Threshold Compatibility and Support-Selection Diagnostics in Neural Discovery of Partial Differential Equations

Zeen Song

cs.LGOct 27, 2026

Threshold Compatibility and Support-Selection Diagnostics in Neural Discovery of Partial Differential Equations

Authors: Zeen Song

Organizations: School of Mathematics, Statistics and Mechanics Beijing University of Technology, Beijing 100124, China

Abstract

Accurate regression need not yield the correct equation after support selection. For a median/sample-standard-deviation threshold in neural PDE discovery, we derive an exact two-signal retention boundary and a complete-vector perturbation bound. Exact advection-diffusion libraries lose diffusion despite coefficient recovery to numerical precision. True-support refits of forty saved sparse states reduce diffusion error without restoring all derivative accuracy. A fixed sequential threshold ridge comparator recovers exact support in sixty ideal controls but selects extra terms in every learned state. All-fold correct-support candidates are rejected by conditional validation; post hoc energy pooling preserves their higher minimum loss in all sixty states. Supplied-support estimates outperform unchanged selected equations in all 240 physical-residual comparisons on four unused analytic trajectories. With coefficients, observation points and weights fixed, exact replacement of the learned library and target reverses their learned-row ordering in 59/60 states. Weighting both losses also fails the specified Burgers diffusion-benefit criterion at every tested budget. These finite diagnostics distinguish threshold incompatibility, refit damage and score rejection; they establish neither automatic correction nor general recovery or predictive stability.

Figures & tables

Fig. 1: Exact-data threshold qualification and saved phase endpoints. (a) All original-spectrum exact observed designs fall below the two-signal boundary; identity and IPW fits recover the same physical coefficients. (b) All twenty sparse-phase diffusion estimates before thresholding; the dashed line marks 0.1. (c,d) Final-refit/sparse observed derivative relative-error ratios after float64 reevaluation of saved float32 parameters; values above one indicate deterioration. These post hoc diagnostics change neither training nor support. Seed order is not time; all outcomes are retained

Fig. 2: True-support intervention and remaining derivative costs. Top: all forty oracle diffusion errors, compared with the original-support value 0.1. Bottom: oracle/sparse observed derivative relative-error ratios; values above one indicate deterioration. Circles denote

u_{xx}

and triangles

u_{xxx}

. Mask indices 1–10 map to seeds 17010–17019 in B1 and 17110–17119 in B2. Both rows use logarithmic vertical axes. Supplied support is an oracle; retained coefficients are learned. All pairs remain, including adverse outcomes and precision flags; shared-condition sampling is described in Section 3.1

Fig. 3: Frozen selector comparison. (a) Exact-support fractions with denominators. Old denotes B1/B2; new denotes B3, which changes spectrum and random conditions together. Each ideal design uses two weights: the two new-spectrum median/SD successes are one mask under both measures. (b) All ten primary learned fits per placement/block, with mean marks and labels; each retains both true terms. Filled markers denote B3; circles/triangles denote

H_{\mathrm{data}}

H_{\mathrm{both}}

. Hatched/solid bars distinguish selectors. Counts reuse shared conditions (Section 3.1); validation is conditional on the fitted representation (Section 3.3)

Fig. 4: Correct-support candidates rejected by the conditional score. (a) All sixty learned states; circles/triangles denote

H_{\mathrm{data}}

H_{\mathrm{both}}

, filled symbols denote B3, and offsets order masks. The logarithmic vertical axis shows positive excess

\Delta/C

: all candidates lie above the acceptance ceiling. (b) The first B1 data-only mask, chosen by identifier, shows minimum loss, the one-SE ceiling (including

10^{-12}

slack), and the best threshold correct in all five training folds. Truth labels are diagnostic; unselected full-data refits were not computed. This heuristic gives no confidence interval

Fig. 5: Ranking reversal under external physical scoring. Each panel includes all sixty states grouped by block/placement, with within-cell offsets in mask order. Circles/triangles denote

