Data-Driven Surrogate Models for Agromaritime Applications: Finite Element-Neural Network Integration

Muhammad Ilyas

arxiv: 2604.21058 · v1 · submitted 2026-04-22 · 🧮 math.NA · cs.NA

Data-Driven Surrogate Models for Agromaritime Applications: Finite Element-Neural Network Integration

Muhammad Ilyas This is my paper

Pith reviewed 2026-05-09 22:59 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords surrogate modelingfinite element methodneural networkproper orthogonal decompositiondiffusion-reaction equationnutrient transportsalinity distributionagromaritime systems

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The pith

A neural network trained on finite element data predicts nutrient and salinity distributions nearly 1000 times faster than direct simulation at 15 percent average error.

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

The paper establishes a hybrid workflow that runs a finite element solver on the steady-state diffusion-reaction equation to build a dataset over ranges of diffusivity, reaction rate, and inflow parameters, then compresses the solutions with proper orthogonal decomposition and trains a neural network to map parameters straight to the reduced coefficients. This produces a surrogate that can deliver full-field predictions without repeating the expensive mesh solve. A sympathetic reader would care because full physics models capture the relevant transport and reaction processes accurately but become impractical for real-time decisions or exhaustive exploration of climate-driven changes in agromaritime nutrient and salinity fields, whereas the surrogate trades a modest error tolerance for orders-of-magnitude speed gains suited to screening.

Core claim

Finite element solutions of the diffusion-reaction equation are generated across varying physical parameters to form a training set; proper orthogonal decomposition reduces each solution to a small set of coefficients; a neural network is then trained to predict those coefficients directly from the input parameters, yielding a surrogate that delivers approximately 956-fold speedup relative to repeated finite element solves while holding mean relative L2 error to 15 percent across held-out test cases, with the accuracy level judged adequate for rapid scenario screening.

What carries the argument

The POD-NN surrogate, which compresses finite element solution fields into low-dimensional coefficients via proper orthogonal decomposition and uses a neural network to learn the direct mapping from diffusivity, reaction, and inflow parameters to those coefficients.

If this is right

Rapid parametric studies and scenario screening become feasible for nutrient transport and salinity distribution without repeated full-order solves.
The method supplies a practical compromise between the physical fidelity of finite element discretizations and the speed of neural network inference.
The surrogate supports efficient exploration of many climate-related threat scenarios in agromaritime management.
Higher error deviations on some test cases indicate the model is best used for initial screening rather than final high-precision calculations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same POD-NN construction could be applied to time-dependent or three-dimensional versions of the governing equation to broaden the range of usable predictions.
Coupling the surrogate with online sensor assimilation might reduce error on site-specific conditions without rebuilding the full training set.
Analogous reduced-order neural surrogates could be built for other environmental PDE models where computational cost currently limits large-scale scenario analysis.

Load-bearing premise

That the chosen two-dimensional steady-state diffusion-reaction equation and the sampled parameter ranges adequately represent the essential nutrient and salinity dynamics encountered in real agromaritime systems, so that an average 15 percent L2 error remains acceptable for screening tasks.

What would settle it

A direct comparison of surrogate predictions against either field sensor data from an actual coastal agricultural site or against new finite element runs on parameter values lying outside the original training envelope, checking whether the relative L2 error stays near or below the reported 15 percent mean.

Figures

Figures reproduced from arXiv: 2604.21058 by Muhammad Ilyas.

**Figure 4.** Figure 4: Leading 4 POD Modes [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Histogram of Neural Network (vs POD) reconstruction error [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: Best (𝜅 = 8.171 × 10 −02 , 𝛽 = 0.815, 𝑞𝑖𝑛 = 0.833) and Worst (𝜅 = 1.101 × 10 −03 , 𝛽 = 0.001, 𝑞𝑖𝑛 = 0.100) NN reconstruction We believe that the mean relative 𝐿 2 -error of approximately 16% arises from two primary sources: (i) the projection error introduced by POD basis truncation, and (ii) the regression error of the neural network approximation. The worst-case deviations exceeding 60% ( [PITH_FULL_IMA… view at source ↗

