An Energy Stable Approach for Learning Derivative Operators from Noisy Data for Maxwells Equations

Ameh Emmanuel Sunday; Victory Obieke

arxiv: 2601.01902 · v6 · submitted 2026-01-05 · 🧮 math.NA · cs.NA

An Energy Stable Approach for Learning Derivative Operators from Noisy Data for Maxwells Equations

Victory Obieke , Ameh Emmanuel Sunday This is my paper

Pith reviewed 2026-05-16 18:23 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords structure-preserving ADMMMaxwell equationsenergy stabilityskew-adjointnessnoisy dataderivative stencilsdata-driven learningperiodic convolution

0 comments

The pith

Reduced parameterization lets SP-ADMM learn energy-conserving derivative operators for Maxwell equations directly from noisy data.

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

The paper develops SP-ADMM to learn spatial derivative stencils for the one-dimensional Maxwell system that remain energy stable even when trained on noisy data. It represents the derivative as a compact periodic convolution stencil and enforces the required skew-adjoint property automatically by parameterizing only the independent positive-side coefficients. This built-in constraint replaces the equality constraints used in standard ADMM and allows the method to recover accurate operators whether the underlying stencil is known or hidden. Across clean and noisy regimes, multiple initial conditions, varying training sizes, and long simulations, SP-ADMM produces the smallest final-time electric-field errors while keeping energy errors at roundoff level. It also matches classical finite differences in accuracy during a layered-medium reflection and transmission test.

Core claim

SP-ADMM enforces skew-adjointness of the learned derivative operator by construction through a reduced parameterization of the stencil coefficients, enabling it to learn accurate operators from noisy data while preserving energy conservation to roundoff accuracy in the underlying Maxwell system, outperforming standard constrained ADMM in error metrics across multiple regimes.

What carries the argument

The reduced parameterization using only independent positive-side stencil coefficients in the compact periodic convolution representation, which enforces the skew-adjoint property required for energy stability by construction.

If this is right

SP-ADMM achieves the smallest final-time electric-field error while preserving energy to roundoff accuracy across clean data, noisy derivative data, multiple initial conditions, different hidden operators, training-set sizes, and long-time simulations.
The learned stencil remains competitive with classical finite differences in physical reflection and transmission settings.
Performance holds under constraint ablations and changes in regularization parameters.
The method works effectively in hidden-operator and noisy-data regimes without post-hoc projection steps.

Where Pith is reading between the lines

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

The same reduced-parameterization idea could apply to learning structure-preserving operators for other wave or transport equations where energy or other invariants must be respected.
Lowering the number of free coefficients may reduce the volume of training data needed to reach a given accuracy level.
The technique suggests a route to embed discrete conservation laws directly into neural or stencil-based PDE solvers without separate penalty terms.
Testing the approach on two- or three-dimensional Maxwell systems would reveal whether the one-dimensional construction scales when skew-adjointness involves more coupled components.

Load-bearing premise

Enforcing skew-adjointness solely through the reduced parameterization of positive-side stencil coefficients is sufficient to guarantee energy stability for the learned operator in the Maxwell system without introducing approximation errors or limiting expressivity.

What would settle it

A long-time simulation in which the learned operator produces energy drift larger than roundoff or final-time electric-field errors exceeding those of classical finite differences in the layered-medium test.

Figures

Figures reproduced from arXiv: 2601.01902 by Ameh Emmanuel Sunday, Victory Obieke.

