DeepFSI: Differentiable Hybrid FSI

• DeepFSI is a differentiable hybrid neural framework that combines IBM-based fluid dynamics with trainable ConvLSTM residual blocks for simulating fluid-structure interaction.
• The framework integrates numerical discretization with learned pressure and velocity corrections, ensuring stable long-horizon predictions and improved parameter generalization.
• By leveraging JAX automatic differentiation, DeepFSI enables end-to-end gradient flow through coupled fluid and structural dynamics, outperforming purely data-driven alternatives.

Searching arXiv for DeepFSI and closely related FSI papers to ground the article in the current literature. DeepFSI is a differentiable hybrid neural modeling framework for fluid-structure interaction (FSI) that integrates discretized immersed-boundary-method (IBM) physics with sequential neural networks in a single end-to-end trainable program. It was introduced for incompressible-flow FSI problems involving both rigid-body vortex-induced vibration (VIV) and flexible-body flow-induced deformation (FID), with the central premise that classical FSI solvers are too expensive for long-horizon and many-query workloads, while purely data-driven sequence models often suffer from instability, poor parameter generalization, and autoregressive error accumulation (Fan et al., 2023). In DeepFSI, fluid and structural dynamics are advanced through physics-encoded recurrent units, trainable ConvLSTM modules supply learned pressure and velocity corrections, and JAX automatic differentiation enables gradient propagation through the full rollout trajectory, so the model behaves as a differentiable solver rather than a detached predictor (Fan et al., 2023).

1. Definition and problem class

DeepFSI addresses two-way coupled FSI in which fluid and solid states co-evolve through moving interfaces and exchanged forces. The formulation considered in the original work couples incompressible Navier–Stokes dynamics for the fluid with rigid or flexible structural dynamics, using IBM-style Lagrangian–Eulerian interpolation and force spreading for coupling (Fan et al., 2023). The fluid equations are written as

$\nabla\cdot\mathbf{u} = 0,$

$\frac{\partial{\mathbf{u}}{\partial t} = -(\mathbf{u}\cdot\nabla)\mathbf{u}+\nu\nabla^2\mathbf{u}-\frac{1}{\rho}\nabla p+\mathbf{f},$

while the flexible-structure model is expressed as

$$\mu_s \frac{\partial^2 \mathbf{w}}{\partial t^2}+EI \frac{\partial^4 \mathbf{w}}{\partial \mathbf{X}^4}=\mathbf{q}.$$

The IBM coupling is given through the reciprocal interpolation and spreading relations

$$\mathbf{\phi}(\mathbf{x},t)=\int_{\gamma_\mathbf{X}} \mathbf{\Phi}(\mathbf{X},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{X},$$

$$\mathbf{\Phi}(\mathbf{X},t)=\int_{\gamma_\mathbf{x}} \mathbf{\phi}(\mathbf{x},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{x}.$$

Within this setting, DeepFSI is neither a purely numerical solver nor a black-box surrogate. Its defining characteristic is the insertion of trainable recurrent components into a coarse-grid numerical FSI update while preserving the governing discretized physics and full differentiability of the rollout (Fan et al., 2023). This places it in a broader line of work on learning-assisted FSI, but its particular emphasis is end-to-end differentiable programming rather than only partitioned coupling or offline reduced-order modeling.

2. Architectural composition

The DeepFSI architecture is organized as a sequence-to-sequence recurrent model composed of repeated FSI-physics-integrated recurrent units. Each unit advances the coupled state using previous-step velocity $\mathbf{u}_t$, pressure $p_t$, and structural response $\mathbf{w}_t$, combining non-trainable discretized physics with trainable ConvLSTM residual blocks (Fan et al., 2023).

At the fluid level, the model uses a fractional-step Navier–Stokes discretization. The intermediate velocity is computed through a coarse-grid update, and IBM forcing is injected through a volume-of-solid mask $\varepsilon^t(\mathbf{x})$. The formulation reported for the intermediate velocity and direct IBM forcing is

$$\mathbf{u}^{*} = -\mathbf{u}^t -{\Delta t}\left[\nabla\cdot(\mathbf{u}^{t}\otimes\mathbf{u}^{t})+\nu \nabla^2\mathit{\mathbf{u}^{t}}+\mathbf{f}^{t}\right],$$

$\frac{\partial{\mathbf{u}}{\partial t} = -(\mathbf{u}\cdot\nabla)\mathbf{u}+\nu\nabla^2\mathbf{u}-\frac{1}{\rho}\nabla p+\mathbf{f},$0

