Shared Physics Responses Recover Hidden Rankings in Neural Operator Libraries

Hanbing Liang; Fujun Liu

cs.LGAug 20, 2026

Shared Physics Responses Recover Hidden Rankings in Neural Operator Libraries

Authors: Hanbing Liang, Fujun Liu

Organizations: Nanophotonics and Biophotonics Key Laboratory of Jilin Province, School of Physics, Changchun University of Science and Technology, Changchun 130022, P.R. China

Abstract

Selecting the optimal neural-operator prediction during deployment is challenging when high-fidelity reference solutions are unavailable. We demonstrate that under a squared Hilbert-space loss, ranking a finite model library depends strictly on the low-dimensional span of candidate differences, allowing us to score all models simultaneously using a single anchor-based linearized response of the governing equation. This shared physical diagnostic accurately recovered over 99.6% of pairwise preferences and 99.0% of optimal checkpoints across diverse Fourier and convolutional operator libraries for fluid, reaction-diffusion, and wave dynamics. Furthermore, the corrected physical proxy frequently outperformed the best individual candidates, and we establish computable sufficient conditions that rigorously certify exact decisions for strongly monotone discretizations. By exploiting the local dynamical response rather than raw defect magnitude, this framework enables the reliable and highly efficient deployment of scientific surrogates without requiring ground-truth data.

Figures & tables

Figure 1: One shared physics response turns a finite surrogate library into a reference-free decision. a , Candidate states $v_{1},\ldots,v_{N}$ are aggregated by a reference-free rule $\mathcal{C}$ into the library anchor $w$ . b , One linearized physics response gives the corrected state $s=w+\delta_{w}$ , which the task map $Q_{a}$ converts to the proxy $q$ . c , Distances $S_{i}(q)=\lVert q-Q_{a}(v_{i})\rVert_{M}^{2}$ rank the candidates and select $v_{k^{\star}}$ ; the dashed branch denotes optional direct use of $q$ .

Figure 2: One shared physics response recovers hidden neural-operator choices. a , The same eight candidate states are shown in three horizontal lanes, ordered from best to worst by hidden-reference task error, distance to the shared proxy and assembled-defect energy. Lines connect candidate identities across lanes. Within each frame, the black curve shows

v_{i}

on a common state scale, whereas the magenta curve shows

v_{i}-w

about the grey zero line on a separate magnified scale shared by all candidates. b , Tolerant top-1 and non-tie pairwise recovery in eight libraries. Thin segments join the raw-residual and shared-response results within each library; large markers show pooled recovery with input-clustered 95% confidence intervals.

Figure 3: Shared physics recovers candidate gaps and yields low-regret choices. a , Hidden-reference pair gaps plotted against gaps estimated from the shared, input-shuffled and reversed responses (49,323 non-tie pairs; common normalized axes, bins and density scale). Matching signs indicate correct ordering; the diagonal indicates agreement in gap magnitude. b , Empirical cumulative distributions of span-normalized selection regret for 1,920 primary input–library cases. Zero denotes selection of the hidden-reference best candidate; curves nearer the upper left are better. c , The same regret measure for a separate reaction–diffusion population (1,920 cases), comparing the shared proxy, raw residual and two candidate-wise variational majorants; the two majorant curves nearly coincide.

Figure 4: Library composition, candidate separation and admissibility shape reliable recovery. a , Eight candidate states, their arithmetic mean and medoid on the common signed-log scale $\operatorname{sgn}(u)\ln(1+|u|/0.05)$ . Extreme states displace the mean, whereas the medoid remains candidate-valued. b , Non-tie pair recovery across hidden-gap deciles for the shared response, raw residual, validation choice, mean and medoid (49,323 pairs). c , Paired minimum pressures before and after backtracking for 35 shock cases; the horizontal zero line marks physical admissibility.

Figure 5: Routing and panel-level model selection use comparison at different timescales. a , Hidden-reference winner and task-matched proxy route for each of 99 inputs; colours distinguish checkpoints and black ticks mark mismatches. b , Empirical cumulative distributions of span-normalized selection regret for the same inputs. Regret is excess hidden loss divided by the three-model loss span; curves nearer the upper left are better. c , Empirical cumulative distributions of the same regret for panel-level selection. The left facet contains eight libraries with eight panel evaluations per library at

B=24

; the right contains 16 additional libraries with eight evaluations per library at

B=144

. Methods are identified in the panel legends.