**Figure 2.** Figure 2: Final-time electric field E(x, T) for the exact central-difference stencil and the learned stencils for R = 1 under Crank–Nicolson time-stepping. The curves are visually indistinguishable at the scale shown. This experiment serves as a sanity check for the learning framework: for the smallest radius R = 1, the energy-constrained optimization recovers a stencil that is essentially indistinguishable from the… view at source ↗

**Figure 3.** Figure 3: Log–log plot of the relative L 2 error errE(∆x) in E(·, T) versus ∆x for the ADMM– learned stencil. 6.4 Discrete energy conservation for learned Maxwell stencils We next examine how closely the learned stencils preserve the semi-discrete electromagnetic energy under Crank–Nicolson time integration [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: Discrete energy error E n − E0 for Crank–Nicolson time stepping with the exact centraldifference stencil and the structure-preserving learned stencils, shown on a linear (left) and logarithmic (right) scale. (a) Amplification error |µ(θ)| − 1. (b) Phase dispersion arg µ(θ)/θ [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗

**Figure 5.** Figure 5: Fourier dispersion analysis of the semi-discrete Maxwell system with Crank–Nicolson [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

**Figure 6.** Figure 6: Nonstandard vs. standard skew-adjoint operators: learned stencil [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗

**Figure 7.** Figure 7: Noisy training and ill-posed least squares: comparison of unconstrained LS, central [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗

**Figure 8.** Figure 8: Space-time plot of the electric field E(x, t) under noisy training perturbations. Although wLS achieves a very small training error, it violates the skew-adjointness constraints and develops very large high-frequency coefficients. When used in Crank–Nicolson time stepping, the resulting Maxwell semi-discretization is no longer energy-stable, and the discrete energy E n blows up (Figure 7a). In contrast, th… view at source ↗

**Figure 9.** Figure 9: Convergence behavior of PG, NAG, and ADMM for a fixed stencil radius [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗

read the original abstract

We develop a structure-preserving ADMM method, denoted SP-ADMM, for learning energy-stable spatial derivative stencils for Maxwell equations from noisy data. Starting from the source-free Maxwell system, we focus on a one-dimensional reduction whose energy conservation depends on the skew-adjointness of the spatial derivative operator. The learned derivative is represented by a compact periodic convolution stencil. Unlike standard constrained ADMM, which learns the full stencil and imposes skew-adjointness through equality constraints, SP-ADMM enforces skew-adjointness by construction through a reduced parameterization using only the independent positive-side stencil coefficients. Numerical experiments show that SP-ADMM is especially effective in hidden operator and noisy-data regimes. Across clean data, noisy derivative data, multiple initial conditions, different hidden skew-adjoint operators, training-set sizes, regularization parameters, constraint ablations, and long-time simulations, SP-ADMM achieves the smallest final-time electric-field error while preserving energy to roundoff accuracy. A layered-medium Maxwell propagation test further shows that the learned structure-preserving stencil remains competitive with classical finite differences in a physical reflection/transmission setting. Overall, SP-ADMM provides a data-driven way to learn accurate Maxwell stencils while retaining the energy-conserving structure of the underlying equations.

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's main advance is a reduced parameterization that bakes skew-adjointness into the stencil for ADMM-based learning of Maxwell derivative operators, guaranteeing energy conservation to roundoff without extra constraints.

read the letter

The key point is that SP-ADMM reparameterizes the periodic convolution stencil so only the positive-side coefficients are free variables, with the negative side set as their negation and the center fixed at zero. This exactly spans the space of skew-adjoint operators for the chosen stencil width, so the learned derivative automatically satisfies the discrete energy identity for the 1D Maxwell system to machine precision. That is a genuine simplification over standard constrained ADMM, and the stress-test confirms it introduces no approximation error or loss of expressivity within the stencil family. The experiments then show this pays off in practice: smaller final-time field errors than baselines across clean data, noisy derivative data, multiple initial conditions, hidden operators, varying training sizes, and long-time runs, plus a layered-medium test where it stays competitive with classical finite differences on reflection and transmission. The energy stays conserved to roundoff in all those cases, which matches what the parameterization promises. The soft spots are mostly in the presentation. The abstract gives no numerical error values, no explicit baseline methods or tables, and almost no detail on how the noisy data was generated or what the regularization schedule looked like. Without those, it is hard to judge how large the practical improvement actually is or how sensitive the results are to implementation choices. The full manuscript would need to supply the code or at least the exact experimental protocol for reproducibility. This work is aimed at people who build or use structure-preserving discretizations for wave problems, especially in electromagnetics where long-time stability matters and data may be noisy. A reader already working on operator learning or ADMM for PDEs would pick up the parameterization trick and the stability guarantee quickly. I would send it to peer review. The central construction is clean and the reported behavior is consistent with the math, even if the experiments need more concrete numbers and controls to be fully convincing.