**Figure 7.** Figure 7: True FEM vs NN reconstruction sample (κ = 5.639 × 10 −02 , β = 0.559, 𝑞𝑖𝑛 = 0.603) Remark: The histogram in [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

read the original abstract

Predicting nutrient transport and salinity distribution is crucial for mitigating climate-related threats to agromaritime systems. Traditional PDE-based models can capture the physics of nutrient dispersion, salinity and water quality. However, they face challenges in scalability and adaptability to real-time problems. In this article, we develop a hybrid approach that combines finite element discretisations with neural network integration to enable efficient and adaptive data-informed predictions. We use a finite element solver for the steady-state diffusion-reaction equation to generate a dataset across varying diffusivity, reaction and inflow conditions. We then build a proper orthogonal decomposition (POD), which reduces dimensionality, and a neural network (NN) that maps parameters to reduced coefficients. A numerical study presented on a simplified model demonstrates the proof-of-concept for predicting nutrient transport and salinity distribution. Numerical experiments show that the NN surrogate achieve a speed-up of approximately 956x compared to a regular FEM solver while maintaining an accuracy of mean relative L2-errors of 15% across the test set, with occasional higher deviations, which is sufficient for rapid scenario screening and parametric studies. These results highlight the method's potential as a fast and accurate surrogate for nutrient and salinity prediction, offering a balance between FEM reliability and NN adaptability for sustainable agromaritime management.

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 applies standard POD-NN to a simple 2D nutrient model and gets a 956x speedup at 15% mean L2 error, but never checks whether that error level keeps screening decisions intact.

read the letter

The work generates training data by solving the steady-state diffusion-reaction equation with finite elements across ranges of diffusivity, reaction rate, and inflow. It then builds a POD basis and trains a neural network to predict the reduced coefficients from the input parameters. The reported outcome is a 956-fold speedup over the full FEM solve with a mean relative L2 error of 15% on the held-out test cases, plus the note that some cases deviate more. That concrete metric pair is the main deliverable and it is presented plainly without over-claiming generality. The pipeline itself follows well-known reduced-order modeling steps, so the novelty is mainly the choice of application domain rather than any new algorithm or theory. The authors are clear that the model is simplified and two-dimensional, which keeps the scope honest. The soft spot is exactly the one the stress-test flagged: there is no follow-up check on whether the observed L2 deviation changes integrated nutrient loads, pushes salinity across decision thresholds, or alters screening classifications relative to the full-order solver. Without that, the claim that 15% error is “sufficient for rapid scenario screening” rests on an untested assertion. The paper also gives no information on training/validation split details, hyperparameter search, or sensitivity to the number of retained POD modes. This is a practical proof-of-concept for people already working on agromaritime or similar environmental transport problems who want a fast surrogate for parametric sweeps. It is not aimed at readers looking for new methodology or high-fidelity validation. I would send it to peer review so referees can press on the missing decision-level error analysis and on whether the simplified physics is representative enough for the intended use. The thinking is straightforward and the numerical evidence is at least reproducible in outline.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a hybrid data-driven surrogate for the steady-state diffusion-reaction equation modeling nutrient transport and salinity in agromaritime systems. It employs finite element methods to generate data, applies proper orthogonal decomposition (POD) to reduce dimensionality, and uses a neural network to predict the POD coefficients from input parameters. Numerical experiments on a simplified model report a 956x speedup over standard FEM with a mean relative L2 error of 15%, claimed to be sufficient for rapid scenario screening and parametric studies.

Significance. If the reported accuracy level is shown to be adequate for the intended screening applications, this work could provide a valuable tool for efficient exploration of parameter spaces in environmental modeling, combining the accuracy of physics-based FEM with the speed of machine learning surrogates. The proof-of-concept demonstrates feasibility but requires further validation to establish broader impact.

major comments (2)

[Abstract] The assertion in the abstract that a mean relative L2-error of 15% (with occasional higher deviations) 'is sufficient for rapid scenario screening and parametric studies' lacks supporting evidence. No analysis is presented demonstrating that this error does not significantly affect integrated quantities like total nutrient loads or the identification of critical salinity thresholds compared to the full-order FEM solution.
[Numerical experiments] The manuscript does not specify the training/validation/test data splits, the number of POD modes retained, the neural network architecture and hyperparameters, or sensitivity of results to the chosen parameter ranges, undermining assessment of the robustness of the reported 956x speedup and error metrics.

minor comments (2)

Provide more details on the specific form of the diffusion-reaction equation, including boundary conditions and the physical interpretation of the parameters.
Include error bars or variance measures for the reported L2 errors to better characterize performance across the test set.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. The comments highlight important areas for clarification and strengthening of the presentation. We address each major comment point by point below and will revise the manuscript accordingly to improve reproducibility and support for our claims.

read point-by-point responses

Referee: [Abstract] The assertion in the abstract that a mean relative L2-error of 15% (with occasional higher deviations) 'is sufficient for rapid scenario screening and parametric studies' lacks supporting evidence. No analysis is presented demonstrating that this error does not significantly affect integrated quantities like total nutrient loads or the identification of critical salinity thresholds compared to the full-order FEM solution.

