Neural Architectures as Functional Priors in Physics-Informed Control Problems

Antonio L\'opez Montes; Fernando Carlos L\'opez Hern\'andez; Sonia Rubio Herranz

arxiv: 2606.19368 · v1 · pith:ZDV24YT2new · submitted 2026-06-11 · 🧮 math.NA · cs.LG· cs.NA· math.OC

Neural Architectures as Functional Priors in Physics-Informed Control Problems

Sonia Rubio Herranz , Fernando Carlos L\'opez Hern\'andez , Antonio L\'opez Montes This is my paper

Pith reviewed 2026-06-27 06:09 UTC · model grok-4.3

classification 🧮 math.NA cs.LGcs.NAmath.OC

keywords neural architecturesfunctional priorsphysics-informed neural networksoptimal controldynamical systemsRLC circuitDuffing systemFourier networks

0 comments

The pith

Neural architectures function as distinct functional priors, producing controls with different spectral and smoothness properties in physics-informed ODE control problems.

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

This paper investigates the influence of neural network architecture on the solutions to control problems for ordinary differential equations using physics-informed neural networks. By comparing multilayer perceptrons and Fourier-based architectures on a linear RLC circuit and a nonlinear Duffing system, it demonstrates that architecture choices lead to systematically different control trajectories despite identical problem formulations and training setups. The differences manifest in spectral content, smoothness, energy use, and phase space behavior. A reader would care because this indicates that the network structure itself biases the form of the control, enabling potential specialization where certain architectures better suit oscillatory or efficient controls. The emergence of functional specialization suggests architecture selection can be leveraged to match desired solution characteristics.

Core claim

The authors establish that in PINN approaches to controlling linear and nonlinear dynamical systems, the choice between multilayer perceptrons and Fourier-based KAN-like networks generates qualitatively distinct controls. These controls differ in spectral structure, smoothness, energy distribution, and phase-space behavior, even when governing equations, loss functions, initial and target states, and all training parameters are the same. Fourier-based architectures tend to yield trajectories with richer oscillatory content, while smoother architectures produce more regular and energetically efficient controls. This points to an implicit functional specialization where architectures handle di

What carries the argument

Neural architectures serving as implicit functional priors that bias the learned control functions toward particular frequency contents and regularity properties.

If this is right

Fourier-based architectures systematically favor controls with higher oscillatory content.
Smoother low-frequency architectures generate controls that are more energetically efficient.
The phenomenon of functional specialization emerges when architectures have freedom to shape control structure.
These effects appear consistently in both linear electrical circuits and nonlinear oscillators.

Where Pith is reading between the lines

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

Hybrid networks might combine the strengths of different architectures for improved control performance.
The specialization effect could be exploited in designing architecture-aware control strategies for more complex systems.
Similar architecture-dependent biases may appear in other physics-informed learning tasks beyond control.

Load-bearing premise

The observed differences in controls stem from the distinct functional priors induced by the architectures rather than from differences in optimization dynamics or random initialization.

What would settle it

If repeated optimizations with varied random seeds for a single architecture produce a range of control properties comparable to those seen across different architectures, the attribution to architectural priors would be undermined.

Figures

Figures reproduced from arXiv: 2606.19368 by Antonio L\'opez Montes, Fernando Carlos L\'opez Hern\'andez, Sonia Rubio Herranz.

**Figure 2.** Figure 2: Phase-space trajectories corresponding to the different control strategies. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: Fourier spectra associated with the different control strategies. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: displays the controls obtained with the different neural architectures together with the classical nonlinear baseline. Although all configurations are trained under the same physical constraints and optimization objectives, they generate qualitatively different control profiles. The differences observed in the linear RLC problem persist in the nonlinear setting. Variations in smoothness, oscillatory behavi… view at source ↗

**Figure 5.** Figure 5: Phase portrait Spectral Analysis and Quantitative Metrics [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

**Figure 6.** Figure 6: shows the Fourier spectra associated with the learned controls together with the classical nonlinear reference solution [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

read the original abstract

In this work we investigate the role of neural architectures as implicit functional priors in control problems governed by ordinary differential equations. Rather than focusing on highly complex problems, our objective is to investigate architecture-dependent effects in controlled dynamical systems within the simplest physically interpretable settings possible. In particular, we study a controlled linear RLC electrical circuit and a nonlinear Duffing-type dynamical system. Both systems are analyzed first through classical optimal-control formulations and later through PINN-based approaches. We compare different combinations of multilayer perceptrons (MLPs) and Fourier-based KAN-like architectures, and analyze their influence on the resulting controls. The numerical experiments suggest that different architectural choices systematically generate qualitatively distinct controls, even under identical governing equations, loss functionals, initial and target states, training parameters and physical constraints. Significant differences appear in the spectral structure, smoothness, energy distribution, and phase-space behavior of the learned solutions. A central observation of this work is the emergence of a functional specialization phenomenon when the neural architectures are allowed sufficient freedom to shape the structure of the learned controls. More specifically, in the systems considered here, Fourier-based architectures tend to produce trajectories with richer oscillatory content, whereas smoother low-frequency-biased architectures tend to generate more regular and energetically efficient controls. This suggests that different functional components of the control problem may be handled more efficiently by different neural architectures, leading to an implicit specialization between state representation and control generation.

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 reports architecture-dependent control outputs in two simple PINN problems but provides no evidence that the differences exceed what random seeds produce on a single architecture.

read the letter

The central observation is that MLP and Fourier-based networks, when used inside PINNs for an RLC circuit and a Duffing oscillator, yield controls with visibly different frequency content, smoothness, and energy profiles even though the equations, loss, initial states, and training settings are the same. The authors frame this as architectures functioning as distinct functional priors and note that the Fourier versions favor richer oscillations while smoother nets favor lower-energy solutions.

