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arxiv: 2604.25957 · v1 · submitted 2026-04-27 · 🧮 math.NA · cs.NA

On Physics-Based Loss Scaling for MF-PINNs applied to the neutron diffusion equation

Minh-Hieu Do (SERMA) , Fran\c{c}ois Madiot (SERMA) , Karim Ammar (SERMA) , Nicolas G\'erard Castaing (SERMA) This is my paper

Pith reviewed 2026-05-08 02:16 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords physics-informed neural networksmixed formulationneutron diffusion equationloss scalingcross sectionsPINNsnumerical methodsk-eigenvalue problem
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The pith

A loss function scaled by material cross sections makes MF-PINNs converge faster and more accurately on the neutron diffusion equation.

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

The paper introduces Physics-Based Loss Scaling for mixed-formulation physics-informed neural networks solving the neutron diffusion equation. A new loss function is derived by scaling residual terms according to the material cross sections; this version is mathematically equivalent to the standard MF-PINN loss yet produces quicker training and smaller errors. The claim is supported by tests on fixed-source and k-eigenvalue problems, covering one-group through multigroup cases and two-dimensional through three-dimensional geometries. The scaling requires no extra hyperparameters and preserves the physical solution.

Core claim

The authors show that scaling the loss terms of an MF-PINN by the material cross sections yields a loss function that is equivalent to the unscaled classical form but accelerates convergence and raises accuracy when the network is trained on the neutron diffusion equation. This equivalence and the observed gains are verified across fixed-source and criticality problems, single-group to multi-group models, and two- to three-dimensional domains.

What carries the argument

Physics-Based Loss Scaling (PBLS), a reweighting of the mixed-formulation residual terms by the local material cross sections that balances their magnitudes during gradient descent without altering the underlying PDE solution.

If this is right

  • The scaled loss applies unchanged to both fixed-source and k-eigenvalue formulations.
  • Accuracy and speed gains persist from one-group to multigroup neutron models.
  • The same scaling works for two-dimensional and three-dimensional spatial configurations.
  • No new hyperparameters are required, keeping the training procedure as simple as the original MF-PINN.

Where Pith is reading between the lines

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

  • A similar coefficient-based scaling may improve training stability for other diffusion-type PDEs solved by physics-informed networks.
  • The approach points toward a systematic way to choose loss weights from the physical parameters already present in the governing equations.
  • Tests on domains with strong material discontinuities would reveal whether the scaling remains effective when cross sections vary sharply.

Load-bearing premise

The cross-section scaling balances every loss component uniformly for any material properties, geometry, and problem type without introducing bias or needing per-case adjustments.

What would settle it

Training an MF-PINN on an identical neutron diffusion problem with and without the proposed cross-section scaling and finding no improvement in iteration count to reach a given error tolerance or in final solution accuracy would refute the performance claim.

Figures

Figures reproduced from arXiv: 2604.25957 by Fran\c{c}ois Madiot (SERMA), Karim Ammar (SERMA), Minh-Hieu Do (SERMA), Nicolas G\'erard Castaing (SERMA).

Figure 1
Figure 1. Figure 1: Fully connected neural network with scaled loss for MF-PINNs view at source ↗
Figure 2
Figure 2. Figure 2: The geometry of the IAEA pool reactor problem. view at source ↗
Figure 3
Figure 3. Figure 3: The relative error of the neutron flux and current of the one-group view at source ↗
Figure 4
Figure 4. Figure 4: Neutron flux and the deviation from the reference solution for the view at source ↗
Figure 5
Figure 5. Figure 5: The simplified C5G7 Geome￾try view at source ↗
Figure 6
Figure 6. Figure 6: The relative error of the neutron flux and current of the simplified view at source ↗
Figure 7
Figure 7. Figure 7: Neutron flux and the deviation from the reference solution for the view at source ↗
Figure 8
Figure 8. Figure 8: The simplified C5G7 Geome￾try view at source ↗
Figure 9
Figure 9. Figure 9: The relative error of the neutron flux and neutron current of the view at source ↗
Figure 10
Figure 10. Figure 10: Neutron flux and the deviation from the reference solution for the view at source ↗
Figure 11
Figure 11. Figure 11: The TWIGL geometry view at source ↗
Figure 12
Figure 12. Figure 12: The relative error of the neutron flux and current of the TWIGL 2D view at source ↗
Figure 13
Figure 13. Figure 13: Neutron flux and the deviation from the reference solution for the view at source ↗
Figure 14
Figure 14. Figure 14: The TWIGL-3D geometry 0 100000 200000 300000 400000 500000 600000 Iteration 10 3 10 2 10 1 UL-Tanh-Random SL-Tanh-Random UL-Sin-Sobol SL-Sin-Sobol (a) ∆¯ ϕ 0 100000 200000 300000 400000 500000 600000 Iteration 10 2 10 1 10 0 UL-Tanh-Random SL-Tanh-Random UL-Sin-Sobol SL-Sin-Sobol (b) ∆¯ p view at source ↗
Figure 15
Figure 15. Figure 15: The relative error of the neutron flux and current of the TWIGL 3D view at source ↗
Figure 16
Figure 16. Figure 16: Neutron flux and the deviation from the reference solution for the view at source ↗
read the original abstract

Physics-Based Loss Scaling (PBLS) is introduced for Mixed-Formulation PINNs (MF-PINNs) applied to the neutron diffusion equation. In particular, we propose a new \textit{scaled} loss function based on the material cross sections, which is equivalent to the classical MF-PINN loss, but accelerates the convergence and improves accuracy of MF-PINNs. Several numerical experiments on both the fixed source and the k-eigenvalue problem, from one-group to multigroup cases and from two-dimensional (2D) to three-dimensional (3D) configurations, illustrate the efficiency of the proposed scaling method.

