Floating-Body Hydrodynamic Neural Networks

• Floating-Body Hydrodynamic Neural Networks (FHNN) are physically structured models that integrate explicit hydrodynamic parameters such as added mass and drag into neural formulations.
• They leverage interpretable physics by embedding fluid-mechanical structure into the network design, enabling stable long-horizon predictions in fluid–structure interactions.
• Various instantiations—from physics-structured translational models to operator-based and inverse PINN approaches—demonstrate improved efficiency and accuracy over black-box methods.

Floating-Body Hydrodynamic Neural Networks (FHNN) are a family of neural formulations for fluid–structure interaction in which hydrodynamic effects are not treated as an unconstrained black-box state-derivative regressor, but are instead represented through explicit physical structure: interpretable added mass and drag parameters, streamfunction-based incompressible flows, geometry-resolved surface loads, differentiable field surrogates, operator maps from wave histories to structural responses, or inverse identification of Cummins coefficients from motion data. Across the recent literature, the term encompasses at least five closely related but distinct instantiations: a physics-structured translational model for rigid bodies in two-dimensional incompressible flow (Zhang et al., 17 Sep 2025), a differentiable immersed-body surrogate for deformable swimmers (Nava et al., 2022), a geometry-aware per-surface load model for partially submerged platforms (Waheed et al., 18 May 2026), neural-operator digital twins for offshore floating responses (Cao et al., 2023), and physics-informed inverse models that estimate hydrodynamic coefficients from movement data (Schou et al., 6 Jul 2026).

1. Conceptual scope and defining characteristics

FHNN emerged in response to a specific difficulty in fluid–structure interaction: realistic floating-body motion is governed by dissipative hydrodynamics, including added mass, linear and quadratic drag, and background flow, yet direct end-to-end regression of state derivatives often yields poor interpretability and unstable long-horizon prediction. The defining FHNN move is therefore to constrain the hypothesis space with fluid-mechanical structure rather than to learn arbitrary acceleration laws. In the canonical rigid-body formulation, the network predicts directional added masses, drag coefficients, and a streamfunction, then integrates analytic equations of motion. This design is intended to preserve incompressibility, separate hydrodynamic mechanisms, and stabilize rollouts (Zhang et al., 17 Sep 2025).

The term is not used uniformly across papers. In some works, FHNN refers narrowly to the 2025 physics-structured model for a rigid body translating in two-dimensional incompressible flow. In other works, the same label is applied more broadly to geometry-aware per-surface surrogates for floating and partially submerged platforms, or to PINN formulations embedding the Cummins equations. This suggests that FHNN is best understood as an architectural principle rather than a single standardized network class: hydrodynamic learning is coupled to explicit mechanics, and the learned quantities retain physical meaning.

A common misconception is that any neural surrogate for marine dynamics is automatically an FHNN. The literature distinguishes FHNN from purely black-box Neural ODEs and from energy-based models such as Hamiltonian Neural Networks and Lagrangian Neural Networks. The former offer flexibility but obscure physical quantities such as mass or drag and tend to degrade over long horizons; the latter are naturally suited to conservative systems but not to dissipative dynamics with unsteady ambient flow (Zhang et al., 17 Sep 2025). A second misconception is that all FHNNs are inherently free-surface models. The differentiable swimmer simulator of 2022 is explicitly single-phase, fully immersed, and without gravity or buoyancy; it is therefore an FHNN for submerged bodies, not yet for a true air–water-interface floating-body problem (Nava et al., 2022).

2. Physics-structured FHNN for two-dimensional translational dynamics

The most explicit definition of FHNN appears in the 2025 formulation for rigid bodies translating in a two-dimensional incompressible flow. The state is

$s=(x,\,y,\,v_x,\,v_y)^\top,$

and the background flow is represented by a scalar streamfunction,

$$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$

which guarantees $\nabla\!\cdot\!\boldsymbol{u}=0$ by construction. Relative motion is defined as

$$\boldsymbol{v}_{\mathrm{rel}}=\boldsymbol{v}-\boldsymbol{u}(\boldsymbol{x}),\qquad \sigma=\|\boldsymbol{v}_{\mathrm{rel}}\|+\varepsilon,$$

with $\varepsilon$ preventing degeneracy when body and flow velocities coincide (Zhang et al., 17 Sep 2025).