They keep the test cases deliberately small, which is the right choice for isolating the effect, and they include classical optimal-control solutions as a reference. That combination lets a reader see the architecture influence without the usual complications of high-dimensional or stiff systems.

The soft spot is the missing check against optimization stochasticity. The abstract states that training parameters are identical but says nothing about fixed seeds, multiple independent runs, or a comparison of within- versus between-architecture variance. In non-convex PINN training this matters; if the reported spectral and energy differences sit inside the spread obtained from different initializations of one architecture, the functional-prior interpretation does not follow. No quantitative tables or statistical tests are described either.

The work is aimed at people already running PINNs on low-dimensional control problems and wondering whether network family matters. It is not a large theoretical step, but the empirical angle could be worth checking if the full paper supplies the reproducibility controls that are absent from the abstract. I would send it to review so the authors can add those runs and clarify the size of the effect.

Referee Report

2 major / 1 minor

Summary. The paper studies neural architectures as implicit functional priors in PINN formulations of optimal control for ODEs. It compares MLPs and Fourier-based KAN-like networks on a linear RLC circuit and a nonlinear Duffing oscillator, reporting that identical problem data, losses, and training settings nevertheless yield controls with systematically different spectral content, smoothness, energy distribution, and phase-space structure. The central claim is the emergence of functional specialization, with Fourier architectures favoring oscillatory content and smoother architectures favoring energetically efficient, low-frequency solutions.

Significance. If the reported differences are shown to be architecture-driven rather than optimization artifacts, the work would provide concrete evidence that network inductive bias shapes the reachable control manifold in physics-informed settings. The choice of minimal, physically interpretable test problems is appropriate for isolating the effect. The manuscript does not yet supply the quantitative controls (multiple independent runs, intra- versus inter-architecture variance statistics, or fixed-seed protocols) needed to substantiate the specialization claim.

major comments (2)

[Abstract / Numerical experiments] Abstract and numerical-experiments section: the assertion that 'different architectural choices systematically generate qualitatively distinct controls' under 'identical ... training parameters' is load-bearing for the functional-prior thesis, yet no mention is made of fixed random seeds, multiple independent optimizations per architecture, or statistical comparison of intra- versus inter-architecture variance. Without such controls, observed spectral and smoothness differences could lie within the range produced by stochastic gradient noise alone.
[Abstract] The claim of 'emergence of a functional specialization phenomenon' requires evidence that the observed specialization is reproducible and exceeds what a single architecture produces across random initializations. The current description supplies only qualitative descriptions of 'richer oscillatory content' and 'more regular ... controls' without quantitative metrics or error bars.

minor comments (1)

[Methods] Notation for the two architectures (MLP versus Fourier KAN-like) should be introduced with explicit layer counts, activation functions, and frequency scaling parameters so that the 'identical training parameters' statement can be verified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which correctly identify the need for stronger statistical validation of the architecture-driven specialization claims. We agree that the current presentation relies on representative single-run results and will revise the manuscript to include multiple independent optimizations, fixed-seed protocols, and quantitative intra- versus inter-architecture comparisons.

read point-by-point responses

Referee: [Abstract / Numerical experiments] Abstract and numerical-experiments section: the assertion that 'different architectural choices systematically generate qualitatively distinct controls' under 'identical ... training parameters' is load-bearing for the functional-prior thesis, yet no mention is made of fixed random seeds, multiple independent optimizations per architecture, or statistical comparison of intra- versus inter-architecture variance. Without such controls, observed spectral and smoothness differences could lie within the range produced by stochastic gradient noise alone.

Authors: We acknowledge this limitation. The reported results used single representative runs per architecture to highlight qualitative distinctions under matched problem data and training settings. In the revised version we will rerun all experiments with multiple independent random seeds (at least 10 per architecture), report means and standard deviations of spectral content, total variation, and control energy, and include statistical comparisons (e.g., t-tests or ANOVA) demonstrating that inter-architecture differences exceed intra-architecture variance. These additions will appear in the numerical experiments section and be referenced from the abstract. revision: yes
Referee: [Abstract] The claim of 'emergence of a functional specialization phenomenon' requires evidence that the observed specialization is reproducible and exceeds what a single architecture produces across random initializations. The current description supplies only qualitative descriptions of 'richer oscillatory content' and 'more regular ... controls' without quantitative metrics or error bars.

Authors: We agree that the specialization claim requires quantitative reproducibility evidence. The revision will augment the abstract and results with explicit metrics (Fourier coefficient norms, integrated control effort, trajectory regularity indices) together with error bars computed across the multiple runs. These will show that the richer oscillatory content of Fourier architectures and the lower-energy regularity of smoother architectures are statistically distinguishable from the variability obtainable from any single architecture under different initializations. revision: yes

Circularity Check

0 steps flagged

No circularity: claim rests on direct numerical comparisons without self-referential derivations

full rationale

The paper reports empirical observations from PINN experiments on two dynamical systems, comparing MLP and Fourier-based architectures under fixed equations, losses, states, and training parameters. No derivations, fitted parameters renamed as predictions, or self-citation chains are present in the abstract or described methodology. The functional-specialization claim is an interpretation of observed differences in spectral structure and energy, not a quantity defined by construction from the same data. External validity concerns (e.g., stochasticity) exist but do not constitute circularity per the defined criteria.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; full manuscript required for ledger construction.

pith-pipeline@v0.9.1-grok · 5799 in / 1182 out tokens · 26766 ms · 2026-06-27T06:09:44.005567+00:00 · methodology

discussion (0)

Reference graph

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