arxiv: 2604.07129 · v1 · submitted 2026-04-08 · ⚛️ physics.flu-dyn · cs.LG· cs.NA· math.NA· physics.ao-ph

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

Elsa Cardoso-Bihlo , Alex Bihlo This is my paper

Pith reviewed 2026-05-10 18:00 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cs.LGcs.NAmath.NAphysics.ao-ph

keywords differentiable physicsshallow water equationscoastal hydrodynamicsautomatic differentiationinverse modelingtopology optimizationbathymetry inversionwave run-up

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The pith

A fully differentiable non-hydrostatic shallow-water solver allows end-to-end optimization for coastal hydrodynamics tasks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a hydrodynamic solver implemented entirely within a reverse-mode automatic differentiation framework. This treats the time-marching loop of the depth-integrated non-hydrostatic shallow-water equations as one continuous computational graph. Consequently, problems such as learning neural corrections for wave propagation, optimizing breakwater topology, training wave-cancellation controllers, and inverting bathymetry from observations all reduce to standard gradient-based optimization. A reader would care because this removes the traditional barrier of deriving separate adjoint codes for each inverse task in coastal engineering.

Core claim

By implementing the full time-marching physics loop inside a reverse-mode automatic differentiation system, the solver makes forward simulation and inverse optimization interchangeable operations within a single framework, demonstrated on neural model correction, continuous topology optimization, in-the-loop neural control, and direct inversion of bathymetry and landslide sources.

What carries the argument

AegirJAX, the fully differentiable implementation of the non-hydrostatic shallow-water equations that embeds the entire simulation time loop as a single computational graph supporting reverse-mode differentiation.

Load-bearing premise

The full time-marching loop of the non-hydrostatic shallow-water solver can be stably embedded as a single computational graph in reverse-mode automatic differentiation without prohibitive memory costs, numerical instability, or loss of physical accuracy.

What would settle it

Running long time integrations where the memory footprint grows prohibitively or where computed gradients fail to match finite-difference checks would show the approach is not yet practical.

Figures

Figures reproduced from arXiv: 2604.07129 by Alex Bihlo, Elsa Cardoso-Bihlo.

**Figure 2.** Figure 2: Neural-augmented correction of the Monai Valley benchmark. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: Zero-shot generalization of the neural correction on the Conical Island benchmark. The [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Optimization of a rigid breakwater gate to minimize harbor inundation. [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: Continuous topology optimization for breakwater design under a strict material volume [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 6.** Figure 6: Zero-shot active wave cancellation using a policy-in-the-loop recurrent neural network. [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

**Figure 7.** Figure 7: Recovery of the submerged trapezoidal bathymetry [ [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗

**Figure 8.** Figure 8: Joint inversion of submarine landslide kinematics and spatial deformation. [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗

**Figure 9.** Figure 9: Inverse source localization of a passively advected tracer in the Monai Valley domain. [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper delivers a JAX-native non-hydrostatic shallow-water solver that supports end-to-end gradients for four coastal tasks, but leaves the memory and stability details for long runs unaddressed in the framing.

read the letter

The main takeaway is that they built the full time-marching non-hydrostatic shallow-water solver inside JAX so the entire forward loop becomes one differentiable graph. This removes the need for hand-derived adjoints and lets the same code handle neural corrections for dispersive waves, continuous topology optimization of breakwaters, in-the-loop RNN wave control, and direct inversion of bathymetry or landslide sources from sensor data. That combination is not a routine wrapper; it is a concrete implementation that treats the physics as part of the computational graph rather than a black box. The four demonstrations show the framework can switch between forward simulation and inverse problems without rewriting the solver each time, which is the practical payoff for coastal engineering workflows. The equations are the standard depth-integrated non-hydrostatic form, and the choice to stay inside JAX keeps everything in one autodiff system. That is the part worth noting. The soft spot is the lack of any reported numbers on accuracy, wall-clock cost, or memory use once the time horizon grows. The stress-test concern about O(T) memory scaling and potential gradient instability from the pressure correction step is not answered in the abstract or the high-level description. If the implementation relies on plain reverse-mode AD without checkpointing or custom adjoints, realistic coastal durations would hit memory walls or lose accuracy. The paper would be stronger with a short section showing how they managed the vjp for hundreds of steps and what the observed gradient norms look like. Readers who already work with JAX and need to couple a coastal model to optimization or learning will find the most immediate use. The work is concrete enough on the solver side and the task list that it should go to peer review so referees can check the code, the scaling behavior, and whether the claimed inversions actually converge on held-out data.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces AegirJAX, a fully differentiable hydrodynamic solver for the depth-integrated non-hydrostatic shallow-water equations implemented entirely in JAX to enable reverse-mode automatic differentiation through the time-marching loop. It claims this framework unifies forward simulation and inverse optimization, with demonstrations on four tasks: neural corrections for dispersive wave model misspecifications, continuous topology optimization of breakwaters, in-the-loop training of recurrent networks for active wave cancellation, and direct inversion of bathymetry and submarine landslide kinematics from sensor data.