The learned hydrodynamic quantities are directional added masses $m_{ax}(r,\sigma)$ and $m_{ay}(r,\sigma)$, linear and quadratic drag coefficients $c_l(r,\sigma)$ and $c_q(r,\sigma)$, and the streamfunction $\psi(x,y)$. The coefficient network is rotation-invariant in the specific sense used in the paper: it takes scalar invariants $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$0 and $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$1 as inputs rather than raw orientation variables. The effective mass matrix is diagonal,

$$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$2

and the hydrodynamic force is decomposed into quadratic and linear drag,

$$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$3

The full dynamics are then

$$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$4

Physical plausibility is enforced through a bounded nonnegative parameter map,

$$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$5

with $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$6 for $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$7. Training combines derivative matching, one-step RK4 rollout matching, and flow smoothness regularization: $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$8

$$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$9

$\nabla\!\cdot\!\boldsymbol{u}=0$0

RK4 is used in the step loss, while long-horizon evaluation uses a high-accuracy ODE solver (Zhang et al., 17 Sep 2025).

The reported quantitative behavior is central to the FHNN claim. On synthetic vortex datasets, FHNN achieves up to an order-of-magnitude lower error than a black-box Neural ODE. At $\nabla\!\cdot\!\boldsymbol{u}=0$1, FHNN RMSE(pos) is $\nabla\!\cdot\!\boldsymbol{u}=0$2 versus $\nabla\!\cdot\!\boldsymbol{u}=0$3 for the Neural ODE, RMSE(vel) is $\nabla\!\cdot\!\boldsymbol{u}=0$4 versus $\nabla\!\cdot\!\boldsymbol{u}=0$5, and FDE(pos) is $\nabla\!\cdot\!\boldsymbol{u}=0$6 versus $\nabla\!\cdot\!\boldsymbol{u}=0$7. At $\nabla\!\cdot\!\boldsymbol{u}=0$8, FHNN RMSE(pos) is $\nabla\!\cdot\!\boldsymbol{u}=0$9 versus $$\boldsymbol{v}_{\mathrm{rel}}=\boldsymbol{v}-\boldsymbol{u}(\boldsymbol{x}),\qquad \sigma=\|\boldsymbol{v}_{\mathrm{rel}}\|+\varepsilon,$$0, RMSE(vel) is $$\boldsymbol{v}_{\mathrm{rel}}=\boldsymbol{v}-\boldsymbol{u}(\boldsymbol{x}),\qquad \sigma=\|\boldsymbol{v}_{\mathrm{rel}}\|+\varepsilon,$$1 versus $$\boldsymbol{v}_{\mathrm{rel}}=\boldsymbol{v}-\boldsymbol{u}(\boldsymbol{x}),\qquad \sigma=\|\boldsymbol{v}_{\mathrm{rel}}\|+\varepsilon,$$2, and FDE(pos) is $$\boldsymbol{v}_{\mathrm{rel}}=\boldsymbol{v}-\boldsymbol{u}(\boldsymbol{x}),\qquad \sigma=\|\boldsymbol{v}_{\mathrm{rel}}\|+\varepsilon,$$3 versus $$\boldsymbol{v}_{\mathrm{rel}}=\boldsymbol{v}-\boldsymbol{u}(\boldsymbol{x}),\qquad \sigma=\|\boldsymbol{v}_{\mathrm{rel}}\|+\varepsilon,$$4. Against HNN, LNN, and DHN, the difference is larger: at $$\boldsymbol{v}_{\mathrm{rel}}=\boldsymbol{v}-\boldsymbol{u}(\boldsymbol{x}),\qquad \sigma=\|\boldsymbol{v}_{\mathrm{rel}}\|+\varepsilon,$$5, FHNN errors are about $$\boldsymbol{v}_{\mathrm{rel}}=\boldsymbol{v}-\boldsymbol{u}(\boldsymbol{x}),\qquad \sigma=\|\boldsymbol{v}_{\mathrm{rel}}\|+\varepsilon,$$6 while HNN, LNN, and DHN are around $$\boldsymbol{v}_{\mathrm{rel}}=\boldsymbol{v}-\boldsymbol{u}(\boldsymbol{x}),\qquad \sigma=\|\boldsymbol{v}_{\mathrm{rel}}\|+\varepsilon,$$7; at $$\boldsymbol{v}_{\mathrm{rel}}=\boldsymbol{v}-\boldsymbol{u}(\boldsymbol{x}),\qquad \sigma=\|\boldsymbol{v}_{\mathrm{rel}}\|+\varepsilon,$$8, LNN diverges catastrophically, HNN accumulates multi-meter drift, and DHN remains unstable. Multi-horizon plots show FHNN stable to $$\boldsymbol{v}_{\mathrm{rel}}=\boldsymbol{v}-\boldsymbol{u}(\boldsymbol{x}),\qquad \sigma=\|\boldsymbol{v}_{\mathrm{rel}}\|+\varepsilon,$$9, with accuracy sustained to roughly $\varepsilon$0–$\varepsilon$1 before error grows rapidly by $\varepsilon$2–$\varepsilon$3 (Zhang et al., 17 Sep 2025).