A distinctive aspect is that the expensive pressure Poisson solve is not performed explicitly. Instead, DeepFSI uses a trainable ConvLSTM block as a learned pressure operator for the modified pressure projection,

$\frac{\partial{\mathbf{u}}{\partial t} = -(\mathbf{u}\cdot\nabla)\mathbf{u}+\nu\nabla^2\mathbf{u}-\frac{1}{\rho}\nabla p+\mathbf{f},$1

with the neural approximation written as

$\frac{\partial{\mathbf{u}}{\partial t} = -(\mathbf{u}\cdot\nabla)\mathbf{u}+\nu\nabla^2\mathbf{u}-\frac{1}{\rho}\nabla p+\mathbf{f},$2

The corrected velocity is then refined by a second ConvLSTM block: $\frac{\partial{\mathbf{u}}{\partial t} = -(\mathbf{u}\cdot\nabla)\mathbf{u}+\nu\nabla^2\mathbf{u}-\frac{1}{\rho}\nabla p+\mathbf{f},$3

$\frac{\partial{\mathbf{u}}{\partial t} = -(\mathbf{u}\cdot\nabla)\mathbf{u}+\nu\nabla^2\mathbf{u}-\frac{1}{\rho}\nabla p+\mathbf{f},$4

On the structural side, DeepFSI keeps numerical time stepping rather than replacing the solid solver with a learned module. The structural dynamics are discretized in the form

$\frac{\partial{\mathbf{u}}{\partial t} = -(\mathbf{u}\cdot\nabla)\mathbf{u}+\nu\nabla^2\mathbf{u}-\frac{1}{\rho}\nabla p+\mathbf{f},$5

with fluid forces recovered from IBM forcing and time integration handled by fourth-order Runge–Kutta for rigid bodies and Newmark-beta for flexible bodies (Fan et al., 2023). This hybridization strategy preserves an explicit mechanics prior in the solid branch while allowing the fluid update to absorb unresolved coarse-grid errors through learned recurrent corrections.

3. Differentiable-programming formulation

DeepFSI’s main methodological claim is that all modules are implemented in JAX so that automatic differentiation can propagate gradients through the entire unrolled trajectory (Fan et al., 2023). This is the critical distinction from weakly coupled hybrid models in which physics modules are present but gradients are blocked, producing offline or teacher-forced behavior rather than genuinely coupled sequence-level optimization.

The optimization objective is a rollout-level loss combining pressure, velocity, and structure errors: $\frac{\partial{\mathbf{u}}{\partial t} = -(\mathbf{u}\cdot\nabla)\mathbf{u}+\nu\nabla^2\mathbf{u}-\frac{1}{\rho}\nabla p+\mathbf{f},$6 where the pressure, velocity, and structure terms are given by

$\frac{\partial{\mathbf{u}}{\partial t} = -(\mathbf{u}\cdot\nabla)\mathbf{u}+\nu\nabla^2\mathbf{u}-\frac{1}{\rho}\nabla p+\mathbf{f},$7

$\frac{\partial{\mathbf{u}}{\partial t} = -(\mathbf{u}\cdot\nabla)\mathbf{u}+\nu\nabla^2\mathbf{u}-\frac{1}{\rho}\nabla p+\mathbf{f},$8

$\frac{\partial{\mathbf{u}}{\partial t} = -(\mathbf{u}\cdot\nabla)\mathbf{u}+\nu\nabla^2\mathbf{u}-\frac{1}{\rho}\nabla p+\mathbf{f},$9

The experiments set $$\mu_s \frac{\partial^2 \mathbf{w}}{\partial t^2}+EI \frac{\partial^4 \mathbf{w}}{\partial \mathbf{X}^4}=\mathbf{q}.$$0 (Fan et al., 2023).

Because full-trajectory backpropagation is memory-intensive, the reported training procedure partitions long rollouts into subchains while passing hidden states between them, thereby preserving temporal continuity and allowing sequence-to-sequence optimization on long trajectories (Fan et al., 2023). This suggests that DeepFSI should be understood as a practical differentiable-programming implementation rather than only a conceptual coupling of neural and physical modules.

