Implicit Neural Representation of Waveform Measurements in Power Systems Waveform Data Analysis

Hamed Mohsenian-Rad; Narges Ehsani; Vishwanath Saragadam

REVIEW 4 major objections 5 minor 21 references

Implicit Neural Representation of Waveform Measurements in Power Systems Waveform Data Analysis

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A double-hidden-layer sinusoidal INR can represent real power-system voltage and current waveforms with 1–2% mean-squared error and 4–6× parameter compression, preserving transient and frequency content that an equal-size…

desk verdict First genuine INR-to-power-waveform application with a solid two-layer vs one-layer empirical case; the headline compression ratio is inflated until you count bytes. read the letter →

arxiv 2505.09789 v2 pith:LD3FU4D5 submitted 2025-05-14 eess.SP

Narges Ehsani , Vishwanath Saragadam , Hamed Mohsenian-Rad This is my paper · ORCID

classification eess.SP

keywords waveformanalyticsimplicitneuralrepresentationsinusoidalactivationpowersystemmonitoringsynchro-waveformmeasurementsthree-phaseoscillationmodelcompression

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that a small neural network can serve as a compact, continuous mathematical model of raw power-system voltage and current waveforms, replacing bulky waveform captures with a few thousand parameters. The key claim is that the network needs two hidden layers with sinusoidal activations: a one-hidden-layer version is essentially a Fourier series and misses sub-cycle transient distortions, whereas the two-layer version captures them, giving roughly three times better accuracy at the same model size. The authors test this on 30 real-world event captures from a 480-volt three-phase sensor, reporting 1–2% mean-squared error and four to six times compression, and they show a shared-layer version can model three phases or synchronized waveforms together with fewer parameters. The motivation is that high-rate waveform monitoring, especially synchro-waveform streaming, can exceed a gigabyte per sensor per day, so a faithful compact representation would make waveform analytics practical.

What carries the argument

The load-bearing object is the double-hidden-layer sinusoidal MLP, an implicit neural representation mapping time $t$ to a waveform value $x(t)$. The single-layer model reduces to a sum of sinusoids, i.e., a truncated Fourier series, while the two-layer model composes sinusoids of sinusoids, which the authors connect to a larger reachable frequency space. The second machinery piece is the shared-hidden-layer multi-output extension, where several correlated waveforms, such as three phases or synchro-waveforms at multiple locations, are produced from one network, reducing parameter count when correlations exist. Sinusoidal activation is what lets both networks represent periodic signals compactly; replacing it with ReLU raises MSE to 18.88% on the initial case study, so the activation choice is central to the representation.

What would settle it

Collect a diverse set of event captures from multiple sensors, voltage levels, and event types, such as voltage sags, capacitor-switching transients, and inverter trips, and rerun the equal-parameter one-layer versus two-layer comparison; the central claim is falsified if the two-layer MSE advantage over one-layer drops to the run-to-run noise level, or if a one-layer model matches two-layer accuracy on the sub-cycle portion of any capture.

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Extended reading notes

Core claim

The central claim is that an implicit neural representation with two sinusoidal hidden layers is the right model class for power-system waveform measurements. With one hidden layer the output has the form $x(t) = \sum_{i=1}^{h} a_{2,i} \sin(a_{1,i}t + b_{1,i}) + b_2$, so the learned parameters play exactly the roles of Fourier frequencies, magnitudes, and phases, and the model cannot represent non-stationary sub-cycle events. With two hidden layers the nested composition $x(t) = \sum_{j=1}^{h_2} a_{3,j} \sin(\sum_{i=1}^{h_1} a_{2,i,j}\sin(a_{1,i}t + b_{1,i}) + b_{2,j}) + b_3$ no longer resembles a Fourier transform; the second layer introduces spectral diversity that lets the network represent transient distortions. On a real 62-cycle capture containing a sub-cycle oscillatory event, the same-parameter-count comparison moves MSE from 2.40% (one layer) to 0.82% (two layers). Across 30 voltage and current event signatures, the two-layer INR reconstructs waveforms with MSE between 0.77% and 2.85%, and in the oscillation case studies it reproduces the DFT spectrum of both a 900 Hz single-mode oscillation and a dual-mode $60$ Hz $\pm f_{\mathrm{sideband}}$ modulation while compressing the raw three-phase data by $4$–$6\times$.

Load-bearing premise

All reported numbers come from 30 waveform captures taken by a single 480-volt three-phase sensor, and if those captures are not representative of grid waveforms in general, the claimed accuracy and the two-layer advantage will not generalize.

