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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, Johann Jäger, Siming Bayer
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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: this https URL
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PHYSICS-INFORMED BUT NOT PHYSICS-CONSISTENT: ERROR GEOMETRY AND SUBSPACE PROJECTION FOR NEURAL AC POWER FLOW Changhun Kim1, Timon Conrad2, Redwanul Karim1, Karan Pahlajani1, Julian Oelhaf1, David Riebesel2 Tomás Arias-Vergara1, Andreas Maier1, Johann Jäger2, Siming Bayer1 †thanks: Corresponding author: changhun.kim@fau.de. This work was supported by the GridAssist project under the “OptiNetD” initiative of the German Federal Ministry for Economic Affairs and Energy (BMWE) and by HPC resources from NHR@FAU, partially funded by the German Research Foundation (DFG). 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%67.0\%, 37.8%37.8\%, 40.5%40.5\%, and 68.9%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: github.com/Kimchangheon/neural-acpf-error-geometry Index Terms: AC power flow, physics-informed neural network, graph neural network, error geometry, subspace projection †address: 1Pattern Recognition Lab, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany 2Institute of Electrical Energy Systems, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany 1 Introduction AC power flow (AC-PF) is a fundamental tool for steady-state power-system analysis [30]. It is repeatedly solved in congestion management [6] and contingency studies [1], and underlies optimal power flow (OPF) [10, 22] and state-estimation workflows [3]. The need for both computational efficiency and physical consistency has motivated neural surrogates including direct data-driven approximation [8, 18], physics-informed training [23, 14, 28], feasibility-seeking mechanisms [21, 9], and learned projection-based correction [5]. More recent cross-grid and foundation-model efforts–including GridSFM [29], gridfm-graphkit and GENCO [12, 25], and LUMINA [17]–extend this direction with global context and topology-aware representations of heterogeneous grid components [7] designed to operate across diverse grids and operating conditions [15]. However, accurate voltage prediction does not necessarily imply physical consistency [19, 26]. Physics-informed objectives can reduce power-flow residuals [20, 13], but remain soft training objectives and do not by themselves enforce exact satisfaction of the nonlinear AC equations [16]. Consequently, neural surrogates with small voltage errors can still exhibit non-negligible power-balance violations. We refer to this discrepancy as the accuracy–consistency gap and ask which properties of the remaining prediction error govern its physical impact. We hypothesize that error direction, rather than magnitude alone, is an important factor: similarly sized voltage errors can induce substantially different AC residuals depending on their orientation across buses. Recent work has examined directional effects of warm-start errors in the distinct context of Newton–Raphson (NR) convergence [2]. This motivates studying the geometry of neural-surrogate errors in the voltage-state space. 