Authors: We acknowledge that the current version of the manuscript does not include explicit quantitative analysis demonstrating the impact of the reported error level on integrated quantities or threshold identification. To address this, we will add a dedicated subsection to the numerical experiments that computes and compares total nutrient loads and salinity threshold crossings between the POD-NN surrogate and the full-order FEM solutions across the test cases. This will provide the supporting evidence for the sufficiency claim in the context of rapid screening applications. revision: yes
Referee: [Numerical experiments] The manuscript does not specify the training/validation/test data splits, the number of POD modes retained, the neural network architecture and hyperparameters, or sensitivity of results to the chosen parameter ranges, undermining assessment of the robustness of the reported 956x speedup and error metrics.

Authors: We agree that these implementation details were omitted from the current manuscript, which limits the ability to fully assess robustness. In the revised version, we will explicitly report the data split ratios used for training, validation, and testing; the number of POD modes retained (selected via an energy criterion); the neural network architecture, including layer sizes, activation functions, and training hyperparameters; and results from sensitivity tests varying the parameter ranges. These additions will enable better evaluation of the speedup and accuracy metrics. revision: yes

Circularity Check

0 steps flagged

No circularity: surrogate metrics derived from independent FEM test data and timing measurements

full rationale

The paper generates a dataset by solving the steady-state diffusion-reaction PDE with a finite-element solver across parameter ranges, applies POD to obtain reduced coefficients, and trains an NN to map input parameters to those coefficients. Reported quantities are then obtained by (a) evaluating the trained NN on held-out parameter values and computing relative L2 error against the corresponding FEM solutions, and (b) measuring wall-clock times of the NN versus the full FEM solver. Neither the error metric nor the speedup factor is defined by the NN weights or by any self-citation; both are external, independently computed observables. No step in the chain equates a claimed prediction to an input quantity by algebraic construction or by renaming a fitted parameter.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Only abstract available; ledger populated from standard assumptions implicit in FEM and NN surrogate literature.

free parameters (2)

Neural network architecture and hyperparameters
Number of layers, neurons, training epochs, and regularization chosen to achieve the reported accuracy on the generated dataset.
Number of POD modes retained
Truncation level selected to balance dimensionality reduction against reconstruction error.

axioms (2)

standard math Finite element discretization converges to the true solution of the steady-state diffusion-reaction PDE under the chosen mesh and boundary conditions.
Invoked when generating the training dataset from the FEM solver.
standard math Proper orthogonal decomposition provides an optimal low-rank approximation in the L2 sense for the snapshot matrix.
Used to reduce the high-dimensional FEM solutions before NN training.

pith-pipeline@v0.9.0 · 5520 in / 1367 out tokens · 39273 ms · 2026-05-09T22:59:54.562184+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Accurate modelling of these processes is critical to ensure resilient agricultural and coastal practices, particularly in regions where freshwater and marine environments interact

Introduction Sustainable agromaritime systems face significant challenges related to the management of water quality, salinity transport, nutrient dispersion, and associated flow dynamics. Accurate modelling of these processes is critical to ensure resilient agricultural and coastal practices, particularly in regions where freshwater and marine environmen...

work page 2016
[2]

The research framework is illustrated in Figure 1

RESEARCH METHODS We will develop a quick prediction model for a salinity transport and nutrient dispersion by integrating machine learning techniques with diffusion -reaction numerical simulations. The research framework is illustrated in Figure 1. Figure 2: Workflow for the Hybrid FEM-NN Surrogate Model Construction 2.1 Governing Equation and Weak Formul...

work page 2012
[3]

High-Fidelity Simulation: Generate the snapshot matrix 𝑈 by solving the FEM system (3) for a wide range of parameters

work page
[4]

Dimensionality Reduction: Compute the POD basis Ψ𝑚 and project all snapshots to obtain the coefficient dataset {(μ(𝑖), 𝑎(𝑖))}

work page
[5]

Surrogate Training: Train a neural network 𝒩𝒩 to approximate the mapping μ ↦ 𝑎

work page
[6]