Figure 6: Bound-to-margin ratios remain below the certification threshold. a , Empirical cumulative distribution of 672 ratios

B_{ij}/|G_{ij}(q)|

; the vertical line marks the sufficient threshold of one. b , Largest ratio in each of 24 cases formed by six physical inputs and four libraries. All ratios are evaluated against the unique zero of the stated same-grid discrete operator.

Appendix figures & tables25 assets

Supplementary material from the paper’s appendix.

Appendix

Study	Candidates	Inputs / cases	Pair decisions	Primary rule
Primary two-PDE	8 trained libraries ( $8$ each)	480 / 1,920	53,760	$10^{-6}$ reference non-ties; tolerant top-1
Additional libraries	16 libraries ( $8$ each)	480 / 3,840	107,520	Strict pairs; exact top-1
Sine–Gordon	8 libraries ( $8$ each)	240 / 1,920	53,760	Strict pairs and top-1; unresolved counted as errors
PDEBench	1 public U-Net library ( $3$ models)	99 / 99	297	Strict pairs; exact top-1

Appendix

Supplementary Table 1: Instance-wise study populations. An input–library case evaluates one candidate library on one input; populations are analysed separately.

Setting	Method	Pairwise result	Top-1 result
Two-PDE, eight libraries	Propagated	$49{,}122/49{,}323=99.5925\%$ [99.5355, 99.6482]	$1{,}901/1{,}920=99.0104\%$ [98.5938, 99.4271]
Two-PDE, eight libraries	Raw residual	$33{,}457/49{,}323=67.8325\%$ [67.1886, 68.4727]	$819/1{,}920=42.6563\%$ [40.7813, 44.4792]
Two-PDE, 16 libraries	Propagated	$99.7063\%$ [99.6187, 99.7870] ( $107{,}205/107{,}520=99.7070\%$ )	$98.9549\%$ [98.3333, 99.4792] ( $3{,}800/3{,}840=98.9583\%$ )
Two-PDE, 16 libraries	Raw residual	$70.4352\%$ [69.5879, 71.2891] ( $75{,}733/107{,}520=70.4362\%$ )	$43.4816\%$ [38.9063, 48.0215] ( $1{,}671/3{,}840=43.5156\%$ )
Sine–Gordon	Propagated	$52{,}942/53{,}760=98.4784\%$ [98.2013, 98.7333]	$1{,}835/1{,}920=95.5729\%$ [94.4792, 96.5625]
Sine–Gordon	Raw residual	$27{,}837/53{,}760=51.7801\%$ [50.6622, 52.9018]	$250/1{,}920=13.0208\%$ [11.4583, 14.6875]

Appendix

Supplementary Table 2: Instance-wise hidden-reference results. Values are estimates with 95% clustered-bootstrap intervals; pooled count ratios appear in parentheses when they differ from the clustered estimator. The first two-PDE comparison uses the

10^{-6}

tolerant top-1 rule; the 16-library, Sine–Gordon and PDEBench comparisons use exact checkpoint identity. Exact primary top-1 was

1{,}899/1{,}920

for propagation and

688/1{,}920

for raw residual (Supplementary Section S5). PDEBench is an exact count for one public library. Populations are not pooled.

Selector	Truth-free construction	Pair correct	Top-1 correct
ID validation	Input-independent ordering by in-distribution validation loss	24,814/49,323	420/1,920
Mean centrality	Squared task-space distance to the candidate mean	28,134/49,323	329/1,920
Medoid centrality	Squared task-space distance to the candidate medoid	27,689/49,323	335/1,920
Wrong-input propagation	Propagated coordinate from another input in the same stratum	33,885/49,323	884/1,920
Reversed propagation	Input-aligned correction with its direction reversed	5,080/49,323	59/1,920

Appendix

Supplementary Table 3: Mechanism controls on the same 1,920 primary input–library cases. Pair counts condition on the 49,323 hidden-reference non-ties; top-1 uses the

10^{-6}

tolerant winner set. The wrong-input control cyclically reassigns

q

within PDE and shift stratum, and the reversed control replaces

w+\delta_{w}

w-\delta_{w}

Supplementary Figure 1: Population-level candidate-rank concordance in the primary population. Each tile cross-tabulates a candidate’s hidden-reference rank and its diagnostic rank for the shared response, raw residual, input-shuffled response and reversed response, using one common colour scale. The diagonal concentration describes complete-order structure beyond the primary tolerant top-1 and non-tie pair endpoints; the tiles are dependent views of the same cases.