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. Goodwin et al. [11] characterized the differential geometry of the AC-PF solution manifold, while Park et al. [24] and Chevalier et al. [4] demonstrated low-dimensional structure in AC-OPF and PF solutions, respectively. Motivated by this evidence of low-dimensional solution structure, we ask whether the relationship between neural prediction errors and a training solution subspace can be quantified to explain physical inconsistency, and whether this structure can be exploited to reduce power-balance violations. To address these questions, we make three contributions. (i) We decompose calibrated prediction errors of four neural surrogates into in- and off-subspace components using a low-rank singular value decomposition (SVD) basis fitted to training AC-PF solutions, and show through fixed-norm interventions and local Jacobian analysis that off-subspace errors are substantially more residual-sensitive. (i) Building on this finding, we propose calibrated solution-subspace projection (CSP), a lightweight post-hoc output correction that uses only training data, removes systematic bias, and suppresses off-subspace prediction components without retraining. (i) We evaluate CSP on frozen PIGNN-GC, GridSFM, gridfm-graphkit, and LUMINA models on the 2224-bus GBnetwork, where it lowers AC power-balance residuals and voltage-magnitude error in all four models and provides a better initialization for one Newton update. We further assess its generality on 30 additional systems and its robustness under topology and load-distribution shifts. 2 Error Geometry and CSP 2.1 AC power-balance consistency and calibration For an N-bus grid, let x=[V,θ]∈ℝ2Nx=[V,θ] ^2N denote the voltage state, with V¯=V⊙eȷθ V=V e θ. Given operating condition u, a surrogate predicts x^=fϕ(u) x=f_φ(u), from which (P,Q)(P,Q) are recomputed using the network admittance model. For setpoints (Pset,Qset)(P^set,Q^set), we define r(x^,u)=[ΔPΔQ]=[Pset−PQset−Q],PB=ΔP2+ΔQ2.r( x;u)= bmatrix P\\ Q bmatrix= bmatrixP^set-P\\ Q^set-Q bmatrix, = P^2+ Q^2. PB uses ΔP P at PV/PQ and ΔQ Q at PQ buses, zero elsewhere; prescribed PV/slack magnitudes and the slack angle are restored before scoring. Mean/Max PB aggregate all bus–scenario pairs on a common 100-MVA per unit (p.u.) base. Before geometric analysis, we remove train-only prediction bias: b^=s∈tr[x^s−xs⋆],x^C=x^−b^, b=E_s [ x_s-x_s ], x^C= x- b, where s indexes scenarios, ⋆ denotes the converged NR reference, and C denotes calibration. b b is frozen at test time; angle bias is estimated as the per-bus circular mean of wrapped prediction errors. 2.2 Error geometry and training solution subspace projection From training Newton–Raphson solutions, let XV,Xθ∈ℝStr×NX_V,X_θ ^S_tr× N stack centered magnitudes and circularly centered angles: XV=WVΣVUV⊤,Xθ=WθΣθUθ⊤,Pk=UkUk⊤,X_V=W_V _VU_V , X_θ=W_θ _θU_θ , P_k=U_kU_k , (1) where Uk=diag(UV,k,Uθ,k)U_k=diag(U_V,k,U_θ,k) contains the leading k state-space directions per channel; k=16k=16 denotes 16 per channel. For calibrated error e=x^C−x⋆e= x^C-x , e∥=Pke,e⟂=(I−Pk)e,x^−x⋆=b^+e∥+e⟂.e =P_ke, e =(I-P_k)e, x-x = b+e +e . (2) Here, b b is the train-estimated mean bias, while e∥e and e⟂e partition the remaining error into in- and off-subspace components. Around a converged state with fixed topology and bus types, r(x⋆+e,u)=r(x⋆,u)+J(x⋆,u)e+(‖e‖22),r(x +e;u)=r(x ;u)+J(x ;u)e+O(\|e\|_2^2), (3) where r(x⋆,u)=0r(x ;u)=0 and J is the AC residual Jacobian; hence equal-norm errors can yield different residuals by orientation. Motivated by this orientational sensitivity, CSP first removes b b and then projects onto the dominant training-solution directions: x^CSP=x¯tr+Pk(x^−b^−x¯tr) x^CSP= x_tr+P_k ( x- b- x_tr ) (4) where x¯tr x_tr is the training-solution mean; both x¯tr x_tr and PkP_k remain fixed at test time. Projection can remove prediction error but also discard valid reference variation. Let eC=x^C−x⋆e^C= x^C-x and eCSP=x^CSP−x⋆e^CSP= x^CSP-x . Since PkP_k is orthogonal, ‖eCSP‖22−‖eC‖22=‖(I−Pk)(x⋆−x¯tr)‖22−‖(I−Pk)eC‖22.