Significance. If the implementation successfully embeds the full non-hydrostatic time-marching loop with stable gradients and manageable memory cost, the work could meaningfully advance end-to-end differentiable physics for coastal engineering inverse problems, reducing reliance on hand-derived adjoints. The breadth of the four claimed demonstrations suggests potential for high impact in scientific machine learning applications, though this hinges on unshown validation.

major comments (2)

[Abstract] Abstract: the four demonstration tasks are enumerated but the text supplies no quantitative results, error metrics, convergence checks, baseline comparisons, or implementation details, so the central claim that the framework successfully enables these tasks cannot be verified from the provided material.
[Solver description] The viability of treating the entire forward physics loop (depth-integrated non-hydrostatic SWE with free-surface evolution and pressure correction) as one unbroken computational graph in reverse-mode AD is load-bearing for all applications, yet no discussion appears of memory mitigation (e.g., checkpointing), custom vector-Jacobian products, or stability analysis for the dispersive and non-hydrostatic terms that could introduce stiffness and gradient amplification over realistic coastal time horizons.

minor comments (1)

[Abstract] The acronym AegirJAX is introduced without etymology or relation to prior coastal solvers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed review of our manuscript on AegirJAX. We address each major comment below and have revised the manuscript to strengthen the presentation of results and implementation details.

read point-by-point responses

Referee: [Abstract] Abstract: the four demonstration tasks are enumerated but the text supplies no quantitative results, error metrics, convergence checks, baseline comparisons, or implementation details, so the central claim that the framework successfully enables these tasks cannot be verified from the provided material.

Authors: We agree that the original abstract was high-level and did not include quantitative indicators. In the revised manuscript we have expanded the abstract to summarize key quantitative outcomes for each of the four tasks (e.g., relative L2 errors for neural corrections, convergence of the topology optimization objective, wave-cancellation attenuation factors, and bathymetry inversion RMSE). Full error metrics, baseline comparisons (including non-differentiable adjoint and finite-difference approaches), convergence diagnostics, and implementation specifics (JAX version, hardware, checkpointing settings) are now explicitly referenced in the abstract and detailed in Sections 4–5 and the supplementary material. revision: yes
Referee: [Solver description] The viability of treating the entire forward physics loop (depth-integrated non-hydrostatic SWE with free-surface evolution and pressure correction) as one unbroken computational graph in reverse-mode AD is load-bearing for all applications, yet no discussion appears of memory mitigation (e.g., checkpointing), custom vector-Jacobian products, or stability analysis for the dispersive and non-hydrostatic terms that could introduce stiffness and gradient amplification over realistic coastal time horizons.

Authors: The referee correctly highlights that memory cost and gradient stability are central to the practicality of the approach. The original manuscript described the JAX implementation but did not elaborate on these engineering aspects. We have added a new subsection (Section 3.4) that (i) details the checkpointing strategy used to keep memory linear in the number of time steps rather than quadratic, (ii) describes custom vector-Jacobian products implemented for the non-hydrostatic pressure Poisson solve to avoid full Jacobian materialization, and (iii) presents numerical experiments quantifying gradient amplification for the dispersive and non-hydrostatic terms over the time horizons employed in the demonstrations (up to several thousand steps). We note that for significantly longer simulations additional techniques such as adjoint checkpointing or reduced-order modeling may be required, but the reported results demonstrate stable gradients for the coastal-engineering time scales considered. revision: yes

Circularity Check

0 steps flagged

No circularity: AegirJAX is a new JAX implementation of the non-hydrostatic SWE solver with independent demonstrations.

full rationale

The paper introduces a fully differentiable solver by embedding the depth-integrated non-hydrostatic shallow-water equations and time-marching loop directly into reverse-mode AD in JAX. No derivation reduces a claimed result to its own fitted parameters or inputs by construction. The four demonstration tasks (neural corrections, topology optimization, RNN-in-the-loop cancellation, bathymetry inversion) are applications of the framework rather than re-derivations of its core equations. No self-citation is invoked as a uniqueness theorem or to justify the central differentiability claim; the implementation itself supplies the end-to-end graph. The framework is therefore self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on two domain assumptions: that the chosen depth-integrated non-hydrostatic equations remain adequate across the demonstrated regimes, and that reverse-mode AD can be applied to the entire time-marching loop without prohibitive cost or instability. No free parameters or new physical entities are explicitly introduced in the abstract.

axioms (2)

domain assumption Depth-integrated non-hydrostatic shallow-water equations are sufficient to capture the target wave propagation and run-up phenomena
Stated as the basis for the AegirJAX solver.
domain assumption Reverse-mode automatic differentiation can be applied to the full time-marching physics loop as a single continuous graph
Core technical premise required for the end-to-end differentiability claim.

invented entities (1)

AegirJAX no independent evidence
purpose: Fully differentiable hydrodynamic solver
New software framework introduced to realize the solver-in-the-loop paradigm.

pith-pipeline@v0.9.0 · 5516 in / 1479 out tokens · 49546 ms · 2026-05-10T18:00:56.021576+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

By implementing the solver entirely within a reverse-mode automatic differentiation framework, AegirJAX treats the time-marching physics loop as a continuous computational graph.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the entire unrolled fluid simulation is a differentiable computational graph

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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