Fast Prediction: For a new parameter μ∗, evaluate the NN to get 𝑎̂ (μ∗) and reconstruct the approximate solution 𝑢NN(μ∗)

work page
[7]

This integration of FEM, POD, and NN provides a lightweight surrogate model that accelerates PDE simulations while preserving essential physical structures for parameterised PDEs

Evaluation: The performance evaluated by comparing the approximate solution 𝑢NN(μ∗) with FEM reference solution 𝒖𝒉 using relative L2-errors of representative fields. This integration of FEM, POD, and NN provides a lightweight surrogate model that accelerates PDE simulations while preserving essential physical structures for parameterised PDEs

work page
[8]

We define the problem on Ω = [0,10] × [0,5] with gridsize 200 × 100

Results and Discussion In this section, we will illustrate the numerical framework showing several snapshots along the way. We define the problem on Ω = [0,10] × [0,5] with gridsize 200 × 100. We have the Dirichlet boundary condition as: 𝑢 = 𝑞in on the left inflow boundary ( 𝑥 = 0), and 𝑢 = 0 on all other boundaries. We also set the source term as 𝑓 = sin...

work page
[9]

Conclusion We developed and evaluated a hybrid FEM -NN framework for nutrient and salinity transport modelling. By training a neural network on FEM -generated datasets and compressing the solution space with POD, we achieved surrogate predictions with mean relative errors of 15% while reducing computational time by approximately 956 times faster compared ...

work page doi:10.1016/j.gsd.2024.101172 2024

[1] [1]

Accurate modelling of these processes is critical to ensure resilient agricultural and coastal practices, particularly in regions where freshwater and marine environments interact

Introduction Sustainable agromaritime systems face significant challenges related to the management of water quality, salinity transport, nutrient dispersion, and associated flow dynamics. Accurate modelling of these processes is critical to ensure resilient agricultural and coastal practices, particularly in regions where freshwater and marine environmen...

work page 2016

[2] [2]

The research framework is illustrated in Figure 1

RESEARCH METHODS We will develop a quick prediction model for a salinity transport and nutrient dispersion by integrating machine learning techniques with diffusion -reaction numerical simulations. The research framework is illustrated in Figure 1. Figure 2: Workflow for the Hybrid FEM-NN Surrogate Model Construction 2.1 Governing Equation and Weak Formul...

work page 2012

[3] [3]

High-Fidelity Simulation: Generate the snapshot matrix 𝑈 by solving the FEM system (3) for a wide range of parameters

work page

[4] [4]

Dimensionality Reduction: Compute the POD basis Ψ𝑚 and project all snapshots to obtain the coefficient dataset {(μ(𝑖), 𝑎(𝑖))}

work page

[5] [5]

Surrogate Training: Train a neural network 𝒩𝒩 to approximate the mapping μ ↦ 𝑎

work page

[6] [6]

Fast Prediction: For a new parameter μ∗, evaluate the NN to get 𝑎̂ (μ∗) and reconstruct the approximate solution 𝑢NN(μ∗)

work page

[7] [7]

This integration of FEM, POD, and NN provides a lightweight surrogate model that accelerates PDE simulations while preserving essential physical structures for parameterised PDEs

Evaluation: The performance evaluated by comparing the approximate solution 𝑢NN(μ∗) with FEM reference solution 𝒖𝒉 using relative L2-errors of representative fields. This integration of FEM, POD, and NN provides a lightweight surrogate model that accelerates PDE simulations while preserving essential physical structures for parameterised PDEs

work page

[8] [8]

We define the problem on Ω = [0,10] × [0,5] with gridsize 200 × 100

Results and Discussion In this section, we will illustrate the numerical framework showing several snapshots along the way. We define the problem on Ω = [0,10] × [0,5] with gridsize 200 × 100. We have the Dirichlet boundary condition as: 𝑢 = 𝑞in on the left inflow boundary ( 𝑥 = 0), and 𝑢 = 0 on all other boundaries. We also set the source term as 𝑓 = sin...

work page

[9] [9]

Conclusion We developed and evaluated a hybrid FEM -NN framework for nutrient and salinity transport modelling. By training a neural network on FEM -generated datasets and compressing the solution space with POD, we achieved surrogate predictions with mean relative errors of 15% while reducing computational time by approximately 956 times faster compared ...

work page doi:10.1016/j.gsd.2024.101172 2024