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arxiv: 2604.22784 · v1 · submitted 2026-04-03 · 💻 cs.LG

Learning Without Adversarial Training: A Physics-Informed Neural Network for Secure Power System State Estimation under False Data Injection Attacks

Solon Falas , Markos Asprou , Charalambos Konstantinou , Maria K. Michael This is my paper

Pith reviewed 2026-05-13 19:35 UTC · model grok-4.3

classification 💻 cs.LG
keywords physics-informed neural networkspower system state estimationfalse data injection attackscyber-physical securitydynamic loss weightingrobust estimationneural network training
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The pith

A physics-informed neural network estimates power system states accurately even under stealthy false data injection attacks without adversarial training.

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

The paper develops a PINN model for power system state estimation that stays accurate when attackers inject false data crafted to evade detection. It replaces adversarial training and manual loss tuning with a dynamic weighting scheme based on homoscedastic uncertainty that automatically balances the data-fit and physics-residual terms. This matters because state estimation underpins grid control decisions and cyber attacks pose increasing risks to digitalized power systems. Tests on the IEEE 118-bus system against families of stealth-constrained AC attacks show lower error and greater stability than fixed-weight PINN baselines.

Core claim

The central claim is that embedding power-flow equations into a neural network and training it with dynamic loss weighting derived from homoscedastic uncertainty produces a PSSE estimator that resists stealth-constrained AC false data injection attacks without any adversarial examples, while delivering lower mean absolute error on voltage magnitudes and phase angles than fixed-weight PINN variants on the IEEE 118-bus test system.

What carries the argument

Dynamic loss-weighting formulation based on homoscedastic uncertainty that learns the relative scaling of supervised data-fit and physics-residual terms during training.

If this is right

  • The estimator remains robust to state distortion, load redistribution, line overloading, and residual-constrained stealth attacks.
  • Accuracy measured by MAE on voltage magnitudes and phase angles exceeds that of fixed-weight PINN models under the tested attack scenarios.
  • Training requires no generation of adversarial attack examples to achieve the reported robustness.
  • Sensitivity to manual selection of loss-term weights is reduced by the automatic uncertainty-based scaling.

Where Pith is reading between the lines

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

  • The same uncertainty-driven weighting could be applied to other physics-constrained learning tasks where adversarial data is expensive to generate.
  • If the approach scales to larger or real-time grids, it could reduce the computational burden of maintaining separate attack detectors in control centers.
  • Extending the model to include additional physics constraints such as transient dynamics might further improve detection of time-varying attacks.

Load-bearing premise

That the homoscedastic uncertainty mechanism automatically learns effective relative scaling between the data and physics loss terms without needing adversarial examples or manual tuning.

What would settle it

If the dynamic-weight PINN model produces higher or equal MAE on voltage magnitudes and phase angles compared with fixed-weight PINN variants when both are tested on the IEEE 118-bus system against the same set of stealth-constrained FDIA families, the claim of superior accuracy and stability is falsified.

Figures

Figures reproduced from arXiv: 2604.22784 by Charalambos Konstantinou, Maria K. Michael, Markos Asprou, Solon Falas.

Figure 1
Figure 1. Figure 1: Simple FDIA attack on the IEEE 118-bus system (Zone 2). Left: mean absolute active- and reactive-power residuals by bus. Right: distributions of [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Proposed dynamic weighting versus fixed and frozen baselines. Left: total loss convergence (log-scale) over [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Dynamic weight evolution: trajectories of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Average MAE across zones for each model per FDIA type. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: aggregates accuracy metrics across all FDIA types and zones, separating voltage and angle errors. Dynamic weighting reduces average V MAE by 96.0% versus fixed and 17% versus frozen, average θ MAE by 75% versus fixed [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Worst-case MAE across [V, θ] for each model. TABLE I AVERAGE MAE ACROSS FDIA TYPES FOR PRIOR MODELS Model Simple Load Line State Estimation FDIA Redistribution Overload Corruption Dynamic PINN 5.3 × 10−3 1.85 × 10−2 2.03 × 10−2 1.29 × 10−2 [9] 1.40 × 10−2 9.46 × 10−2 5.37 × 10−2 4.51 × 10−2 [7] 6.53 × 10−1 6.51 × 10−1 6.63 × 10−1 6.50 × 10−1 structure, and attack semantics differ, IEEE 14 values are not tr… view at source ↗
read the original abstract

State estimation is a cornerstone of power system control-center operations, and its robust operation is increasingly a cyber-physical security concern as modern grids become more digitalized and communication-intensive. Neural network-based approaches have gained attention as alternatives to conventional model-based state estimation methods. Physics-Informed Neural Networks (PINNs), which embed power-flow consistency into the learning objective, have shown improved accuracy over existing approaches. This work proposes a PINN-based model for Power System State Estimation (PSSE) that protects the estimation process against the stealth-constrained AC False Data Injection Attacks (FDIAs) considered in this study. The model is developed without adversarial training. Instead, a dynamic loss-weighting formulation based on homoscedastic uncertainty learns the relative scaling of supervised data-fit and physics-residual terms during training, reducing sensitivity to manual weight tuning. Robustness is evaluated on the IEEE 118-bus system using representative stealthy-FDIA families including state distortion, load redistribution, line overloading, and residual-constrained stealth corruption. Performance is measured using Mean Absolute Error (MAE) on voltage magnitudes and phase angles. Results demonstrate higher accuracy and stability than existing fixed-weight PINN variants.