H_{\mathrm{data}}

H_{\mathrm{both}}

; filled markers identify B3. Both panels use logarithmic vertical axes for selected/supplied-support squared-residual ratios; dashed lines mark equality. (a) Original learned rows and residual measure. (b) Equal mean of four unused-trajectory scores under uniform physical measure. Oracle support is supplied, but coefficients use only original learned rows. External data tune no model; these are correlated comparisons (Section 3.1), not independent recovery trials

Block	Placement	Mean $e_{D}$	Median $e_{D}$	$e_{D}$ improved	$u_{xxx}$ error worsened
1	H_data	0.0022844	0.00197657	10/10	8/10
1	H_both	0.00149751	0.0014515	10/10	8/10
2	H_data	0.00167691	0.0012384	10/10	8/10
2	H_both	0.00179883	0.00182292	10/10	9/10

Table S1: Matched true-support intervention in two separate condition blocks.

Field	Max diff. / truth	Max diff. / learned	Data mean error	Both mean error	Wins
$u$	4.32749e-07	4.32777e-07	0.0723199	0.065128	8/10
$u_{t}$	9.74597e-06	7.36233e-06	0.710358	0.590171	7/10
$u_{x}$	1.03922e-05	6.90312e-06	0.569706	0.449637	6/10
$u_{xx}$	0.00180266	1.10373e-05	30.8901	18.1893	6/10
$u_{xxx}$	0.682347	2.50027e-05	3867.31	1160.75	6/10

Table S2: Float64 reevaluation of saved parameters: both precision scales.

Block	Placement	Rejected	Min. excess	Max. excess	Min. candidates
1	H_data	10/10	3.02195	56.4414	1
1	H_both	10/10	3.33454	26.6163	4
2	H_data	10/10	1.17593	23.7894	4
2	H_both	10/10	1.03995	28.7687	4
3	H_data	10/10	5.29253	64.2488	5
3	H_both	10/10	6.40998	47.6414	7

Table S3: Conditional score rejection of all-fold correct-support candidates.

Block / placement	Reversals	Learned: S; O	Exact: S; O	External: S; O
B1 / data	9/10	6.98e-05; 0.00105	0.000885; 5.97e-05	0.00415; 7.71e-05
B1 / both	10/10	5.75e-05; 0.00114	0.00104; 4.85e-05	0.00341; 5.91e-05
B2 / data	10/10	4.96e-05; 0.000565	0.000535; 7.98e-05	0.00256; 0.000136
B2 / both	10/10	3.82e-05; 0.000646	0.000647; 0.000144	0.00369; 0.000156
B3 / data	10/10	2.71e-05; 0.000394	0.000527; 0.000148	0.00423; 0.000176
B3 / both	10/10	2e-05; 0.000431	0.000551; 0.00011	0.00373; 0.000114

Table S4: Fixed-equation same-point diagnostic. Entries are medians of normalized squared-residual scores for selected (S) and supplied-support (O), computed separately across ten states. Each S; O entry gives two scores, not their ratio. External scores first average four trajectory risks within each state. Reversals compare LL with EE. All six cells have zero ties and invalid states.

Fig. S1: Post hoc fixed-equation diagnostic. (a) All noisy observations and learned field values for the first B1 $H_{\mathrm{data}}$ state, chosen by identifier, against exact values at those rows; these are field measurements, not derivative targets. (b) All sixty selected/supplied-support ratios on learned LL versus exact EE rows, with coordinates/weights fixed. (c) Mixed-source LE versus EL. (d) EE versus the equal mean of four unused-trajectory scores. In (b)–(d), axes are logarithmic, dotted lines mark equality, and ratios below one favor selected. Colors denote blocks; circles/triangles denote $H_{\mathrm{data}}$ / $H_{\mathrm{both}}$ . All states remain, including the labeled B1 seed-17018 exception. CSVs retain all three estimators and absolute scores; these residuals are not forward prediction errors

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