\|e^CSP\|_2^2-\|e^C\|_2^2=\|(I-P_k)(x - x_tr)\|_2^2-\|(I-P_k)e^C\|_2^2. (5) Hence, CSP improves state accuracy when the removed off-subspace prediction-error energy exceeds the discarded reference variation. CSP is a lightweight, data-driven post-hoc output correction rather than a projection onto the nonlinear AC-feasible set; it therefore does not guarantee exact feasibility, but can improve physical consistency when the removed off-subspace components are residual-sensitive under the local AC Jacobian. 3 Experiments and Results Our primary study uses 37,022 converged pandapower Newton–Raphson operating points on the 2224-bus GBnetwork, split 1:1:1 for training/validation/test. The dataset combines global load/generation scaling with independent load P/QP/Q and active-generation perturbations; the backbone topology and PV-voltage control schedule remain fixed. We evaluate PIGNN-GC, our global-context extension of PIGNN-Attn-LS [16] retaining its line-search, GridSFM [29], gridfm-graphkit [12], and LUMINA [17]; GridSFM is also evaluated cross-grid on 31 systems. All models are trained from scratch using model-specific stable initialization, and CSP is evaluated within each frozen model. gridfm-graphkit uses its released architecture/configuration without a pretrained checkpoint. Both GridSFM and LUMINA retain their released backbones and are adapted from AC-OPF to AC-PF by fixing generator dispatch, predicting only voltage states with PF-specific residuals, and applying known-variable masking. All models use pre-solution pandapower [30, 27] internal PPC-derived inputs. Training and inference use one NVIDIA H100 96-GB GPU; inference timing uses the largest memory-fitting batch to maximize GPU utilization. Calibration statistics and solution bases use training data only. From k∈4,8,16,32,64k∈\4,8,16,32,64\, validation Mean PB selects k⋆=16k =16 without degrading calibrated voltage-magnitude RMSE and is frozen for test and distribution-shift experiments. Unless otherwise stated, mechanistic, ablation, and cross-grid analyses use model seed 42, whereas Tables 1 and 3 aggregate seeds 41–43. Voltage RMSE alone can be small even when a model largely predicts a static per-bus profile rather than following scenario-dependent voltage variation [29]. To isolate this scenario-dependent response, we center each bus as V~is=Vis−V¯i V_is=V_is- V_i and V^~is=V^is−V^¯i V_is= V_is- V_i, and report Rw2=corr2(V~is,V^~is),aw=Cov(V~is,V^~is)Var(V~is).R_w^2=corr^2 ( V_is, V_is ), a_w= Cov ( V_is, V_is )Var ( V_is ). (6) Rw2R_w^2 is the squared correlation between the within-bus-centered prediction and reference across all bus–scenario pairs, while awa_w measures the scale of the predicted variation relative to the true variation. Table 1: GBnetwork results at k=16k=16. All models report mean ± standard deviation over three model seeds. C denotes calibration, P16P_16 projection, and CSP16 their composition. Inference time is GPU-synchronized forward execution plus the corresponding output transform, reported per scenario. Model Output V RMSE θ RMSE Rw2↑R_w^2 awa_w Mean PB Max PB Inference ↓ (∘) ↓ ↓ ↓ (ms/scen.) Per-bus mean Baseline .02341 6.847 .000 .000 .12084 5.88 – PIGNN-GC Raw .02850±.00232.02850±.00232 1.108±.1751.108±.175 .671±.050.671±.050 .546±.072.546±.072 .31592±.03240.31592±.03240 195.89±52.54195.89±52.54 10.034±.02310.034±.023 C .01393±.00105.01393±.00105 .777±.161.777±.161 .671±.050.671±.050 .546±.072.546±.072 .16661±.02352.16661±.02352 190.00±37.53190.00±37.53 10.028±.02310.028±.023 P16P_16 .01624±.00254.01624±.00254 1.091±.1661.091±.166 .829±.045.829±.045 .541±.071.541±.071 .07594±.00422.07594±.00422 5.07±.855.07±.85 10.036±.02310.036±.023 CSP16 .01222±.00134.01222±.00134 .835±.141.835±.141 .829±.045.829±.045 .541±.071.541±.071 .05497±.00257.05497±.00257 3.93±.073.93±.07 10.034±.03210.034±.032 GridSFM Raw .02868±.01652.02868±.01652 2.294±.3022.294±.302 .768±.151.768±.151 .724±.149.724±.149 .26867±.13004.26867±.13004 9.10±4.029.10±4.02 25.045±.20725.045±.207 C .01086±.00403.01086±.00403 1.781±.0551.781±.055 .768±.151.768±.151 .724±.149.724±.149 .08251±.01093.08251±.01093 7.29±3.537.29±3.53 25.015±.09625.015±.096 P16P_16 .01814±.01145.01814±.01145 2.244±.3252.244±.325 .936±.032.936±.032 .724±.149.724±.149 .07630±.01942.07630±.01942 5.08±.495.08±.49 25.087±.27925.087±.279 CSP16 .00782±.00307.00782±.00307 1.807±.0531.807±.053 .936±.032.936±.032 .724±.149.724±.149 .05136±.00143.05136±.00143 4.94±.424.94±.42 24.577±.09124.577±.091 gridfm- Raw .00892±.00154.00892±.00154 .385±.042.385±.042 .905±.030.905±.030 .811±.045.811±.045 .09282±.01763.09282±.01763 19.82±8.6519.82±8.65 5.917±.0195.917±.019 graphkit C .00753±.00118.00753±.00118 .367±.042.367±.042 .905±.030.905±.030 .811±.045.811±.045 .07685±.01307.07685±.01307 18.88±8.6918.88±8.69 5.730±.0045.730±.004 P16P_16 .00617±.00099.00617±.00099 .503±.018.503±.018 .959±.008.959±.008 .801±.044.801±.044 .04593±.00156.04593±.00156 5.36±.565.36±.56 5.723±.0115.723±.011 CSP16 .00606±.00090.00606±.00090 .496±.021.496±.021 .959±.008.959±.008 .801±.044.801±.044 .04571±.00135.04571±.00135 5.08±.185.08±.18 5.779±.0125.779±.012 LUMINA Raw .02653±.00031.02653±.00031 2.703±.3132.703±.313 .443±.014.443±.014 .346±.012.346±.012 1.29896±.850561.29896±.85056 664.19±448.00664.19±448.00 7.548±0.3697.548±0.369 C .01780±.00020.01780±.00020 1.442±.0601.442±.060 .443±.014.443±.014 .346±.012.346±.012 .21432±.01186.21432±.01186 20.65±15.8220.65±15.82 7.380±0.1557.380±0.155 P16P_16 .01898±.00049.01898±.00049 2.160±.2682.160±.268 .906±.012.906±.012 .338±.012.338±.012 .07641±.00128.07641±.00128 5.47±.565.47±.56 7.981±0.8687.981±0.868 CSP16 .01570±.00024.01570±.00024 1.282±.0581.282±.058 .906±.012.906±.012 .338±.012.338±.012 .06662±.00074.06662±.00074 4.61±.194.61±.19 7.891±0.5677.891±0.567 3.1 Off-subspace error and directional sensitivity. Off-subspace error. Off-subspace error is substantial across models and is positively associated with Mean PB. Although the rank-16 basis explains 98.8%/99.7% of training voltage-magnitude/angle variation, the median scenario-wise fractions of total squared error lying outside the respective subspaces are 26.3/55.0/56.8/27.1% for magnitude and 10.3/2.7/39.6/31.7% for angle, for PIGNN-GC/GridSFM/gridfm-graphkit/LUMINA, respectively. After accounting for differences in total error norms, scenarios with larger off-subspace fractions have higher Mean PB, with partial Spearman correlations of .659/.635/.594/.833 for magnitude and .789/.906/.809/.860 for angle (all 95% CIs exclude zero). Fixed-norm intervention. At fixed error norm, suppressing off-subspace error consistently lowers Relative Mean PB across all four surrogates, whereas suppressing in-subspace error has the opposite effect (Fig. 2), isolating error orientation from error magnitude. We test this with an oracle diagnostic using the known reference state, varying the voltage-magnitude error orientation at fixed ℓ2 _2 norm: e~(α)=‖eV‖2eV∥+αeV⟂‖eV∥+αeV⟂‖2, e(α)=\|e_V\|_2 e_V +α e_V \|e_V +α e_V \|_2, (7) where phase angles remain at their calibrated predictions. Decreasing α suppresses off-subspace error; the control analogously suppresses the in-subspace component by exchanging eV∥e_V and eV⟂e_V . Jacobian sensitivity. Off-subspace directions are substantially more residual-sensitive than in-subspace directions under the local AC Jacobian, consistent with the fixed-norm intervention. Using directional gain g(d)=‖Jd‖2/‖d‖2g(d)=\|Jd\|_2/\|d\|_2, 32 random