Cell	Prop. pair	Raw pair	Prop. top-1	Raw top-1	Prop. mean regret
Burgers CNO e1	98.7928	71.7288	96.6667	48.7500	$2.37\times 10^{-6}$
Burgers CNO e2	98.8074	72.4806	96.6667	42.5000	$3.24\times 10^{-6}$
Burgers FNO e1	99.7283	86.8246	99.5833	71.6667	$1.63\times 10^{-7}$
Burgers FNO e2	99.6694	89.5116	99.1667	67.9167	$6.39\times 10^{-6}$
RD CNO e1	100.0000	53.7785	100.0000	10.8333	0
RD CNO e2	100.0000	33.4425	100.0000	0.0000	0

Appendix

Supplementary Table 4: Results for the eight two-PDE libraries. Pair columns condition on hidden-reference non-ties; top-1 is tolerant. Mean regret is the unnormalised excess hidden task loss of the selected checkpoint over the best candidate, averaged across inputs. Each cell has 240 inputs and at most 6,720 pair opportunities.

Cell	Prop. pair	Raw pair	Prop. top-1	Raw top-1	Prop. mean regret
Burgers CNO r1	99.4345	78.5565	97.5000	54.1667	$2.66\times 10^{-6}$
Burgers CNO r2	99.1815	78.7351	97.9167	57.9167	$1.26\times 10^{-6}$
Burgers CNO r3	99.2857	77.2917	98.7500	65.4167	$4.16\times 10^{-7}$
Burgers CNO r4	98.8690	73.5268	95.4167	38.7500	$3.42\times 10^{-6}$
Burgers FNO r1	99.8512	91.5476	99.5833	65.4167	$7.86\times 10^{-7}$
Burgers FNO r2	99.6875	88.0952	98.3333	76.2500	$6.39\times 10^{-6}$

Appendix

Supplementary Table 5: Results for the 16 two-PDE libraries. Mean regret is the unnormalised excess hidden task loss of the selected checkpoint over the best candidate, averaged across inputs. Each cell has 240 inputs and 6,720 strict pair opportunities. Strict top-1 is exact checkpoint agreement.

Cell	Prop. pair	Raw pair	Prop. top-1	Raw top-1	Pair indet.
CNO r1	99.1667	44.3006	97.0833	20.0000	0
CNO r2	98.8542	51.0268	95.8333	8.3333	1
CNO r3	98.7798	51.6071	97.0833	7.9167	1
CNO r4	99.3899	47.8869	98.3333	13.7500	2
FNO r1	97.9167	57.4405	94.5833	15.4167	0
FNO r2	97.7530	48.0804	92.5000	11.2500	2

Appendix

Supplementary Table 6: Sine–Gordon conservative cell results. Reference-indeterminate events are retained as errors. Each cell has 240 inputs and 6,720 pair opportunities.

Method	Strict pair	Exact top-1	Mean regret	Selected checkpoint
Propagated physics	$297/297$	$99/99$	0	Autoregressive
Raw residual	$0/297$	$0/99$	0.282935	One-step
Full-trajectory-medoid centrality	$198/297$	$0/99$	0.141394	Push-forward-20

Appendix

Supplementary Table 7: Descriptive results for the fixed public PDEBench U-Net library on inputs 0901–0999. Each input contributes three strict pair comparisons. No population interval is attached because candidate ordering and method correctness are constant across inputs. Mean regret is the unnormalised excess terminal task loss over the best checkpoint, averaged across inputs.

$B$	Propagated physics				Raw residual
	NRegret	Exact	Tol.	Top-2	NRegret	Exact	Tol.	Top-2
3	0.1269	37	40	46	0.2660	30	32	33
6	0.1335	35	39	42	0.2506	27	30	31
12	0.0481	46	50	54	0.2148	29	33	37
24	0.0351	49	52	55	0.1866	31	37	39

Appendix

Supplementary Table 8: Route-level results across nested selection-panel sizes. The main comparison uses

B=24

; the full curve shows the observed panel-size dependence.

Method	Non-tie pairwise	Tolerant top-1
Propagated ∗	$5{,}007/5{,}031=99.5230\%$	$190/192=98.9583\%$
Candidate-specific	$5{,}010/5{,}031=99.5826\%$ [99.3993, 99.7616]	$190/192=98.9583\%$

Appendix

Supplementary Table 9: Correction control on the deterministic subset. For the propagated row ( ∗ ), the pair entry is pair-cross-fit whereas the top-1 entry uses the ordinary shared response; the two columns therefore have different propagated constructions. The candidate-specific pair interval is a 95% input-cluster bootstrap interval.