3. Differentiable immersed-body FHNN and the HydroNet lineage

A distinct but closely related lineage is the 2022 differentiable fluid–structure interaction framework combining DiffPD for solid mechanics, a U-Net hydrodynamic surrogate called HydroNet, and an immersed boundary method. In FHNN terminology, HydroNet functions as a hydrodynamic surrogate that maps boundary geometry and motion to pressure and velocity fields, and hence to hydrodynamic forces on an immersed body. The setting is strictly two-dimensional planar flow, with incompressible Navier–Stokes solved at inference by a neural surrogate trained with physics-constrained losses rather than with an explicit pressure Poisson solve (Nava et al., 2022).

The fluid model uses the incompressible Navier–Stokes equations,

$\varepsilon$4

with Dirichlet boundary condition

$\varepsilon$5

In two dimensions, velocity is represented through a scalar curl or streamfunction field $\varepsilon$6,

$\varepsilon$7

so incompressibility is satisfied identically. HydroNet takes as input the current curl field $\varepsilon$8, pressure $\varepsilon$9, a soft boundary mask $m_{ax}(r,\sigma)$0, and a boundary velocity field $m_{ax}(r,\sigma)$1, and predicts $m_{ax}(r,\sigma)$2 and $m_{ax}(r,\sigma)$3 on a $m_{ax}(r,\sigma)$4 MAC grid for a $m_{ax}(r,\sigma)$5 domain with $m_{ax}(r,\sigma)$6. The network is trained without supervised CFD labels using

$m_{ax}(r,\sigma)$7

$m_{ax}(r,\sigma)$8

Spatial derivatives are discretized by finite differences on the MAC grid, and the time derivative uses forward difference. No ground-truth DNS data are required (Nava et al., 2022).

The immersed boundary coupling is pressure-dominated for the water-like viscosities used in the experiments. The continuous net hydrodynamic force and torque are

$m_{ax}(r,\sigma)$9

but in practice the viscous term is neglected. The discrete force per surface element uses a Gaussian-smoothed delta kernel,

$m_{ay}(r,\sigma)$0

and the total external hydrodynamic force is obtained by summing over elements. Differentiable rasterization provides the soft boundary mask and boundary velocities, gradients flow through HydroNet, IBM force spreading, and the DiffPD implicit solve, and long episodes use gradient checkpointing. The framework thereby supports direct gradient-based optimization of control parameters such as the angular frequency $m_{ay}(r,\sigma)$1 in a carangiform actuation envelope (Nava et al., 2022).

