The research, led by Hao Wu, Kai Pan, and Huatao Chen of Shandong University of Technology, together with Kun Zhao of the Beijing Electro-Mechanical Engineering Institute and Juan L. G. Guirao of the Technical University of Cartagena, addresses a long-standing bottleneck. Accurately simulating water entry requires computational fluid dynamics (CFD) tools that resolve strong nonlinear fluid–structure interaction, free-surface deformation, cavity formation, and turbulence. A single high-fidelity simulation can consume enormous computational resources, making it impractical to sweep across the full space of entry angles and velocities that engineers need to explore when designing a vehicle or planning a mission.

To build their training data, the team simulated the oblique water entry of a delta-wing vehicle using the commercial CFD platform STAR-CCM+. The numerical setup combined the Navier–Stokes equations with the Volume of Fluid method to track the air–water interface, a standard k–ε turbulence model, and a three-degree-of-freedom rigid-body motion model that allowed the vehicle to translate in the vertical plane and pitch about its transverse axis. The researchers examined entry angles from 40 to 60 degrees and entry velocities from 15 to 30 meters per second, a range chosen because prior studies indicate that moderately large oblique angles help minimize instantaneous impact loads while preserving trajectory stability. Mesh independence testing confirmed that the chosen grid resolution reproduced center-of-mass trajectories with a relative error of just over two percent compared with a much finer mesh.

The simulations revealed the tightly coupled physics the model would need to learn. As the vehicle penetrates the surface, free-surface deformation, local pressure peaks, and cavity evolution unfold together in time, directly shaping the tangential force, normal force, and pitching moment acting on the craft. The team found that peak loads rise nearly steadily with entry velocity, and that the normal force becomes especially sensitive to velocity at the steepest 60-degree entry angle. Displacements of the center of mass also grow with velocity, though more gently. These five quantities—two forces, one moment, and two displacement components—formed the prediction targets for the new machine learning framework.

At the heart of the study is a hybrid architecture that merges two strands of modern machine learning: long short-term memory (LSTM) networks and physics-informed neural networks (PINNs). LSTMs are recurrent networks with internal gating structures that allow them to selectively retain and update information over long time horizons, making them well suited to capturing the short-lived but history-dependent transients of water entry. PINNs, first systematically proposed by Raissi and colleagues in 2019, embed physical equations directly into a network’s training loss, forcing predictions to remain consistent with known dynamics. Purely data-driven models, the authors note, can lose physical consistency and robustness when asked to predict conditions not well represented in their training data—a serious limitation for safety-critical tasks such as peak-load estimation.

The innovation lies in how the two components are coupled. The LSTM takes the initial entry conditions and advances the flow of loads in time, outputting the resultant tangential force, normal force, and pitching moment at each step through a linear output layer that avoids imposing artificial bounds on signed, spike-prone quantities. Crucially, the vehicle’s trajectory is not produced by a separate neural output branch. Instead, the LSTM-predicted loads are fed into a discrete Newton–Euler integration layer, which recursively propagates the rigid-body velocities, attitude, and displacements forward in time. This ensures that the predicted forces and the predicted motion remain physically consistent with one another—a property that standalone data-driven models often lack.

Training combines three loss terms: a load-supervision term that matches predicted force and moment histories to the CFD references, a trajectory-supervision term that anchors the integrated displacements to the simulated paths, and a Newton–Euler consistency penalty that enforces the discrete load–motion relation. The physical term is weighted at 0.1, a deliberate compromise so that the physics constraint regularizes the model without smoothing away the sharp transient peaks that matter most in impact engineering. To assess generalization honestly, the team used leave-one-condition-out cross-validation at the level of complete operating sequences: each angle–velocity combination was held out in its entirety during one validation run, so the reported accuracy reflects genuine cross-condition prediction within the tested domain rather than interpolation between scattered time points.

The results show that the hybrid model reproduces the full time histories of loads and trajectories with high fidelity across the 40 to 60 degree range. It captures the rapid rise and multi-peak structure of the normal force and pitching moment under strongly transient conditions, with only minor phase shifts near some secondary peaks. Scatter plots of predicted versus simulated characteristic quantities cluster tightly along the diagonal, and error histograms concentrate in low-error intervals without long tails or systematic bias. In head-to-head comparisons under identical network configurations and training data, the standalone LSTM reproduced broad trends but overshot and shifted phase near peaks, while the standalone PINN fit smoothly but missed peak amplitudes and high-frequency detail. The hybrid LSTM–PINN model achieved lower overall errors and better agreement across MSE, MAE, and R-squared metrics.

The authors are careful to frame the scope of their claims. The demonstrated generalization is in-domain: it covers complete held-out sequences within the investigated ranges of 40 to 60 degrees and 15 to 30 meters per second, not extrapolation to new geometries, steeper angles, or experimental conditions. The Newton–Euler constraint is a consistency relation for rigid-body motion rather than a complete hydrodynamic model; effects such as added mass, hydrodynamic damping, and nonlinear buoyancy are learned implicitly from the CFD-derived resultant loads rather than parameterized explicitly. Even so, the framework offers something engineering practice has lacked: a fast, physically grounded surrogate that can sweep the water-entry parameter space in seconds instead of hours, evaluate peak magnitude and peak timing robustly, and feed directly into attitude optimization and real-time control design. Future work, the team says, will expand the operating-condition database, incorporate experimental measurements, and extend the evaluation to multi-attribute criteria including uncertainty, load-impulse error, and computational efficiency—steps that could bring physics-guided machine learning from the simulation lab into the design loop for the next generation of air-and-sea vehicles.

Subject of Research: Physics-guided deep learning prediction of water-entry impact loads and motion trajectories for cross-medium vehicles

Article Title: Prediction of water entry loads and motion trajectories for a delta wing cross medium vehicle using a physics guided LSTM model

Article References: Wu, H., Pan, K., Zhao, K., Guirao, J. L. G., & Chen, H. (2026). Prediction of water entry loads and motion trajectories for a delta wing cross medium vehicle using a physics guided LSTM model. Discover Artificial Intelligence, 6(1), Article 1129. https://doi.org/10.1007/s44163-026-02085-3

Image Credits: AI Generated

DOI: 10.1007/s44163-026-02085-3

Keywords: cross-medium vehicle, water entry, LSTM, physics-informed neural networks, computational fluid dynamics, fluid–structure interaction, delta wing, hydrodynamic loads, motion trajectory prediction, STAR-CCM+, machine learning, ocean engineering