---
title: 'DeepFSI: Differentiable Hybrid FSI'
url: https://www.emergentmind.com/topics/deepfsi
type: topic
---

# DeepFSI: Differentiable Hybrid FSI

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 [2303.12971]. 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 [2303.12971].

## 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 [2303.12971]. 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 [2303.12971]. 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 [2303.12971].

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

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,
\[
\nabla^2p^{t+1} = \frac{\rho\nabla \cdot \left[(1-\varepsilon^t)\mathbf{u}^{*}+\varepsilon^t \mathbf{u}_c^{t}\right]}{\Delta t},
\]
with the neural approximation written as
\[
p^{t+1}=\mathscr{F}_{conv_1}[(1-\varepsilon^t)\mathbf{u}^{*}+\varepsilon^t \mathbf{u}_c^{t}, \mathbf{s}_{conv_1}^t, \boldsymbol{\lambda}; \boldsymbol{\theta}_1] + p^t.
\]
The corrected velocity is then refined by a second ConvLSTM block:
\[
\mathbf{\widetilde{u}}^{t+1}=\mathbf{u}^{*}  -\frac{\Delta t}{\rho}\nabla{p}^{t+1},
\]
\[
\mathbf{u}^{t+1}=\mathscr{F}_{conv_2}(\mathbf{\widetilde{u}}^{t+1},\mathbf{u}^{t},\varepsilon^{t+1}, \varepsilon^{t}, \mathbf{s}_{conv_2}^t, \boldsymbol{\lambda}; \boldsymbol{\theta}_2)+\mathbf{\widetilde{u}}^{t+1}.
\]

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
\[
\mathbf{M} \frac{\partial^{2}\mathit{\mathbf{w}}}{\partial t^{2}}+ \mathbf{C} \frac{\partial \mathit{\mathbf{w}}}{\partial t}+\mathbf{Kw}=\mathbf{Q},
\]
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 [2303.12971]. 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 [2303.12971]. 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:
\[
\mathcal{L}(\boldsymbol{\theta}_1, \boldsymbol{\theta}_2) = \alpha \left[ \mathcal{L}_p (\boldsymbol{\theta_1}) + \mathcal{L}_v (\boldsymbol{\theta_2})\right] + \beta \mathcal{L}_s(\boldsymbol{\theta_1}, \boldsymbol{\theta_2}),
\]
where the pressure, velocity, and structure terms are given by
\[
\mathcal{L}_p(\boldsymbol{\theta}_1) = \frac{1}{N} \sum_{t=0}^{N-1} \lVert p^{t}+\mathscr{F}_{conv_1}[(1-\varepsilon^t)\mathbf{u}^{*}+\varepsilon^t \mathbf{u}_c^{t}, \mathbf{s}_{conv_1}^t, \boldsymbol{\lambda}; \boldsymbol{\theta}_1] -\mathbf{p}^{t+1}_{d} \rVert_{L_2}^{2},
\]
\[
\mathcal{L}_v(\boldsymbol{\theta}_2) = \frac{1}{N} \sum_{t=0}^{N-1} \lVert \widetilde{\mathbf{u}}^{t+1}+\mathscr{F}_{conv_2}(\mathbf{\widetilde{u}}^{t+1},\mathbf{u}^{t},\varepsilon^{t+1}, \varepsilon^{t}, \mathbf{s}_{conv_2}^t, \boldsymbol{\lambda}; \boldsymbol{\theta}_2) -\mathbf{u}^{t+1}_{d} \rVert_{L_2}^{2},
\]
\[
\mathcal{L}_s(\boldsymbol{\theta}_1, \boldsymbol{\theta}_2) = \frac{1}{N} \sum_{t=0}^{N-1}\lVert \mathbf{w}^{t+1} - \mathbf{w}^{t+1}_{d} \rVert_{L_2}^{2}.
\]
The experiments set \(\alpha=\beta=1\) [2303.12971].

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 [2303.12971]. 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 [2505.23940]. 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 \([32,64,64,64,32,1]\), kernel size \(3\times3\), ReLU activation except in the final layer, Adam optimization, initial learning rate \(10^{-4}\), and a CosineDecaySchedule with \(\alpha=10^{-9}\) [2303.12971]. Ground-truth data are produced by high-fidelity FSI simulations on a \(512 \times 256\) grid, while the neural model operates on a \(64 \times 32\) grid, described as \(8\times\) coarser in each dimension. The learning step is taken to be twice the fine-grid numerical time step [2303.12971].

The two benchmark classes are a rigid-body VIV problem and a flexible-body FID problem. The VIV case uses \(Re = 150\), mass ratio \(m^*=2\), natural frequency range \(f_n = 0.1-0.3\), zero damping, training window \(tV/D \in [0,40]\), and forecast horizon to \(tV/D = 90\). The FID case uses \(m^* = 2.75\), \(E = 10^8\) Pa, Poisson ratio \(0.3\), zero damping, \(Re = 150\), training window \(tV/L \in [0,10]\), and forecast horizon to \(tV/L = 30\) [2303.12971].

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 [2303.12971]. 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 [2401.02311]. 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 \(2S\) 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 [2303.12971]. 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 | \(\infty\) | \(\infty\) |

For unseen spring stiffness values in VIV, the paper evaluates both interpolation at \(k=0.14\) and extrapolation at \(k=0.18\). 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 [2303.12971]. 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 | \(\infty\) | \(\infty\) |

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 [2303.12971]. 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 [2303.12971]. 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 [2102.09095]. 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 \(240\text{–}250\times\), but future-horizon divergence appears after about 25–30 predicted steps without feedback correction [2102.09095]. 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 [2105.06785]. 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 [2105.06785].

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 [2401.02311]. 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 [2505.23940]. 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 [2303.12971]. 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 [2401.02311]; the POD-RNN/CRAN framework is a partitioned reduced-order approach [2102.09095]; the preCICE-coupled tube model is a partitioned neural prototype [2105.06785]; and Diff-FlowFSI is a differentiable CFD platform [2505.23940].

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 [2303.12971].

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 [2303.12971]. 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.

Source: https://www.emergentmind.com/topics/deepfsi