Method	Non-tie pairwise	Tolerant top-1
Shared propagated	$21{,}378/21{,}458=99.6272\%$	$189/192=98.4375\%$
Candidate-specific	$21{,}370/21{,}458=99.5899\%$	$190/192=98.9583\%$
Raw residual	$14{,}235/21{,}458=66.3389\%$	$67/192=34.8958\%$

Appendix

Supplementary Table 10: Pooled-16 accuracy control. Pairwise accuracy conditions on hidden-reference non-ties; top-1 uses the

10^{-6}

tolerant winner set. The raw-residual row uses the same candidates and inputs.

Category	Count
No flip	37,782
Diagnostic and truth flip	15,534
Diagnostic-only flip	256
Truth-only flip	188

Appendix

Supplementary Table 11: Checkpoint-re-inference flip decomposition over 53,760 unordered pair opportunities in the primary two-PDE study.

	Burgers		Reaction–diffusion
$N$	S	C	S	C
2	0.653	1.293	0.0472	0.0930
4	0.551	2.164	0.0475	0.1887
8	0.540	4.337	0.0483	0.3764
16	0.541	8.664	0.0486	0.7455

Appendix

Supplementary Table 12: Median measured physics-diagnostic component time in seconds for the pooled-library scaling study. S denotes one shared solve and C one solve per candidate. These measurements are separate from the eight-candidate accuracy control and use fewer inputs than the pooled-16 accuracy control.

Supplementary Figure 2: Complete within-cell pairwise contrasts on matched candidates and inputs. Panels show 16 additional-library cells under strict pairs, eight primary cells under reference non-ties, and eight Sine–Gordon cells under conservative strict pairs. B, RD and SG denote the three PDEs; ens. and rep. identify ensembles and library replicates. Shared inputs induce dependence, so no cellwise intervals are shown and the three population estimands remain separate.

Transition	Cells	Pairs	Strict flips	Tol. transitions	Top-1 changes
Burgers 192 $\to$ 384	4	26,874	0	0	0
RD 64 $\to$ 128	4	26,835	10	63	2
RD 128 $\to$ 256	4	26,818	0	25	0
RD 256 $\to$ 512, new inputs	4	26,793	0	6	0

Appendix

Supplementary Table 13: Fixed-candidate grid transitions. The pair column counts comparisons that are non-ties on at least one endpoint grid; pairs tied on both grids are excluded from strict-flip rates. Strict flips compare positive/negative ordering, whereas tolerance transitions also include movement into or out of the pre-specified tie region.

Population	Group	Pairs	Pair coverage	Top-1 coverage	Max closure error
Earlier	FNO	26,880	98.1734%	93.8542%	$4.72\times 10^{-16}$
Earlier	CNO	26,880	32.7195%	6.0417%	$9.16\times 10^{-16}$
Checkpoint re-inference	FNO	26,880	98.0394%	93.4375%	$9.02\times 10^{-16}$
Checkpoint re-inference	CNO	26,880	32.1205%	5.6250%	$9.02\times 10^{-16}$

Appendix

Supplementary Table 14: Exact cross-resolution diagnostic stability. Coverage is the fraction of fine-grid diagnostic decisions passing the exact stability screen; all uncovered decisions abstain. Agreement of every covered decision follows algebraically and does not measure hidden-reference ranking.

Supplementary Figure 3: Coverage of the exact cross-resolution rank-or-abstain stability screen in two distinct cross-resolution populations. Each contains 26,880 pair decisions and 960 top-1 cases per architecture. Blue decisions pass the screen and grey decisions abstain. Coverage is high for FNO and low for CNO, especially for top-1. The screen measures diagnostic stability between two computed grids.

Quantity	Result
Library–input cases	24
Candidate pairs	672
Pair inequalities satisfied	672/672
Strict top-1 conditions satisfied	24/24
Tolerant top-1 conditions satisfied	24/24
Largest bound/proxy-margin ratio	0.09494

Appendix

Supplementary Table 15: Float64 evaluation of the reaction–diffusion sufficient conditions. A failed sufficient condition returns abstention.

Comparison	Result
State bounds exceeding approximate-zero difference	24/24
State bound/observed-error ratio	1.000018–1.000211
Pair bounds exceeding approximate-zero drift	672/672
Disagreements among selected pair signs	0/672
Disagreements among strict top-1 selections	0/24
Disagreements among tolerant top-1 selections	0/24

Appendix

Supplementary Table 16: Comparison with a high-accuracy numerical approximation to the same-grid operator zero.