The reported efficiency and accuracy are important because they situate HydroNet as a practical FHNN precursor. HydroNet warmup per episode is approximately $m_{ay}(r,\sigma)$2. A $m_{ay}(r,\sigma)$3-frame forward pass takes approximately $m_{ay}(r,\sigma)$4, with DiffPD at approximately $m_{ay}(r,\sigma)$5 and HydroNet at approximately $m_{ay}(r,\sigma)$6. The backward pass takes approximately $m_{ay}(r,\sigma)$7 with gradient checkpointing, whereas a comparable COMSOL Multiphysics episode takes approximately $m_{ay}(r,\sigma)$8, corresponding to a speedup of about $m_{ay}(r,\sigma)$9. Validation against COMSOL shows a monotonic relationship in travelled distance versus controller frequency across $c_l(r,\sigma)$0, the same optimum at $c_l(r,\sigma)$1, and an average mismatch of approximately $c_l(r,\sigma)$2 over a $c_l(r,\sigma)$3 episode. The framework also reproduces qualitative flow features such as Kármán vortex streets and the Magnus effect (Nava et al., 2022).

Its limitations are equally explicit. The model is two-dimensional, single-phase, and fully immersed; there is no free surface, no gravity or buoyancy in the fluid solver, and no air–water interface. The paper therefore characterizes the method as an FHNN for immersed bodies, not yet for true floating-body problems. Extension to a true floating-body FHNN would require two-phase flow, free-surface representation, gravity and buoyancy, and stronger incompressibility control, potentially through a differentiable pressure-projection layer (Nava et al., 2022).

4. Geometry-aware per-surface FHNN for floating and partially submerged bodies

A more direct floating-body instantiation appears in the geometry-aware per-surface surrogate derived from hydrodynamics estimation for amphibious ground vehicles and explicitly generalized to floating and partially submerged bodies. Here FHNN is built around signed-distance-field-based submergence resolution, shared per-surface networks, and aggregation of local loads into net forces and moments. The body is decomposed into $c_l(r,\sigma)$4 semantically meaningful patches, and a body-specific SDF determines which portions are wet and how deeply submerged. For each surface patch, the features include submerged fraction, normalized mean submerged depth, type one-hot, centroid position, average normal, area, and projected area relative to the body-frame velocity direction (Waheed et al., 18 May 2026).

The submergence features are defined from sampled surface points using

$c_l(r,\sigma)$5

$c_l(r,\sigma)$6

$c_l(r,\sigma)$7

Global features include body-frame speed and velocity components, density, mean submerged depth, reference dimensions, and the dimensionless groups

$c_l(r,\sigma)$8

Inputs are expressed in a body-fixed right-handed frame, per-surface features are kept in physical units, and targets are normalized by density as $c_l(r,\sigma)$9 and then z-scored for training (Waheed et al., 18 May 2026).

The per-surface predictor is a shared MLP with two hidden layers of width $c_q(r,\sigma)$0 and ReLU activations. For each surface,

$c_q(r,\sigma)$1

$c_q(r,\sigma)$2

and the output is the three-component force on that patch in the body frame. Net loads are then reconstructed through

$c_q(r,\sigma)$3

The training loss is

$c_q(r,\sigma)$4

with

$c_q(r,\sigma)$5

$c_q(r,\sigma)$6, and $c_q(r,\sigma)$7 in the relative term. $c_q(r,\sigma)$8 enforces net-force consistency and $c_q(r,\sigma)$9 penalizes non-zero predicted force on dry surfaces (Waheed et al., 18 May 2026).

The training data use high-fidelity CFD with two-phase VOF free-surface modeling, compressive volume-fraction discretization, transient pressure-based RANS, and the $\psi(x,y)$0–$\psi(x,y)$1 SST turbulence model. The reported datasets cover two watertight rigid meshes, Clearpath Husky A200 and Warthog, with Latin Hypercube Sampling over speed, flow angle, density, and water depth. Time steps are $\psi(x,y)$2 for Husky and $\psi(x,y)$3 for Warthog; four seconds of physical time are simulated, the first two seconds are discarded, and the last two are split into twenty quasi-steady sections for sample generation. Bilateral and longitudinal symmetries are used for $\psi(x,y)$4 data augmentation (Waheed et al., 18 May 2026).

