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arxiv: 2601.01902 · v7 · pith:OPKNYLPQnew · submitted 2026-01-05 · 🧮 math.NA · cs.NA

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

Victory C. 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
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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 C. Obieke.

Figure 1
Figure 1. Figure 1: Space-time evolution of the electric field [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
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. 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. Figure 4: Discrete energy error E n − E0 for Crank–Nicolson time stepping with the exact central￾difference stencil and the structure-preserving learned stencils, shown on a linear (left) and loga￾rithmic (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. 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. 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. 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. 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. 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.

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)
  1. §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.
  2. 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.
  3. 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

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