Editorial extensions

If this is right

Waveform captures can be stored and transmitted as INR parameters rather than raw samples, cutting the per-sensor data footprint by roughly four to six times while keeping reconstruction error below a few percent.
Event analysis that relies on the frequency spectrum, such as locating a dominant 900 Hz oscillation mode or resolving sidebands around 60 Hz, can be run on the INR reconstruction without losing the modes visible in raw data.
Modeling three-phase or multi-location waveforms with one shared network is more parameter-efficient than modeling each channel separately, especially when channels are correlated.
Because a one-hidden-layer INR is mathematically a Fourier representation, any waveform-analytics pipeline that needs sub-cycle transient detail should use at least two hidden layers.
The accuracy saturation observed in the sensitivity analysis indicates that for a fixed parameter budget, increasing $h_2$, the second-layer width, buys more accuracy than increasing $h_1$, the first-layer width.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next test is whether the same two-layer architecture yields a fixed-length embedding of an event; if so, INR weights could themselves become features for classifying fault types or locating disturbances, without computing spectra.
The reported compression counts model parameters against raw samples and ignores the cost of encoding those parameters; a fair comparison with Fourier or wavelet codecs would need a bitrate-versus-distortion curve on the same captures.
Because the second layer is what creates spectral diversity, the architecture may also apply to other non-stationary power signals, such as traveling-wave fault records or inverter switching transients, where a Fourier basis is known to be inefficient.
A testable prediction follows from the sensitivity analysis: for a fixed parameter budget, optimal width allocation should favor $h_2$ over $h_1$; one could check this by grid-searching widths on a larger multi-site dataset.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing to know: this is a genuine first application of INRs to power-system waveform measurements, not a repackaging. The paper shows that a two-hidden-layer sinusoidal MLP can represent real voltage/current captures at 1-3% MSE, and that sharing layers across phases cuts parameters by about 2x. The core modeling claim is believable.

New and good: they fit time-to-waveform SIREN-type MLPs, compare one vs double hidden layer, show two layer captures transient sub-cycle events with ~3x lower MSE at equal parameter count, and extend to combined three-phase/synchro-waveform models. Results are on real SEL 735 captures, 30 signatures averaged over 10 runs. The 1 vs 2 layer comparison is a clean design contribution. Citation pattern is appropriate: it builds on Sitzmann, Roddenberry, plus power-waveform background. No code or data, but the experimental setup is transparent enough to reproduce roughly.

Soft spots: headline '4-6x compression' is a parameter-count ratio, not bytes. Raw samples are usually 16-bit; float32 weights run about 1.5-2.2x. If they quantize and entropy code, maybe, but none is specified. They need to state bitrate or bytes. Also all results are from one 480V SEL 735 sensor; the MSE numbers and the two-layer advantage may not generalize to other sensors/event classes. There are no classical baselines (Fourier/wavelet/Slipstream) against the reconstructed quality; the conclusion acknowledges this but only as an apology. Hyperparameters and training details are missing, and the abstract's '<1-2% MSE' doesn't match the 0.77-2.85% range in Section IV. The 'single-layer = Fourier transform' claim is a useful intuition but overstated; a single-layer sine MLP with trained frequencies is not literally DFT and can approximate transients with enough terms. Those are real soft spots, but none of them sinks the central representation claim.

Bottom line: if you want a careful byte-level compression paper, this isn't it. If you want an applied first look at INRs for waveform analytics with some honest empirical design, this is useful. I would send it to peer review, because the application is timely and the one-vs-two-layer result is worth a serious referee's time despite the needed compression correction. My own cite: maybe for context, though I'd wait for code/data.

Referee Report

4 major / 5 minor

Summary. This paper proposes the use of implicit neural representations (INRs) with sinusoidal activations to model power-system voltage and current waveforms. It compares single-hidden-layer and double-hidden-layer INRs, arguing that the former behaves like a Fourier series and cannot capture transients while the latter achieves roughly 3x lower MSE at equal parameter count. It further proposes a shared-hidden-layer INR to represent three-phase and synchro-waveform measurements, and reports 4-6x 'compression' relative to raw samples. Experiments use 30 real-world waveform captures from a 480V SEL 735 sensor, with sensitivity analysis for h1 and h2, comparisons of separate versus combined models, and two oscillation-monitoring case studies.

