Physics-Informed but Not Physics-Consistent: Error Geometry and Subspace Projection for Neural AC Power Flow

Changhun Kim; Timon Conrad; Redwanul Karim; Karan Pahlajani; Julian Oelhaf; David Riebesel; Tomás Arias-Vergara; Andreas Maier

cs.LGOct 5, 2026

Physics-Informed but Not Physics-Consistent: Error Geometry and Subspace Projection for Neural AC Power Flow

Authors: Changhun Kim, Timon Conrad, Redwanul Karim, Karan Pahlajani, Julian Oelhaf, David Riebesel, Tomás Arias-Vergara, Andreas Maier, +2 more

Organizations: Pattern Recognition Lab, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany · Institute of Electrical Energy Systems, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

Abstract

Recent neural power-flow solvers, including emerging foundation models, achieve accurate voltage predictions, yet such accuracy does not necessarily imply physically consistent solutions. Even small complex voltage errors can yield large AC power-balance residuals. We study this accuracy-consistency gap across PIGNN-GC, GridSFM, gridfm-graphkit, and LUMINA on realistic 2224-bus Great Britain network (GBnetwork) scenarios, with cross-grid evaluation of GridSFM over 31 systems. Using a singular value decomposition (SVD) basis fitted to training AC power-flow solutions, we find that neural prediction errors contain substantial components outside the dominant solution subspace. To address this mismatch, calibrated solution-subspace projection (CSP) suppresses off-subspace prediction components after train-only bias calibration, reducing Mean PB by 67.0%, 37.8%, 40.5%, and 68.9% for PIGNN-GC, GridSFM, gridfm-graphkit, and LUMINA, respectively, relative to calibrated predictions, while improving voltage-magnitude accuracy in all four models. These results identify output-error geometry as an important factor in physics-consistent neural AC power flow. Code: https://github.com/Kimchangheon/neural-acpf-error-geometry

Figures & tables

Figure 1: A low-dimensional subspace is derived from the training solutions. CSP first removes train-estimated bias and projects the corrected prediction onto this subspace, suppressing off-subspace errors and improving voltage accuracy and AC power-balance consistency.

Model	Output	$V$ RMSE	$\theta$ RMSE	$R_{\mathrm{w}}^{2}\uparrow$	$a_{\mathrm{w}}$	Mean PB	Max PB	Inference
		$\downarrow$	( ∘ ) $\downarrow$			$\downarrow$	$\downarrow$	(ms/scen.)
Per-bus mean	Baseline	.02341	6.847	.000	.000	.12084	5.88	–
PIGNN-GC	Raw	$.02850{\pm}.00232$	$1.108{\pm}.175$	$.671{\pm}.050$	$.546{\pm}.072$	$.31592{\pm}.03240$	$195.89{\pm}52.54$	$10.034{\pm}.023$
	$C$	$.01393{\pm}.00105$	$\mathbf{.777{\pm}.161}$	$.671{\pm}.050$	$.546{\pm}.072$	$.16661{\pm}.02352$	$190.00{\pm}37.53$	$10.028{\pm}.023$
	$P_{16}$	$.01624{\pm}.00254$	$1.091{\pm}.166$	$\mathbf{.829{\pm}.045}$	$.541{\pm}.071$	$.07594{\pm}.00422$	$5.07{\pm}.85$	$10.036{\pm}.023$
	CSP 16	$\mathbf{.01222{\pm}.00134}$	$.835{\pm}.141$	$\mathbf{.829{\pm}.045}$	$.541{\pm}.071$	$\mathbf{.05497{\pm}.00257}$	$\mathbf{3.93{\pm}.07}$	$10.034{\pm}.032$

Table 1: GBnetwork results at

k=16

. All models report mean

\pm

standard deviation over three model seeds.

C

denotes calibration,

P_{16}

projection, and CSP 16 their composition. Inference time is GPU-synchronized forward execution plus the corresponding output transform, reported per scenario.

Figure 2: Fixed-norm directional intervention on GBnetwork, with Relative Mean PB normalized at $\alpha=1$ .