Referee Report

0 major / 3 minor

Summary. The manuscript proposes SP-ADMM, a structure-preserving ADMM algorithm for learning energy-stable derivative stencils for the 1D Maxwell equations from noisy data. The key innovation is a reduced parameterization of the compact periodic convolution stencil that enforces skew-adjointness by construction using only the positive-side coefficients, thereby guaranteeing discrete energy conservation. Experiments across clean and noisy data, various initial conditions, hidden operators, and long-time simulations show SP-ADMM yielding the smallest final-time electric field errors while maintaining energy to roundoff accuracy, and performing competitively in a layered-medium propagation test.

Significance. If the results hold, this work provides a principled data-driven approach to learning numerical operators that preserve the energy stability essential for long-time accuracy in wave propagation problems like Maxwell's equations. The exact enforcement of skew-adjointness without loss of expressivity (the reduced parameterization spans the full space of skew-adjoint stencils of given width) is a notable strength, as is the demonstration of robustness to noise and the competitiveness with classical finite differences in a physical setting. This could impact fields requiring stable learned discretizations from data.

minor comments (3)

§2.2: The notation for the positive-side coefficients p_k could be introduced with an explicit low-order example (e.g., m=1) to make the reparameterization immediately clear to readers.
Table 2: The caption should state the precise stencil width and regularization parameter values used for each row so that the ablation results can be reproduced without consulting the text.
Figure 4: The long-time energy plot would be more informative if the vertical axis were scaled to show the roundoff-level deviation explicitly (e.g., 10^{-14} to 10^{-16}) rather than a generic log scale.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on SP-ADMM for learning energy-stable derivative stencils from noisy data. The recommendation for minor revision is appreciated, and we note that no specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's derivation enforces skew-adjointness directly via reduced parameterization of positive-side stencil coefficients, which by the 1D Maxwell energy identity guarantees discrete conservation to machine precision for any choice of free coefficients. This is a valid structural enforcement rather than a self-definitional loop or fitted input renamed as prediction. The performance claims (smallest final-time error across regimes) rest on numerical experiments, not on tautological reduction of outputs to inputs. No self-citations, uniqueness theorems from prior author work, or ansatzes smuggled via citation appear in the load-bearing steps; the construction is self-contained against the stated energy identity.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that skew-adjointness of the derivative operator is necessary and sufficient for energy conservation in the 1D Maxwell reduction, plus the modeling choice that a compact periodic convolution stencil can represent the learned operator. No explicit free parameters or new entities are introduced beyond regularization parameters mentioned generically in the experiments.

free parameters (1)

regularization parameters
Referenced as varied in experiments but no specific values or fitting procedure given in the abstract.

axioms (1)

domain assumption Energy conservation in the source-free Maxwell system depends on the skew-adjointness of the spatial derivative operator.
Stated directly as the reason for focusing on skew-adjointness in the 1D reduction.

pith-pipeline@v0.9.0 · 5519 in / 1409 out tokens · 46518 ms · 2026-05-16T18:23:27.699485+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 (J-cost uniqueness) unclear

?

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

Theorem 2 (Characterization of skew-symmetric stencils)... w0=0, w_{-ℓ}=-w_{+ℓ}... equivalent to skew-adjointness... guarantees exact conservation of discrete electromagnetic energy
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3 forcing) unclear

?

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

SP-ADMM enforces skew-adjointness by construction through a reduced parameterization using only the independent positive-side stencil coefficients

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

Works this paper leans on

28 extracted references · 28 canonical work pages · 1 internal anchor

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H E # =A ⋆

Kane S. Yee. Numerical solution of initial boundary value problems involving maxwell’s equa- tions in isotropic media.IEEE Transactions on Antennas and Propagation, 14(3):302–307, 1966. 9 Appendix Appendix: Spectral structure of the learned stencil Theorem 3(Spectral structure and modal energy conservation for the learned stencil).Letw ⋆ ∈ R2R+1 be any so...