Metric	Theory scope	Fixed pair / top-1	Matched pair / top-1 / order
Terminal squared $L^{2}$	Exact squared-Hilbert	297/297; 99/99	297/297; 99/99; 99/99
Trajectory squared $L^{2}$	Exact squared-Hilbert	297/297; 99/99	297/297; 99/99; 99/99
RMSE	Nonlinear $L^{2}$ reduction	297/297; 99/99	297/297; 99/99; 99/99
Normalized RMSE plug-in	Truth-dependent plug-in	296/297; 99/99	297/297; 99/99; 99/99
Conservation error	Linear summary; nonlinear reduction	297/297; 99/99	297/297; 99/99; 99/99
Maximum error	$L^{\infty}$ ; outside theorem	297/297; 99/99	293/297; 99/99; 95/99

Appendix

Supplementary Table 17: Secondary task transfer on the fixed public PDEBench library. Pair and top-1 counts use three candidates and 99 inputs. “Fixed” reuses the original terminal squared-

L^{2}

score; “matched” reuses the same corrected trajectory but changes the candidate-to-proxy distance. The ten metrics are correlated objectives, not independent replications.

Supplementary Figure 4: Partial-correction path. Panels a–c normalize each library’s median error by its value at

\alpha=0

; Sine–Gordon points are the medians of eight library-specific medians and whiskers span their range, while PDEBench contains one public library. Comparison error is the orthogonal projection of truth–proxy error onto the trajectory candidate-difference subspace. Panel d aggregates exact stored pair counts: one terminal squared-

L^{2}

task across eight Sine–Gordon libraries (53,760 pairs) and ten correlated objectives in PDEBench (2,970 pairs). The path reuses one correction.

Supplementary Figure 5: Comparison and reconstruction regimes. a , Deterministic PDEBench terminal sample 0950 shows the reference, shared proxy and exact decomposition of their error. Only $e_{D}=P_{\mathcal{D}}^{M}(t-q)$ changes fixed squared-loss gaps along candidate directions $d_{ij}=y_{i}-y_{j}$ ; maps show field 0, while the projection and norms use both task fields. b , Paired within-population contraction factors show preferential improvement in the decision subspace for PDEBench (9.86-fold full-state versus 16.1-fold decision-direction contraction) and for the two-model flow study (2.95-fold versus 22.2-fold). In the operator-mismatched flow population, comparison was correct in 10/10 cases whereas direct use improved on the best candidate in 4/10. PDEBench uses one public library and 99 inputs; flow factors are medians of ten casewise ratios. Absolute error scales are not compared across PDEs.

Method	Non-tie pairwise	Exact top-1	Tolerant top-1
Raw product residual	25,553/42,761 (59.758%)	451/1,920 (23.490%)	780/1,920 (40.625%)
Common-energy variational	35,371/42,761 (82.718%)	1,111/1,920 (57.865%)	1,430/1,920 (74.479%)
Candidate-adapted $L^{2}$ majorant	35,369/42,761 (82.713%)	1,111/1,920 (57.865%)	1,430/1,920 (74.479%)
Shared corrected proxy	42,761/42,761 (100%)	1,917/1,920 (99.844%)	1,920/1,920 (100%)

Appendix

Supplementary Table 18: Candidate-wise residual estimators on the reaction–diffusion population. Pairwise accuracy conditions on 42,761 hidden-reference non-ties; top-1 denominators contain all 1,920 cases. Inverse actions count the linear inverse actions used by the diagnostic after candidate outputs are available. Times are median post-inference components, not end-to-end latencies.

Quantity	Improved/correct	Median factor
Full trajectory error	$10/10$	2.951
Candidate-difference error	$10/10$	22.238
Perpendicular error	$4/10$	0.894
Operator defect	$10/10$	32.767
Common-state top-1	$6/10$	—
Corrected-proxy top-1	$10/10$	—

Appendix

Supplementary Table 19: Two-dimensional flow comparison and reconstruction. Improvement factors are common-state error divided by corrected-proxy error at

\alpha=1

. Ranking contains one binary pair per input.

Readout	Valid	Pairwise	Top-1	Median error factor
Common state ( $\alpha=0$ )	$35/35$	$74.3\%$	$40.0\%$	1.00
Unit correction ( $\alpha=1$ )	$22/35$	$61.9\%$	$62.9\%$	2.47
Truth-free backtracking	$35/35$	$94.3\%$	$91.4\%$	2.04

Appendix

Supplementary Table 20: One-dimensional shock admissibility. Error factors are common-state full error divided by selected-proxy full error. The

\alpha=1

row counts every inadmissible unit step as an incorrect decision.

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