Significance. The core representation claim is plausible and supported by the experiments: on the presented data, two-layer sine INRs outperform equal-parameter one-layer INRs and ReLU, and the combined three-phase model is more parameter-efficient than three separate models. The use of real waveforms and the averaging over 300 and 900 training runs are strengths. However, the headline compression gain is measured in parameter counts rather than bytes, the dataset is a single-sensor collection, and there is no comparison with existing waveform-modeling or compression methods. With those corrections and scoping, the paper would be a useful contribution to waveform analytics.

major comments (4)

[Abstract; Section IV-D] The claimed '4–6× compression' is not supported by the reported numbers as a storage ratio. Section IV-D compares 23,808 raw 'parameters' with 5,503 INR parameters, but raw waveform samples are not parameters and their storage cost depends on numeric precision. For a 62-cycle capture at 128 samples/cycle, one phase has 7,936 samples; at the typical 16-bit WMU resolution this is 15,872 bytes, while a float32 INR for that phase (h1=h2=50, 2,701 parameters) needs 10,804 bytes, a factor of only 1.47. For the three-phase combined model in Section IV-D, 5,503 float32 weights need 22,012 bytes versus 47,616 raw bytes, a factor of about 2.16. The '4–6×' figure also does not match the stated parameter ratios: 23,808/8,103 = 2.94 and 23,808/5,503 = 4.33. Please report byte-level compression with the actual precision of the INR weights, or explicitly relabel the claim as parameter-count reduction and avoid the word 'compression' in the abstract.
[Section IV-D and Fig. 7] The hyperparameters used for the separate and combined models in Fig. 7 are not stated, and the reported parameter counts are inconsistent with the earlier sensitivity setup. Three separate phase models with h1=h2=50 give 8,103 parameters by Eq. (4), but a combined three-output model with h1=h2=50 gives 2,803 by Eq. (6), not 5,503; the 5,503 figure corresponds to h1=50, h2=100. Please specify the exact h1 and h2 used for each curve and recompute the comparison on an equal-size or matched-accuracy basis.
[Section IV-A and Abstract] The abstract claims 'accuracy (<1–2% MSE)', but Section IV-A states that the MSE varies from 0.77% to 2.85% across the nine displayed examples, and Fig. 5 shows MSE values well above 2% for some configurations. Please either define the range over which the <1–2% claim is made or correct the abstract, because the headline accuracy is part of the central claim.
[Sections IV and V] All quantitative results are based on 30 waveform captures from a single SEL 735 sensor at one 480 V location. Since both the reported MSE values and the roughly 3× advantage of the two-layer INR are averages over this one-sensor dataset, the conclusion that INRs are generally effective for power-system waveforms is not yet established. Please add data from additional sensors, locations, or voltage classes, or explicitly limit the claims to the studied dataset.

minor comments (5)

[Eq. (3)] In Eq. (3), the bias inside the second sine is written as b2,i, but the summation index is j; it should be b2,j (or a separate bias per second-layer neuron). Please fix the notation.
[Section IV-B] The training-details paragraph mentions the largest INR model with h1=50 and h2=70, but Figs. 5(a)–(c) only show h1 up to 30 and h2 up to 50. Please reconcile the text and the figure, or clarify that the compute-time example is outside the plotted sensitivity range.
[Section II and IV] The paper should define what 'paramter' means when comparing to raw samples, and state the numeric precision (e.g., float32 vs 16-bit) used in any reported compression. This is essential for reproducibility and for interpreting the 'model size' claims.
[Section IV-A] The training setup is not fully specified: the optimizer, learning rate, number of epochs, SIREN initialization scale (ω0), and loss function are not reported, even though these are known to be important for INR training. Providing them would strengthen the reproducibility of the results.
[Section V] The limitation that no comparison with state-of-the-art waveform modeling or compression techniques is performed is acknowledged only in the conclusions; since the paper positions INRs against Fourier- and wavelet-style representations, a baseline comparison (even on the same dataset) would substantially strengthen the novelty claim.

Circularity Check

1 steps flagged · score 2.0 of 10

Only mild self-citation of a co-authored frequency-convergence result; the central MSE and compression results are empirically self-contained.

self citation load bearing [Section II-B, paragraph after Eq. (3)]
"Indeed, as explained in [21], INRs with two or more hidden layers provide a larger convergence of the frequency space than INRs with only one hidden layer."

Reference [21] is a co-authored arXiv preprint (Roddenberry, Saragadam, de Hoop, Baraniuk), and V. Saragadam is an author of the present paper. The theoretical premise that two or more hidden layers converge a larger frequency space is imported from that self-citation rather than derived or independently verified in this paper. This premise is used to motivate the paper's core double-hidden-layer design and to explain why it outperforms a one-layer, Fourier-like INR. However, the paper's headline accuracy and roughly 3x improvement are supported by direct real-data comparisons, so the self-citation is not the sole load-bearing evidence; the circularity is therefore limited.