Held-out CFD results show longitudinal-force symmetric MAPE of approximately $\psi(x,y)$5 on both vehicles, vertical-force sMAPE of approximately $\psi(x,y)$6 for Husky and $\psi(x,y)$7 for Warthog, and lateral sMAPE of approximately $\psi(x,y)$8, which the paper attributes to near-zero sign-changing distributions. Single-sample CPU latency has median approximately $\psi(x,y)$9–$$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$00 with $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$01, and end-to-end inference remains below $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$02 per sample, making the approach suitable for control and planning loops at $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$03–$$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$04 (Waheed et al., 18 May 2026).

A major result is the emergence of physical laws not explicitly encoded in the loss. In full-scale wading trials, the predicted drag follows quadratic speed scaling with $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$05 in the abstract and $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$06–$$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$07 across depths in the detailed validation. Buoyancy intercepts scale linearly with depth with $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$08. The paper interprets this as evidence that summing patchwise forces with SDF-resolved submergence reproduces meaningful hydrodynamic structure. For FHNN, the significance is that geometry-aware local decomposition can recover net force laws without imposing them directly (Waheed et al., 18 May 2026).

5. Neural operators and the operator viewpoint in FHNN

Another strand of the literature approaches floating-body hydrodynamics at the level of input–output operators rather than local force laws. In the offshore-structure study, the learned object is the coupled hydrodynamic-structural map from irregular-wave histories to rigid-body motions and mooring-line tension. The equation of motion is given in Cummins form,

$$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$09

with hydrodynamic data generated by WAMIT under linear and second-order potential theory and time-domain responses simulated in Orcaflex. Inputs are irregular-wave elevation histories $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$10, optionally together with initial conditions, and outputs are target responses such as surge, heave, pitch, and mooring top tension (Cao et al., 2023).

The study evaluates DeepONet, FNO, and WNO, together with extensions including POD-DeepONet, DeepONet with history, self-adaptive DeepONet, self-adaptive WNO, and wavelet-DeepONet. The operator mappings are written as

$$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$11

and, for fixed initial conditions,

$$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$12

The paper explicitly states that this combined operator viewpoint is crucial for FHNN design, because it treats the floating-body system as a map from input functionals to output functionals rather than as a pointwise regression problem (Cao et al., 2023).

The comparative findings are sharply differentiated by spectral character. FNO performs best for single-functional inputs and narrow-band responses such as heave, pitch, and mooring tension. DeepONet with historical states performs best for broadband, transient responses such as surge and for varying initial conditions. In zero-initial-condition surge prediction, Case 3 gives DeepONet with history MSE approximately $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$13 versus FNO approximately $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$14, GRU approximately $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$15, and WNO approximately $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$16. In varying-initial-condition surge prediction, Case 8 gives DeepONet with history MSE approximately $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$17 versus FNO approximately $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$18 and GRU approximately $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$19. Increasing the number of historical states $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$20 improves surge prediction, and the study uses $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$21 in most figures. Wavelet-DeepONet and self-adaptive WNO improve surge accuracy over their vanilla counterparts, though SA-WNO is reported to be sensitive to batch size and hyperparameters (Cao et al., 2023).

For digital-twin applications, inference speed is decisive. For $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$22 samples, the trained neural operators require less than $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$23 seconds, whereas Orcaflex with $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$24 parallel threads requires approximately $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$25 minutes. The study therefore reports greater than two orders-of-magnitude improvement in response delivery. This does not make the models hydrodynamic coefficient estimators in the narrow FHNN sense, but it does establish a complementary operator-learning route for real-time floating-body prediction under irregular waves (Cao et al., 2023).