A related but solver-centric extension of the same design philosophy appears in Diff-FlowFSI, a JAX-based GPU-differentiable CFD/FSI platform that exposes JVPs, VJPs, implicit differentiation, and gradient checkpointing for inverse problems and hybrid neural–CFD workflows (Fan et al., 29 May 2025). DeepFSI differs in being a learning-based hybrid solver, whereas Diff-FlowFSI is a differentiable simulation platform; however, both exemplify end-to-end differentiation as a central organizing principle for modern FSI computation.

4. Numerical setup and implementation

The reported DeepFSI implementation uses two ConvLSTM blocks, each with 5 layers, channels $$\mu_s \frac{\partial^2 \mathbf{w}}{\partial t^2}+EI \frac{\partial^4 \mathbf{w}}{\partial \mathbf{X}^4}=\mathbf{q}.$$1, kernel size $$\mu_s \frac{\partial^2 \mathbf{w}}{\partial t^2}+EI \frac{\partial^4 \mathbf{w}}{\partial \mathbf{X}^4}=\mathbf{q}.$$2, ReLU activation except in the final layer, Adam optimization, initial learning rate $$\mu_s \frac{\partial^2 \mathbf{w}}{\partial t^2}+EI \frac{\partial^4 \mathbf{w}}{\partial \mathbf{X}^4}=\mathbf{q}.$$3, and a CosineDecaySchedule with $$\mu_s \frac{\partial^2 \mathbf{w}}{\partial t^2}+EI \frac{\partial^4 \mathbf{w}}{\partial \mathbf{X}^4}=\mathbf{q}.$$4 (Fan et al., 2023). Ground-truth data are produced by high-fidelity FSI simulations on a $$\mu_s \frac{\partial^2 \mathbf{w}}{\partial t^2}+EI \frac{\partial^4 \mathbf{w}}{\partial \mathbf{X}^4}=\mathbf{q}.$$5 grid, while the neural model operates on a $$\mu_s \frac{\partial^2 \mathbf{w}}{\partial t^2}+EI \frac{\partial^4 \mathbf{w}}{\partial \mathbf{X}^4}=\mathbf{q}.$$6 grid, described as $$\mu_s \frac{\partial^2 \mathbf{w}}{\partial t^2}+EI \frac{\partial^4 \mathbf{w}}{\partial \mathbf{X}^4}=\mathbf{q}.$$7 coarser in each dimension. The learning step is taken to be twice the fine-grid numerical time step (Fan et al., 2023).

The two benchmark classes are a rigid-body VIV problem and a flexible-body FID problem. The VIV case uses $$\mu_s \frac{\partial^2 \mathbf{w}}{\partial t^2}+EI \frac{\partial^4 \mathbf{w}}{\partial \mathbf{X}^4}=\mathbf{q}.$$8, mass ratio $$\mu_s \frac{\partial^2 \mathbf{w}}{\partial t^2}+EI \frac{\partial^4 \mathbf{w}}{\partial \mathbf{X}^4}=\mathbf{q}.$$9, natural frequency range $$\mathbf{\phi}(\mathbf{x},t)=\int_{\gamma_\mathbf{X}} \mathbf{\Phi}(\mathbf{X},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{X},$$0, zero damping, training window $$\mathbf{\phi}(\mathbf{x},t)=\int_{\gamma_\mathbf{X}} \mathbf{\Phi}(\mathbf{X},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{X},$$1, and forecast horizon to $$\mathbf{\phi}(\mathbf{x},t)=\int_{\gamma_\mathbf{X}} \mathbf{\Phi}(\mathbf{X},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{X},$$2. The FID case uses $$\mathbf{\phi}(\mathbf{x},t)=\int_{\gamma_\mathbf{X}} \mathbf{\Phi}(\mathbf{X},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{X},$$3, $$\mathbf{\phi}(\mathbf{x},t)=\int_{\gamma_\mathbf{X}} \mathbf{\Phi}(\mathbf{X},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{X},$$4 Pa, Poisson ratio $$\mathbf{\phi}(\mathbf{x},t)=\int_{\gamma_\mathbf{X}} \mathbf{\Phi}(\mathbf{X},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{X},$$5, zero damping, $$\mathbf{\phi}(\mathbf{x},t)=\int_{\gamma_\mathbf{X}} \mathbf{\Phi}(\mathbf{X},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{X},$$6, training window $$\mathbf{\phi}(\mathbf{x},t)=\int_{\gamma_\mathbf{X}} \mathbf{\Phi}(\mathbf{X},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{X},$$7, and forecast horizon to $$\mathbf{\phi}(\mathbf{x},t)=\int_{\gamma_\mathbf{X}} \mathbf{\Phi}(\mathbf{X},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{X},$$8 (Fan et al., 2023).

