physics.flu-dynApr 8, 2026

A solver-in-the-loop framework for end-to-end differentiable coastal hydrodynamics

Authors: Elsa Cardoso-Bihlo, Alex Bihlo

Organizations: Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7, Canada

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Abstract

Numerical simulation of wave propagation and run-up is a cornerstone of coastal engineering and tsunami hazard assessment. However, applying these forward models to inverse problems, such as bathymetry estimation, source inversion, and structural optimization, remains notoriously difficult due to the rigidity and high computational cost of deriving discrete adjoints. In this paper, we introduce AegirJAX, a fully differentiable hydrodynamic solver based on the depth-integrated, non-hydrostatic shallow-water equations. By implementing the solver entirely within a reverse-mode automatic differentiation framework, AegirJAX treats the time-marching physics loop as a continuous computational graph. We demonstrate the framework's versatility across a suite of scientific machine learning tasks: (1) discovering regime-specific neural corrections for model misspecifications in highly dispersive wave propagation; (2) performing continuous topology optimization for breakwater design; (3) training recurrent neural networks in-the-loop for active wave cancellation; and (4) inverting hidden bathymetry and submarine landslide kinematics directly from downstream sensor data. The proposed differentiable paradigm fundamentally blurs the line between forward simulation and inverse optimization, offering a unified, end-to-end framework for coastal hydrodynamics.

Figures & tables

Figure 1: Correction of missing dispersive physics in the Beji–Battjes benchmark [ 6 ] The baseline hydrostatic model (blue) fails to capture the short-wavelength super-harmonics behind the submerged bar. By utilizing higher-order spatial derivatives as dispersion triggers, the neural-augmented solver (red) learns a regime-specific forcing that accurately recovers the experimental wave amplitudes and phases (black) for both non-breaking and spilling wave cases.
Figure 2: Neural-augmented correction of the Monai Valley benchmark. Top row: Wave gauge time-series comparing the baseline hydrostatic solver, the hybrid model, and experimental ground truth. Middle and bottom rows: Snapshots of the surface elevation differences and the corresponding spatial neural momentum corrections ann\mathbf{a}_{\rm nn} at t=16t=16 s. The network injects localized forcing to correct the wave routing and secondary reflections missed by the coarse-grid baseline.
Figure 3: Zero-shot generalization of the neural correction on the Conical Island benchmark. The hybrid model, trained only on extreme wave amplitudes, is evaluated on an unseen intermediate wave, A/h≈0.096A/h\approx 0.096 . The neural-augmented solver successfully interpolates the missing physics, significantly reducing phase errors and amplitude under-predictions compared to the coarse-grid hydrostatic baseline.
Figure 4: Optimization of a rigid breakwater gate to minimize harbor inundation. Top: Initial misaligned state, intermediate optimization step, and final converged gate position. Bottom: Convergence history of the wave gauge energy, the time series of the optimized variables and the top-down spatial trajectory of the gate’s center coordinates guided by analytical gradients into the static wall gap.
Figure 5: Continuous topology optimization for breakwater design under a strict material volume constraint. Top: The initial sloping beach topography and the final optimally distributed breakwater material Δh\Delta h . Bottom: The optimization history demonstrating the minimization of wave impact energy on the facility (left) while adhering to the target material volume budget (right) . The optimizer autonomously discovers that a curved offshore berm optimally refracts the incident wave.
Figure 6: Zero-shot active wave cancellation using a policy-in-the-loop recurrent neural network. The learned policy reads incoming wave states from an upstream sensor array (yellow region) and commands a parameterized actuator (gray region). The RNN accurately anticipates the unseen wave’s phase and amplitude, applying a destructive interference force that reduces the total energy entering the protected harbor (green region) by over 97%.
Figure 7: Recovery of the submerged trapezoidal bathymetry [ 6 ] from downstream wave gauge observations. Left: A continuous neural network parameterization accurately recovers the front and lee slopes of the bar. The deviation on the far right reflects the physical ill-posedness of the unmonitored shadow zone. Right: The resulting wave time-series demonstrating precise agreement with the downstream observations.
Figure 8: Joint inversion of submarine landslide kinematics and spatial deformation. Top: Comparison of the true and reconstructed initial bottom topography at t=0t=0 . Middle and bottom: Reconstructed surface elevation time-series at near-field and far-field gauges. By back-propagating the downstream error, AegirJAX successfully recovers both the spatial footprint of the mass and its log-cosh acceleration profile.
Figure 9: Inverse source localization of a passively advected tracer in the Monai Valley domain. Top: The spatial trajectory of the source optimizer navigating the bathymetry to the ground-truth location, alongside the convergence of the gauge mean squared error. Middle: The recovered concentration time-series at three downstream sensors. Bottom: Snapshots of the simulated tracer plume advecting through the complex velocity field.
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