6. PINN-based inverse FHNN and the limits of current formulations

A further expansion of FHNN recasts the problem as inverse identification: instead of predicting loads or motions directly, the network estimates hydrodynamic coefficients from measured movement data by embedding the Cummins equations into the loss. In the 2026 PINN formulation, the time-domain equation

$$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$26

is rewritten as a first-order system in $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$27 and $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$28, with the causal convolution evaluated numerically by Gaussian quadrature. The method uses two stages: a motion PINN reconstructs state trajectories from displacement data, and a coefficient PINN then learns the radiation impulse response function $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$29 and the infinite-frequency added mass $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$30 from the learned trajectories via time-domain, asymptotic, and Laplace-domain residuals (Schou et al., 6 Jul 2026).

For the one-dimensional heave experiments, the motion surrogates are decoupled MLPs

$$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$31

with Fourier feature embeddings, two hidden layers, $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$32 neurons per layer, tanh activation, and $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$33 Fourier features. The IRF network uses a decaying ansatz

$$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$34

with $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$35, two hidden layers, $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$36 neurons per layer, and $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$37 Fourier features. Hydrodynamic coefficients are scaled by

$$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$38

for training stability. The coefficient loss combines the Cummins residual, two asymptotic residuals, and a Laplace-domain residual, with weights proportional to $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$39 and normalized to sum to $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$40; self-adaptive PINN weighting is applied to the time-domain term (Schou et al., 6 Jul 2026).

The validation cases are free-decay heave motion of a sphere and a box, with known mass $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$41 and hydrostatic restoring $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$42, and $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$43. The data are generated by DTUMotionSimulator, displacement is sampled from $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$44 to $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$45, and fourth-order finite differences provide velocity and acceleration. Three data regimes are tested: dense ($$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$46), noisy ($$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$47 with $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$48 relative noise), and sparse ($$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$49). The normalized estimates of $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$50 demonstrate the data dependence of inverse FHNN. For the sphere, with $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$51, the reference value is $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$52 and the predictions are $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$53 in the dense case, $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$54 in the noisy case, and $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$55 in the sparse case, corresponding to relative errors of $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$56, $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$57, and $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$58. For the box, with $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$59, the reference is $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$60 and the predictions are $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$61, $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$62, and $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$63, giving relative errors of $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$64, $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$65, and $$\boldsymbol{u}(\boldsymbol{x})=\nabla^\perp\psi(\boldsymbol{x}),$$66 (Schou et al., 6 Jul 2026).

These results define several current limits of FHNN as a broader research area. First, not all formulations solve the same problem. The 2025 physics-structured FHNN is two-dimensional and translational only, with no rotational degrees of freedom or Coriolis/centripetal terms (Zhang et al., 17 Sep 2025). The 2022 HydroNet framework is submerged, single-phase, and lacks free-surface, gravity, and buoyancy effects (Nava et al., 2022). The per-surface SDF surrogate is trained in a quasi-steady regime and therefore does not directly represent added-mass transients, radiation damping, or vortex-shedding periodicity without temporal extensions such as RNNs, LSTMs, GRUs, or transformer encoders (Waheed et al., 18 May 2026). The neural-operator offshore study relies mainly on linear and second-order potential theory and does not include significant viscous damping or breaking waves (Cao et al., 2023). The PINN inverse approach assumes linear wave–body interaction, known hydrostatic restoring, and in its experiments only one-dimensional heave free decay (Schou et al., 6 Jul 2026).

Second, “floating-body” should not be conflated with “fully realistic marine hydrodynamics.” The literature repeatedly identifies missing ingredients required for a full free-surface FHNN: two-phase flow, gravity and buoyancy, explicit free-surface boundary conditions, pressure-projection or stronger divergence control, three-dimensional extension, temporal memory for radiation damping, and robust treatment of turbulence, slamming, green water, cavitation, and planing (Nava et al., 2022). A plausible implication is that FHNN is presently less a finished model class than a unifying program for physically structured hydrodynamic learning: one branch emphasizes interpretable reduced-order equations, another geometry-resolved local loads, another operator surrogates for wave-to-response maps, and another inverse recovery of hydrodynamic coefficients from motion records.