Basis	Ref. var.	Ref.	Pred.
	retained (%)	PB	PB
SVD ( $16/16$ )	$98.789/99.683$	.0415	.0569
Laplacian matched ( $1750/2122$ )	$98.819/99.688$	26.15	26.22
Spatial matched ( $\beta=.357/.208$ )	$98.801/99.684$	6.20	6.22

Table 2: Matched basis comparison on GBnetwork for PIGNN-GC. Parentheses denote

V/\theta

ranks or smoothing strengths.

Figure 3: Grid-by-grid physical consistency of GridSFM before and after CSP 16 across 31 systems ordered by bus count. Mean PB is shown on a logarithmic scale.

Model	Start	$V$ RMSE	$\theta$ RMSE	Mean PB
		$(10^{-3})$	( ∘ )	$(10^{-3})$
PIGNN-GC	$C$	$1.80{\pm}.12$	$.119{\pm}.013$	$2.84{\pm}.46$
	CSP 16	$\mathbf{1.17{\pm}.10}$	$\mathbf{.073{\pm}.009}$	$\mathbf{1.48{\pm}.13}$
GridSFM	$C$	$11.86{\pm}6.77$	$.681{\pm}.380$	$2.50{\pm}1.54$
	CSP 16	$\mathbf{.97{\pm}.20}$	$\mathbf{.059{\pm}.013}$	$\mathbf{1.22{\pm}.29}$
gridfm-	$C$	$.943{\pm}.232$	$.0843{\pm}.0236$	$.941{\pm}.325$

Table 3: One Newton update from calibrated or CSP 16 outputs on GBnetwork.

V

RMSE and Mean PB are reported in

10^{-3}

. Values are means

\pm

standard deviations over three model seeds.

Condition	$V$ RMSE	$R_{\mathrm{w}}^{2}\uparrow$	$a_{\mathrm{w}}$	Mean	Med.	p95	p99	Max
Control	.01281	.8476	.4866	.0581	.0131	.2665	.5462	3.58
$N$ –1 (T)	.01314	.8391	.4750	.0603	.0131	.2713	.5632	33.16
$N$ –2 (T)	.01288	.8458	.4870	.0617	.0132	.2713	.5721	27.45
GBcorr (T)	.01966	.4819	.4025	.1003	.0231	.4512	.9746	7.00
GBcorr (R)	.01620	.5150	.4247	.0680	.0174	.3031	.6415	4.99

Table 4: PIGNN-GC with fixed-rank CSP 16 under topology and operating-distribution shifts. T transfers the control calibration and basis, whereas R refits both on condition-matched training data.

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Physics-informed GNN for medium-high voltage AC power flow with edge-aware attention and line search correction operator

Changhun Kim, Timon Conrad, Redwanul Karim, Julian Oelhaf+5

Pattern Recognition Lab, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany · Institute of Electrical Energy Systems, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

Parametric Physics-Informed Neural Network

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Physics-informed graph neural networks (PIGNNs) have emerged as fast AC power-flow solvers that can replace the classic NewtonRaphson (NR) solvers, especially when thousands of scenarios must be evaluated. However, current PIGNNs still need accuracy improvements at parity speed; in particular, the soft constraint on the physics loss is inoperative at inference, which can deter operational adoption. We address this with PIGNN-Attn-LS, combining an edge-aware attention mechanism that explicitly encodes line physics via per-edge biases to form a fully differentiable knownoperator layer inside the computation graph, with a backtracking line-search-based globalized correction operator that restores an operative decrease criterion at inference. Training and testing use a realistic High-/Medium-Voltage scenario generator, with NR used only to construct reference states. On held-out HV cases consisting of 4-32-bus grids, PIGNN-Attn-LS achieves a test RMSE of 0.00033 p.u. in voltage and 0.08 deg in angle, outperforming the PIGNN-MLP baseline by 99.5% and 87.1%, respectively. With streaming micro-batches, it delivers 2-5x faster batched inference than NR on 4-1024-bus grids.

Abstract

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