direction pairs at each of 128 held-out operating points yield a median paired off-/in-subspace gain ratio of 1,563×1,563×. The same gain contrast appears in the actual calibrated prediction errors: the ratios of median off- to in-subspace gains are 22.81×22.81×, 575.83×575.83×, 4.69×4.69×, and 6.51×6.51× for PIGNN-GC, GridSFM, gridfm-graphkit, and LUMINA, respectively. The observed error regime is consistent with the first-order expansion in Eq. (3), with median linearized-to-nonlinear residual-change ratios between 0.9960.996 and 1.0091.009 across all four models. Figure 2: Fixed-norm directional intervention on GBnetwork, with Relative Mean PB normalized at α=1α=1. 3.2 Solution subspace specificity. Our matched-basis comparison shows that preserving the same amount of reference-state variation is insufficient for low PB, indicating that the specific state-space directions retained by the basis are important. We compare SVD16 with graph-Laplacian and spatial low-pass controls, each tuned using training references to match the V/θV/θ variation retained by SVD16. On the test set, all three preserve nearly identical reference variation (Table 2), yet the alternatives yield substantially larger PB for both projected references and corrected predictions. Table 2: Matched basis comparison on GBnetwork for PIGNN-GC. Parentheses denote V/θV/θ ranks or smoothing strengths. Basis Ref. var. Ref. Pred. retained (%) PB PB SVD (16/1616/16) 98.789/99.68398.789/99.683 .0415 .0569 Laplacian matched (1750/21221750/2122) 98.819/99.68898.819/99.688 26.15 26.22 Spatial matched (β=.357/.208β=.357/.208) 98.801/99.68498.801/99.684 6.20 6.22 3.3 Post-training CSP correction Table 1 compares Raw, C, P16P_16, and CSP16 across the four frozen surrogates. Relative to C, CSP reduces Mean PB by 67.0%67.0\%, 37.8%37.8\%, 40.5%40.5\%, and 68.9%68.9\%, respectively, while lowering voltage-magnitude RMSE in all four models. Projection improves Rw2R_w^2 with little change in awa_w; angle accuracy is model dependent. Raw and CSP16 inference times are comparable. Magnitude-only, angle-only, and full projection show model-dependent effects of angle projection. PIGNN-GC achieves the lowest PB with full CSP, whereas GridSFM performs best with magnitude-only projection. For gridfm-graphkit, adding angle projection reduces Mean/Max PB from .0612/18.23.0612/18.23 to .0457/5.08.0457/5.08, but degrades angle RMSE from .367∘.367 to .496∘.496 . For LUMINA, it reduces Mean/Max PB from .1939/38.53.1939/38.53 to .0660/4.51.0660/4.51 while improving angle RMSE from 1.374∘1.374 to 1.215∘1.215 . Thus, better physical consistency from angle projection need not imply better angle accuracy. These model-dependent accuracy changes are explained by Eq. (5). For magnitude, the removed off-subspace prediction-error energy exceeds the discarded reference variation across all four models. For angle, the balance differs across models: in gridfm-graphkit, angle projection discards more valid reference variation than model error and therefore worsens angle RMSE, whereas the opposite holds for LUMINA. Figure 3: Grid-by-grid physical consistency of GridSFM before and after CSP16 across 31 systems ordered by bus count. Mean PB is shown on a logarithmic scale. 