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 / 2 minor

Summary. The paper proposes a physics-informed neural network (PINN) for power system state estimation (PSSE) that achieves robustness to stealth-constrained AC false data injection attacks (FDIAs) without adversarial training. It uses a dynamic loss-weighting scheme based on homoscedastic uncertainty to automatically scale the supervised data-fit and physics-residual terms during training. The method is evaluated on the IEEE 118-bus system against representative attack families (state distortion, load redistribution, line overloading, residual-constrained stealth corruption), with performance reported via mean absolute error on voltage magnitudes and phase angles, claiming higher accuracy and stability than fixed-weight PINN baselines.

Significance. If the dynamic weighting formulation demonstrably learns adaptive scalings that confer the reported robustness gains, the work would offer a practical advance for secure PSSE by eliminating the need for adversarial training while embedding power-flow physics. This could reduce sensitivity to hyperparameter tuning and improve real-time applicability in cyber-physical power systems. The evaluation on multiple stealthy FDIA families on a standard test system strengthens the potential impact, provided the gains are shown to stem specifically from the learned weighting rather than other modeling choices.

major comments (2)
  1. [§4] §4 (Dynamic Loss Weighting): The central claim that homoscedastic uncertainty weighting learns effective relative scaling between supervised and physics-residual terms (thereby conferring stability without adversarial examples) requires explicit verification. The manuscript should report the evolution of the learned uncertainty parameters over training epochs and include an ablation where the final learned weights are frozen and compared directly to the dynamic case; without this, it remains unclear whether the reported MAE improvements on the IEEE 118-bus system are attributable to adaptivity or simply to a favorable fixed weighting.
  2. [§5] §5 (Results on IEEE 118-bus): The performance tables compare against fixed-weight PINN variants, but do not report error bars, number of random seeds, or statistical significance tests for the claimed higher accuracy and stability across the four FDIA families. Given that the soundness assessment notes absence of these details, the robustness conclusion is not yet load-bearing.
minor comments (2)
  1. [Abstract] Abstract and §1: The phrase 'stealth-constrained AC False Data Injection Attacks considered in this study' should be expanded with a brief reference to the specific attack construction (e.g., residual-constrained or optimization-based) to clarify the threat model for readers unfamiliar with the FDIA literature.
  2. [§3] Notation: Ensure consistent use of symbols for voltage magnitudes, phase angles, and the uncertainty parameters across equations and figures; minor inconsistencies in subscripting were noted in the method description.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of our dynamic loss-weighting approach and strengthen the statistical support for the reported results. We address each major comment below and will incorporate the suggested analyses in the revised manuscript.

read point-by-point responses

  1. Referee: [§4] §4 (Dynamic Loss Weighting): The central claim that homoscedastic uncertainty weighting learns effective relative scaling between supervised and physics-residual terms (thereby conferring stability without adversarial examples) requires explicit verification. The manuscript should report the evolution of the learned uncertainty parameters over training epochs and include an ablation where the final learned weights are frozen and compared directly to the dynamic case; without this, it remains unclear whether the reported MAE improvements on the IEEE 118-bus system are attributable to adaptivity or simply to a favorable fixed weighting.

    Authors: We agree that explicit verification of the adaptivity is necessary to support the central claim. In the revised manuscript we will add plots of the learned uncertainty parameters (σ_data and σ_physics) over training epochs for the IEEE 118-bus experiments. We will also include a new ablation study that freezes the weights at their final learned values and directly compares performance against the fully dynamic case across the four FDIA families. These additions will isolate the contribution of the online adaptation. revision: yes

  2. Referee: [§5] §5 (Results on IEEE 118-bus): The performance tables compare against fixed-weight PINN variants, but do not report error bars, number of random seeds, or statistical significance tests for the claimed higher accuracy and stability across the four FDIA families. Given that the soundness assessment notes absence of these details, the robustness conclusion is not yet load-bearing.

    Authors: We accept the need for statistical rigor. In the revised version we will repeat all experiments with at least five independent random seeds, report mean MAE values together with standard-deviation error bars, and include paired t-test results to establish statistical significance of the improvements over fixed-weight baselines for each FDIA family. revision: yes

Circularity Check

0 steps flagged

No significant circularity; dynamic weighting is an independent formulation

full rationale

The paper's central derivation introduces a PINN for PSSE with a dynamic loss-weighting scheme based on homoscedastic uncertainty to balance supervised and physics-residual terms without adversarial training. This weighting is presented as a learned mechanism during training rather than a fitted parameter renamed as a prediction or defined in terms of the target robustness. No load-bearing steps reduce by construction to inputs via self-definition, self-citation chains, or imported uniqueness theorems. The robustness claim rests on empirical evaluation against specific FDIAs on the IEEE 118-bus system, which is externally falsifiable and does not rely on the weighting reducing to a constant or pre-set value by definition. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard power-flow equations embedded in the PINN loss and the homoscedastic uncertainty weighting technique; no new entities or free parameters are explicitly introduced in the abstract.

axioms (2)
  • domain assumption Power-flow equations hold for the underlying system model
    Invoked when embedding physics consistency into the neural network loss
  • domain assumption Homoscedastic uncertainty provides a valid mechanism for dynamic loss weighting
    Used to learn relative scaling of data-fit and physics-residual terms

pith-pipeline@v0.9.0 · 5526 in / 1202 out tokens · 39053 ms · 2026-05-13T19:35:13.209132+00:00 · methodology

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