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.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces Physics-Based Loss Scaling (PBLS) for Mixed-Formulation Physics-Informed Neural Networks (MF-PINNs) applied to the neutron diffusion equation. It proposes a scaled loss function derived from material cross sections that is mathematically equivalent to the classical MF-PINN loss (preserving the same zero set for exact solutions) while accelerating convergence and improving accuracy. This is illustrated through numerical experiments on fixed-source and k-eigenvalue problems spanning one-group to multigroup cases and two- to three-dimensional geometries.

Significance. If the equivalence holds rigorously and the observed gains prove robust, PBLS offers a parameter-free way to improve optimizer behavior for PINNs in heterogeneous media, which could enhance the practicality of neural solvers for neutron diffusion in nuclear engineering applications. The approach leverages the physical coefficients directly, avoiding ad-hoc hyperparameters, and the experiments cover a useful range of problem types.

major comments (2)
  1. [Section 3] Section 3 (derivation of scaled loss): the construction multiplies residual terms by factors from the diffusion and removal operators so that any exact PDE solution remains a zero of the new loss; however, the manuscript should explicitly verify that this scaling preserves equivalence for the mixed formulation (including the current variable) and does not alter the conditioning of the loss landscape in a way that could introduce spurious local minima.
  2. [Section 5] Section 5 (numerical results): while faster convergence and lower errors are reported across test cases, the central claim of improved accuracy would be strengthened by including quantitative error tables (e.g., relative L2 errors or pointwise maxima) comparing scaled vs. unscaled MF-PINNs on identical network architectures and training budgets.
minor comments (3)
  1. [Throughout] Notation for cross sections (Σ_a, Σ_r, etc.) and the mixed variables (flux φ and current J) should be defined once in a nomenclature table or at first use to improve readability.
  2. [Section 5] Figure captions in the results section would benefit from listing the specific material properties and boundary conditions for each test case to aid reproducibility.
  3. [Abstract] The abstract states equivalence and empirical gains but omits any mention of the precise scaling factors or the range of cross-section values tested; a single sentence clarifying these would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. The comments are constructive and we address each one below, indicating the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [Section 3] Section 3 (derivation of scaled loss): the construction multiplies residual terms by factors from the diffusion and removal operators so that any exact PDE solution remains a zero of the new loss; however, the manuscript should explicitly verify that this scaling preserves equivalence for the mixed formulation (including the current variable) and does not alter the conditioning of the loss landscape in a way that could introduce spurious local minima.

    Authors: We thank the referee for highlighting this point. In Section 3 the scaled loss is obtained by multiplying the residual of the neutron balance equation by positive factors derived from the diffusion coefficient and removal cross section (inverse scaling). The mixed formulation augments the system with the current variable satisfying Fick's law; the corresponding residual term in the loss is left unscaled or scaled by the same physical coefficients so that the entire loss vanishes if and only if both the balance and constitutive residuals are zero. Because every scaling factor is strictly positive for physically admissible cross sections, the zero set of the loss is identical to that of the classical MF-PINN loss and therefore coincides exactly with the solutions of the original mixed system. Positive reweighting cannot create additional zeros and hence cannot introduce spurious local minima at non-solutions; it only improves the relative magnitudes of the loss terms, which is the source of the observed faster convergence. We will add a short clarifying paragraph at the end of Section 3 that explicitly states this equivalence for the mixed variables and notes the absence of new minima. revision: yes

  2. Referee: [Section 5] Section 5 (numerical results): while faster convergence and lower errors are reported across test cases, the central claim of improved accuracy would be strengthened by including quantitative error tables (e.g., relative L2 errors or pointwise maxima) comparing scaled vs. unscaled MF-PINNs on identical network architectures and training budgets.

    Authors: We agree that explicit quantitative tables would make the accuracy improvement more transparent. The present manuscript demonstrates the gains primarily through convergence curves and spatial error plots. In the revised version we will insert a new table (or set of tables) in Section 5 that reports, for every test case, the relative L2 error and maximum pointwise error of the scalar flux (and current when relevant) obtained by both the scaled and unscaled MF-PINNs. All entries will be computed on identical network architectures, the same number of training iterations, and the same optimizer settings, thereby providing a direct, quantitative comparison. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in the PBLS derivation

full rationale

The paper constructs the physics-based loss scaling directly from the neutron diffusion operators by multiplying each residual term (diffusion, removal, fission, etc.) by a factor involving the local material cross sections. This reweighting is shown algebraically to leave the zero set unchanged: any function satisfying the original PDE and boundary conditions yields zero loss under both the classical and scaled formulations. The equivalence is therefore an explicit identity derived from the PDE residuals rather than a fitted parameter, a self-referential definition, or a load-bearing self-citation. Subsequent numerical experiments on fixed-source and k-eigenvalue problems serve only to illustrate optimizer behavior and do not retroactively define the scaling itself. The derivation chain remains self-contained against the stated PDE and does not reduce any claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; scaling is described as derived from existing material cross sections without new postulates.

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