The reported baselines are a purely data-driven sequence-to-sequence model, a weakly coupled hybrid model with non-differentiable physics modules, and a pure coarse-grid numerical solver (Fan et al., 2023). This baseline design matters because DeepFSI’s claims concern not only raw predictive accuracy, but specifically the effect of differentiable end-to-end training relative to both black-box learning and non-differentiable physics-assisted learning.

A separate line of FSI operator learning based on Fourier neural operators replaces the PDE-based fluid solver in an IBM pipeline while retaining a conventional finite-difference structural update; the vesicle-dynamics study reports that the best interpolation performance is obtained with steady-state data and multi-step labels (Xiao et al., 2024). That method is often “DeepFSI-like” in spirit, but it uses an FNO surrogate for induced flow rather than DeepFSI’s physics-integrated ConvLSTM recurrent unit. The distinction is methodological: operator learning replaces the flow solver directly, whereas DeepFSI embeds learned corrections within a discretized FSI step.

5. Empirical behavior and reported performance

On rigid-body VIV, DeepFSI is reported to capture the transition to the steady $$\mathbf{\phi}(\mathbf{x},t)=\int_{\gamma_\mathbf{X}} \mathbf{\Phi}(\mathbf{X},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{X},$$9 vortex shedding mode, reproduce the “8”-shaped trajectory, predict streamwise and transverse displacements accurately, match frequency spectra closely, and maintain low error over long rollouts (Fan et al., 2023). The mean relative errors reported for VIV are summarized below.

Model	Fluid	Structure
DeepFSI	0.16	0.02
Purely data-driven	0.67	0.11
Pure solver	3.76	2.49
Weakly coupled	$$\mathbf{\Phi}(\mathbf{X},t)=\int_{\gamma_\mathbf{x}} \mathbf{\phi}(\mathbf{x},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{x}.$$0	$$\mathbf{\Phi}(\mathbf{X},t)=\int_{\gamma_\mathbf{x}} \mathbf{\phi}(\mathbf{x},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{x}.$$1

For unseen spring stiffness values in VIV, the paper evaluates both interpolation at $$\mathbf{\Phi}(\mathbf{X},t)=\int_{\gamma_\mathbf{x}} \mathbf{\phi}(\mathbf{x},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{x}.$$2 and extrapolation at $$\mathbf{\Phi}(\mathbf{X},t)=\int_{\gamma_\mathbf{x}} \mathbf{\phi}(\mathbf{x},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{x}.$$3. DeepFSI is reported to predict vortex evolution and structural response successfully in both settings, whereas the pure solver fails, the weakly coupled model diverges, and the purely data-driven model degrades quickly under parameter shift (Fan et al., 2023). The reported mean relative errors for unseen stiffness are:

Model	Fluid	Structure
DeepFSI	0.21	0.05
Purely data-driven	1.01	0.64
Pure solver	4.00	2.44
Weakly coupled	$$\mathbf{\Phi}(\mathbf{X},t)=\int_{\gamma_\mathbf{x}} \mathbf{\phi}(\mathbf{x},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{x}.$$4	$$\mathbf{\Phi}(\mathbf{X},t)=\int_{\gamma_\mathbf{x}} \mathbf{\phi}(\mathbf{x},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{x}.$$5

On flexible-body FID, DeepFSI is reported to predict free-end displacement history accurately, capture mode evolution, and remain physically consistent under more complex non-periodic dynamics (Fan et al., 2023). The baseline comparisons are particularly important here: the pure solver overestimates free-end displacement, the purely data-driven model may match displacement magnitude but produces nonphysical vibration modes, and the weakly coupled hybrid again diverges. This suggests that the explicit structural-physics encoding is not merely auxiliary but central to stabilizing harder flexible-body cases.

The computational trade-off reported in the same study is that a single learning step costs about 20% of one fine-grid numerical step, with an overall speedup of 8.3× for the same physical time horizon, while training for 300 rollout steps costs 7.7× the wall time of the purely data-driven baseline (Fan et al., 2023). The inference advantage is therefore paired with a more expensive offline optimization procedure.

6. Relation to adjacent FSI-learning frameworks

DeepFSI sits within a heterogeneous literature on neural FSI, but its specific combination of IBM physics encoding, recurrent learned pressure/velocity correction, and end-to-end differentiable rollout distinguishes it from both partitioned ROMs and operator-learning surrogates.