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[1] [1]

Bar-Sinai, S

J. Bar-Sinai, S. Hoyer, J. Hickey, and M. Levine. Learning data-driven discretizations for partial differential equations.Proceedings of the National Academy of Sciences, 116(31):15344– 15349, 2019

work page 2019

[2] [2]

Bokil and Nathan L

Vrushali A. Bokil and Nathan L. Gibson. Analysis of spatial high-order finite difference methods for maxwell’s equations in dispersive media.IMA Journal of Numerical Analysis, 31(3):1322–1350, 2011

work page 2011

[3] [3]

Bokil and Nathan L

Vrushali A. Bokil and Nathan L. Gibson. High spatial order energy stable fdtd methods for maxwell’s equations in nonlinear optical media in one dimension.Journal of Scientific Computing, 77(1):330–371, 2018

work page 2018

[4] [4]

Bokil, Olivia A

Vrushali A. Bokil, Olivia A. Keefer, and Amy C.-Y. Leung. Operator splitting methods for maxwell’s equations in dispersive media with orientational polarization.Journal of Computa- tional and Applied Mathematics, 263:160–188, 2014

work page 2014

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Distributed optimization and statistical learning via the alternating direction method of multipliers.Foun- dations and Trends in Machine Learning, 3(1):1–122, 2011

Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, and Jonathan Eckstein. Distributed optimization and statistical learning via the alternating direction method of multipliers.Foun- dations and Trends in Machine Learning, 3(1):1–122, 2011

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Message passing neural pde solvers.arXiv preprint arXiv:2202.03376, 2022

Johannes Brandstetter, Daniel E. Worrall, and Max Welling. Message passing neural pde solvers.arXiv preprint arXiv:2202.03376, 2022

work page arXiv 2022

[7] [7]

Brunton, Joshua L

Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems.Proceedings of the National Academy of Sciences, 113(15):3932–3937, 2016

work page 2016

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Springer, 2nd edition, 2006

Ernst Hairer, Christian Lubich, and Gerhard Wanner.Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, 2nd edition, 2006. 18

work page 2006

[9] [9]

Springer, 2 edition, 2010

Ernst Hairer and Gerhard Wanner.Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, volume 14 ofSpringer Series in Computational Mathematics. Springer, 2 edition, 2010

work page 2010

[10] [10]

Karniadakis

Alexander Heinlein, Yuwei Chen, and George E. Karniadakis. Multi-fidelity domain- decomposition physics-informed neural operators for time-dependent partial differential equa- tions.arXiv preprint arXiv:2508.03774, 2025

work page arXiv 2025

[11] [11]

Smith, Ayya Alieva, Qing Wang, Michael P

Dmitrii Kochkov, Jamie A. Smith, Ayya Alieva, Qing Wang, Michael P. Brenner, and Stephan Hoyer. Machine learning–accelerated computational fluid dynamics.Proceedings of the Na- tional Academy of Sciences, 118(21):e2101784118, 2021

work page 2021

[12] [12]

LeVeque.Finite Difference Methods for Ordinary and Partial Differential Equations

Randall J. LeVeque.Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM, 2007

work page 2007

[13] [13]

Fourier Neural Operator for Parametric Partial Differential Equations

Zongyi Li, Nikola Borislavov Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhat- tacharya, Andrew M. Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. InProceedings of the International Conference on Learning Rep- resentations (ICLR), 2021. arXiv:2010.08895

work page internal anchor Pith review Pith/arXiv arXiv 2021

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Reynolds averaged turbulence mod- elling using deep neural networks with embedded invariance.Journal of Fluid Mechanics, 807:155–166, 2016

Julia Ling, Andrew Kurzawski, and Jeremy Templeton. Reynolds averaged turbulence mod- elling using deep neural networks with embedded invariance.Journal of Fluid Mechanics, 807:155–166, 2016

work page 2016

[15] [15]

clawno: Conservation law–encoded neural operators.Advances in Neural Information Processing Systems, 37, 2024

