness property is that modal over-parameterization (increasing number of structural modes beyond the minimum needed to capture deformation energy) does not degrade reconstruction accuracy or cause overfitting, unlike classical truncation-regularized approaches. Empirically, error plateaus once the basis captures ≥99% energy and remains stable as further (extraneous) modes are added.

6. Benchmark Performance and Empirical Results

NeuralFluid has been quantitatively validated on two canonical FSI scenarios:

• 2D vortex-induced oscillation (VIO) in a flapping plate: With ≈0.004 particles per pixel (planar LPT data) and 8 modal eigenmodes, vorticity NRMSE is ≈15%, pressure error ≈8%, modal coefficient (amplitude/phase) error for leading modes <5%, and structural tip-displacement error <2 mm over a 34 mm amplitude. The reconstruction error saturates once the modal basis captures ≈99% of deformation energy.
• 3D pulse wave (PW) in a flexible pipe: With ≈0.01 ppp (multi-camera LPT) and up to 14 POD modes, transverse velocity error ≈14%, axial velocity ≈5%, pressure ≈6%, radial surface deflection error ≤9%. Error plateau occurs at ≈8 modes (≥95% energy), independently of further mode count.

In both cases, the system remains robust to over-parameterization: modal error and field reconstruction do not degrade as the number of modes increases beyond the energy threshold.

7. Implications, Limitations, and Application Scope

NeuralFluid establishes that:

• Single-phase, sparse Lagrangian track data is sufficient for fully coupled FSI inference in the presence of moving solid boundaries, with no need for solid constitutive law specification or direct surface tracking, although inclusion of boundary measurements can further improve performance.
• Modal over-parameterization does not necessitate intervention by regularization or manual truncation, resulting in a regularization-free regime advantageous in experiments with uncertain or unknown active deformation mode count.
• The approach is particularly suited for experimental settings in which surface observations are asynchronous, incomplete, or altogether absent, and thus enables a broad class of real-world FSI problems to be quantitatively studied.

Limitations include reliance on informative modal expansions and the assumption of sufficient track coverage away from the interface for reliable field inference. Structural model generalization depends on the completeness of the modal basis.

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Benchmark	Data type / DoF	Velocity error	Pressure error	Structure error	Modal basis	Robustness (over-parameterization)
2D VIO Flapping	≈0.004 ppp, 8 modes	≈15% (vort.)	≈8%	<2 mm tip, <5% modal coeff.	Linear eig., ≥99% energy	Errors plateau, stable extra modes
3D PW Pipe	≈0.01 ppp, 14 modes	14%/5% (T/Ax)	≈6%	≤9% surface bulge	POD, ≥95% energy	Error saturates, stable extra modes

NeuralFluid thus synthesizes physics-constrained neural operators, low-rank boundary parametrization, and Lagrangian data for non-intrusive, regularization-free fluid-structure inference with quantifiable accuracy across canonical FSI flows—even as the modal basis is over-specified (Tang et al., 30 Jun 2025).