full rationale

The paper's derivation chain is largely self-contained. Equation (1) is a sum of sines and Eq. (3) is a two-layer composition of sines; the MSE comparisons in Figs. 2 and 5 are direct fits of the same waveforms, and no fitted parameter is renamed as a prediction. The oscillation-spectrum demonstration compares DFTs of raw and reconstructed data, which is a consistency check rather than a prediction derived from a fit. The claimed 4-6x compression is a parameter-count comparison with raw samples, which is a legitimate model-size statement, even if a practical byte-level ratio would require precision and quantization details. The only circularity-adjacent element is the citation of a co-authored preprint for the frequency-convergence advantage of two hidden layers; this supports the architecture choice, but the empirical results stand independently. Thus no central claim reduces by construction, and the paper warrants a low score of 2 for one minor self-citation that is not ultimately load-bearing.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper contributes no new physical or mathematical entities. Its central claims rest on standard MLP training, borrowed INR frequency-convergence theory, and the representative-data assumption. The main uncharged costs are hand-chosen architecture sizes, unreported training details, and the parameter-count-as-compression convention.

free parameters (2)

Hidden layer sizes h1 and h2 = h1 = 10-50, h2 = 10-70 in sweeps; h1 = 30, h2 = 50 in Fig. 2
Model capacity is chosen by hand, and all accuracy and compression results depend on these architecture sizes.
SIREN initialization scale and training hyperparameters = not reported
Sinusoidal activation networks are sensitive to initialization scale; the paper does not state optimizer, learning rate, epochs, or activation frequency, so these unreported settings act as implicit free parameters in reproducing the results.

assumptions (5)

domain assumption Sinusoidal activation is appropriate because power-system voltage and current waveforms are predominantly periodic.
Stated in Sections I-B and II as the motivation for choosing sigma(x) = sin(x).
standard math A two-hidden-layer INR has larger convergence of the frequency space than a one-hidden-layer INR.
Invoked in Section II-B via reference [21], which is an arXiv preprint by a co-author; the paper relies on this result instead of deriving it.
domain assumption The 30 waveform signatures from one SEL 735 sensor are representative of power-system waveforms generally.
All quantitative conclusions in Section IV are averaged over this single-sensor dataset; no external dataset is used.
ad hoc to paper INR storage cost equals the number of network parameters, and raw waveform storage cost equals the number of samples.
The 4-6x compression claim in Section IV-D is a parameter-count ratio (23,808 raw parameters to 5,503 INR parameters); this ignores weight precision, metadata, and training cost.
domain assumption Gradient-descent training finds a sufficiently accurate fit for each waveform.
The MSE results assume successful optimization; no convergence guarantees are provided.

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Cite this review

Pith. "Pith review of Implicit Neural Representation of Waveform Measurements in Power Systems Waveform Data Analysis." pith.science (2026). https://pith.science/paper/LD3FU4D5

@misc{pith2026250509789,
  author       = {Pith},
  title        = {Pith review of: Implicit Neural Representation of Waveform Measurements in Power Systems Waveform Data Analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LD3FU4D5}},
  note         = {Machine review of arXiv:2505.09789}
}

read the original abstract

There is currently a paradigm shift in several power system monitoring applications, such as incipient fault detection and monitoring inverter-based resources, to transition from traditional phasor analytics to more informative waveform analytics. This paper contributes to this transition by developing a novel approach to modeling voltage and current waveform measurements using implicit neural representations (INRs). INRs are continuous function approximators that are recently used in vision and signal processing. The proposed INR models are specifically designed to meet the requirements of waveform analytics in power systems, such as by using sinusoidal activation functions that capture the periodic nature of voltage and current waveforms. We also propose extended models that can efficiently represent correlated waveforms, such as three-phase waveforms and synchro-waveforms. Real-world case studies demonstrate the effectiveness of the proposed INR models in terms of accuracy (<1-2% MSE) and model size (4-6x compression). We also investigate the application of INR models in oscillation monitoring, for single mode oscillations and dual mode modulated oscillations.

Figures

Figures reproduced from arXiv: 2505.09789 by the authors.

**Figure 3.** The proposed model to enhance efficiency in modeling t [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** shows examples of real-world waveform measurements, including raw data (blue) and reconstructed waveforms (red) based on INR with h1 = h2 = 50. The measurements are obtained from a three-phase SEL 735 power quality sensor at 480V (line-to-line). The sampling rate of the waveform measurements is 128 samples per cycle, i.e., 128 × 60 = 7680 samples per second. Each waveform capture includes 62 cycles of waveform meas… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Various waveform event signatures. Each example sho [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: Sensitivity analysis of the INR models to the number o [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

**Figure 7.** Figure 7: Analysis of the frequency spectrum (in the right colu [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]

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Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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High-resolution time-sy nchronized grid monitoring devices,

A. Silverstein and J. Follum, “High-resolution time-sy nchronized grid monitoring devices,” North American Synchrophasor Initia tive Technical Report NASPI-2020-TR-004, Mar. 2020

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