A hybrid partitioned deep learning methodology for moving-interface FSI couples a POD-RNN for moving interface kinematics with a convolutional recurrent autoencoder network (CRAN) for the flow field, linked by snapshot-field transfer and load recovery on a structured Eulerian grid (Gupta et al., 2021). That framework is partitioned across fluid and structural surrogates and demonstrates acceptable interface tracking and nonlinear wake prediction for a freely oscillating cylinder. Its emphasis is reduced-order decomposition and load recovery rather than solver-level differentiability. The reported speed-up is about $$\mathbf{\Phi}(\mathbf{X},t)=\int_{\gamma_\mathbf{x}} \mathbf{\phi}(\mathbf{x},t)\delta(\mathbf{x}-\mathbf{X})d\gamma_\mathbf{x}.$$6, but future-horizon divergence appears after about 25–30 predicted steps without feedback correction (Gupta et al., 2021). Relative to DeepFSI, this indicates a stronger reduction-model orientation and a weaker focus on end-to-end gradient flow through physics modules.

Another partitioned strategy assigns independent CNN–LSTM networks to fluid and solid subdomains and couples them through preCICE with quasi-Newton acceleration for a 1D elastic-tube problem (Totounferoush et al., 2021). That work explicitly positions itself as a preliminary step toward using neural networks as accurate initial guesses for classical FSI solvers. The relation to DeepFSI is therefore conceptual rather than architectural: both decompose the coupled problem, but the preCICE-based framework preserves a classical strong-coupling loop between separate neural subsolvers, whereas DeepFSI incorporates numerical physics and learned modules within a single differentiable recurrence (Totounferoush et al., 2021).

The FNO-based vesicle solver mentioned above provides an additional contrast. It uses a Fourier neural operator to predict induced fluid velocity fields in a 3D immersed-boundary setting and then updates the vesicle position by finite differences. The best interpolation performance is obtained by training with steady-state data and multi-step labels, and the resulting solver captures transient relaxation, deformation, and convergence to a new steady state (Xiao et al., 2024). This suggests a family resemblance to DeepFSI in hybrid fluid-surrogate plus classical-structure coupling, but not identity of method.

Finally, Diff-FlowFSI provides a fully differentiable CFD/FSI platform rather than a learned hybrid surrogate. It uses a vectorized finite-volume solver, direct-forcing IBM, and JAX automatic differentiation for parameter inference and neural-CFD integration (Fan et al., 29 May 2025). A plausible implication is that such platforms could serve as training-data generators, differentiable backends, or alternative substrates for future DeepFSI-like architectures.

7. Scope, misconceptions, and limitations

A common misconception is to treat “DeepFSI” as a generic label for any deep-learning-based FSI method. In the literature summarized here, “DeepFSI” refers specifically to the differentiable hybrid neural modeling framework introduced for IBM-based incompressible FSI in JAX (Fan et al., 2023). Other systems may be “DeepFSI-like” in the sense that they combine learned and classical FSI components, but they are methodologically distinct. The FNO vesicle solver is an operator-learning IBM hybrid (Xiao et al., 2024); the POD-RNN/CRAN framework is a partitioned reduced-order approach (Gupta et al., 2021); the preCICE-coupled tube model is a partitioned neural prototype (Totounferoush et al., 2021); and Diff-FlowFSI is a differentiable CFD platform (Fan et al., 29 May 2025).

The original DeepFSI paper is explicit about several limitations. The demonstrated cases are two-dimensional. The framework depends on high-fidelity labeled training data generated offline. The operating regime is a coarse-grid surrogate of fine-grid dynamics, so performance depends on the fidelity and representativeness of training simulations. Training is memory-intensive and slower than purely data-driven alternatives. The architecture is specialized to IBM-based FSI and ConvLSTM recurrent modeling, and strongly nonlinear three-dimensional moving-boundary problems are not demonstrated (Fan et al., 2023).

These limitations define the present scope of the method. At the same time, the empirical results suggest that differentiable trajectory-level training materially improves stability and parameter generalization compared with both black-box sequence learning and non-differentiable hybridization (Fan et al., 2023). In that sense, DeepFSI represents a specific synthesis of numerical FSI, neural sequence modeling, and differentiable programming: a solver-like learned system in which physical operators are preserved, learned corrections are localized to numerically expensive or dissipative components, and optimization is performed over the coupled rollout as a whole.