3.4 Complementarity with one-step Newton refinement. CSP provides a consistently better initialization for one-step Newton refinement, improving both state accuracy and post-update physical consistency across all four models. Starting from C or CSP16, we apply one Newton update (NR1NR_1) to non-slack angles and PQ voltage magnitudes, with validation-selected damping factor; η=1η=1 is selected for all models, with no linear-solve failures. Relative to C initialization, CSP lowers magnitude and angle RMSE in all models and reduces post-update Mean PB by 21.121.1–96.4%96.4\% (Table 3). The benefit is particularly pronounced for LUMINA: C+NR1C+NR_1 shows a large one-seed overshoot despite a successful linear solve, whereas CSP16 initialization remains consistent across seeds. Table 3: One Newton update from calibrated or CSP16 outputs on GBnetwork. V RMSE and Mean PB are reported in 10−310^-3. Values are means ± standard deviations over three model seeds. Model Start V RMSE θ RMSE Mean PB (10−3)(10^-3) (∘) (10−3)(10^-3) PIGNN-GC C 1.80±.121.80±.12 .119±.013.119±.013 2.84±.462.84±.46 CSP16 1.17±.101.17±.10 .073±.009.073±.009 1.48±.131.48±.13 GridSFM C 11.86±6.7711.86±6.77 .681±.380.681±.380 2.50±1.542.50±1.54 CSP16 .97±.20.97±.20 .059±.013.059±.013 1.22±.291.22±.29 gridfm- C .943±.232.943±.232 .0843±.0236.0843±.0236 .941±.325.941±.325 graphkit CSP16 .496±.085.496±.085 .0277±.0041.0277±.0041 .742±.080.742±.080 LUMINA C 13.71±9.2313.71±9.23 .671±.038.671±.038 46.08±64.1546.08±64.15 CSP16 1.040±.0161.040±.016 .0802±.0031.0802±.0031 1.653±.0361.653±.036 3.5 Cross-grid breadth and distribution shift Cross-grid breadth. CSP reduces GridSFM Mean PB across all 31 systems (Fig. 3), spanning 4–9,241 buses and sub-kV to 750-kV levels. The set includes 28 pandapower/PYPOWER-format systems from IEEE, PEGASE, RTE, and GB benchmark families, plus three Common Information Model (CIM)-based networks. For each system, calibration and subspace statistics are fitted from its training data, using nominal k=16k=16 per channel and the available SVD components on smaller systems. PV-voltage setpoints vary for most systems, but are fixed for GBnetwork with conflicting controllers. Topology and spatial shifts. CSP remains stable in most statistics under N–1/N–2 topology shifts, with degradation concentrated in isolated worst-case violations, whereas GBcorr causes a broader performance drop. Under the topology shifts, voltage accuracy, tracking, and bulk PB remain close to control, while larger maxima motivate worst-case assessment beyond average error. GBcorr applies correlated, spatially varying load scaling across 14 zones while preserving total demand and generation settings. Transferring both calibration and basis gives Mean PB of 0.10030.1003 p.u.; refitting calibration alone reduces this to 0.09440.0944, basis alone to 0.07400.0740, and both to 0.0680.068. Although the transferred/refitted bases retain 93.55%/98.53%93.55\%/98.53\% of GBcorr magnitude variation, projecting converged GBcorr references yields Mean PBs of 0.0868/0.04380.0868/0.0438 p.u., respectively. This reference-only diagnostic isolates the effect of basis transfer from surrogate error and helps explain the improvement from basis refitting. Table 4: PIGNN-GC with fixed-rank CSP16 under topology and operating-distribution shifts. T transfers the control calibration and basis, whereas R refits both on condition-matched training data. Condition V RMSE Rw2↑R_w^2 awa_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 4 CONCLUSION This work identifies output-error geometry as an important factor in the gap between voltage accuracy and physical consistency in neural AC power-flow surrogates. Across four architecturally diverse neural surrogates, we show that small voltage errors can contain off-subspace components that induce large AC power-balance violations. Building on this insight, calibrated solution-subspace projection suppresses these components, reducing voltage-magnitude error and power-balance residuals without retraining; the residual reduction extends to 30 additional grids. 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