Cheng Liu, Johannes Brandstetter, Nils Thuerey, and Kiwon Um. clawno: Conservation law–encoded neural operators.Advances in Neural Information Processing Systems, 37, 2024

work page 2024

[16] [16]

Conservation-preserved fourier neural operator through adaptive correction.Computer Methods in Applied Mechanics and Engineering, 2025

Yuxuan Liu, Harsh Patel, Ana Renganathan, and Lu Lu. Conservation-preserved fourier neural operator through adaptive correction.Computer Methods in Applied Mechanics and Engineering, 2025. To appear

work page 2025

[17] [17]

Pic 2o-sim: A physics- inspired causality-aware dynamic convolutional neural operator for ultra-fast photonic device fdtd simulation.arXiv preprint arXiv:2406.17810, 2024

Yunfan Ma, Mingfei Wu, Zhihao Peng, Baile Zhang, and Zhenglong Xu. Pic 2o-sim: A physics- inspired causality-aware dynamic convolutional neural operator for ultra-fast photonic device fdtd simulation.arXiv preprint arXiv:2406.17810, 2024

work page arXiv 2024

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Karniadakis

Yunan Mao, Yiming Li, Luca Magri, and George E. Karniadakis. Towards general neural surrogate solvers with specialized neural accelerators: Domain-decomposition neural operators for large-scale pdes.arXiv preprint arXiv:2405.02351, 2024

work page arXiv 2024

[19] [19]

Oxford University Press, 2003

Peter Monk.Finite Element Methods for Maxwell’s Equations. Oxford University Press, 2003

work page 2003

[20] [20]

K. W. Morton and D. F. Mayers.Numerical Solution of Partial Differential Equations: An Introduction. Cambridge University Press, 1994

work page 1994

[21] [21]

Parish and Karthik Duraisamy

Eric J. Parish and Karthik Duraisamy. A paradigm for data-driven predictive modeling using field inversion and machine learning.Journal of Computational Physics, 305:758–774, 2016

work page 2016

[22] [22]

Karniadakis

Maziar Raissi, Paris Perdikaris, and George E. Karniadakis. Hidden physics models: Machine learning of nonlinear partial differential equations.Journal of Computational Physics, 357:125– 141, 2018

work page 2018

[23] [23]

R. D. Richtmyer and K. W. Morton.Difference Methods for Initial-Value Problems. Wiley- Interscience, 2 edition, 1967. 19

work page 1967

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Rudy, Steven L

Samuel H. Rudy, Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Data-driven discovery of partial differential equations.Science Advances, 3(4):e1602614, 2017

work page 2017

[25] [25]

Hagness.Computational Electrodynamics: The Finite-Difference Time-Domain Method

Allen Taflove and Susan C. Hagness.Computational Electrodynamics: The Finite-Difference Time-Domain Method. Artech House, 3rd edition, 2005

work page 2005

[26] [26]

Teixeira

Fernando L. Teixeira. A summary review on 25 years of progress and future challenges in fdtd and fetd techniques.ACES Journal, 25(1):1–14, 2010

work page 2010

[27] [27]

Solver-in-the- loop: Learning from differentiable physics to interact with iterative pde-solvers

Kiwon Um, Robert Brand, Yun (Raymond) Fei, Philipp Holl, and Nils Thuerey. Solver-in-the- loop: Learning from differentiable physics to interact with iterative pde-solvers. InAdvances in Neural Information Processing Systems (NeurIPS), volume 33, 2020

work page 2020

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H E # =A ⋆

Kane S. Yee. Numerical solution of initial boundary value problems involving maxwell’s equa- tions in isotropic media.IEEE Transactions on Antennas and Propagation, 14(3):302–307, 1966. 9 Appendix Appendix: Spectral structure of the learned stencil Theorem 3(Spectral structure and modal energy conservation for the learned stencil).Letw ⋆ ∈ R2R+1 be any so...

work page 1966