FIDE: Fiber-Informed Differentiable Experimental Design for Reference-Relative Law Dynamics

Pietro Zanotta; Ján Drgoňa

cs.LGOct 29, 2026

FIDE: Fiber-Informed Differentiable Experimental Design for Reference-Relative Law Dynamics

Authors: Pietro Zanotta, Ján Drgoňa

Organizations: Department of Civil and Systems Engineering Johns Hopkins University Baltimore, MD, USA

Abstract

Aggregate measurements generally leave an evolving population law underdetermined, so different measurement systems can be equally adequate for a scientific task while constraining different aspects of the law. We study how to resolve this residual design freedom in baseline-preserving settings, where an existing dynamical model remains the working baseline outside quantities explicitly valued by the task. Fiber-Informed Differentiable Experimental Design (FIDE) completes each candidate's aggregate constraints by information projection relative to a common frozen reference, then ranks scientifically adequate designs by the dynamical revision required to realize the resulting completed paths. The resulting Full action is the minimum mean-squared velocity correction needed to realize a completed path; equivalently, its square root is the sharp reference-relative expectation-rate discrepancy over a gradient-normalized Sobolev class. This defines a conditional minimum-revision principle for baseline-preserving design rather than a generic truth-recovery objective. Experiments show that this preference changes sensor selection, can help or harm withheld downstream predictions depending on baseline reliability, can be used prospectively from aggregate predictions, and remains computationally usable for a $32$ -dimensional configuration law.

Figures & tables

Selector	Selection risk increase (%)	Held-out Full (cross-eval.)	Truth-relative hidden-law $\mathrm{SW}_{2}^{2}$
Law	$0$	$29.4178$	$0.096460$
Tangent	$0.491$	$37.3317$	$0.094144$
Residual $L^{2}$	$0.441$	$36.0548$	$0.094173$
Forward KL	$0.303$	$33.0647$	$0.094352$
Static $W_{2}$	$0.294$	$32.9924$	$0.094357$
Full	$0.604$	$19.5359$	$0.102205$

Table 1: Criterion-specific Gaussian designs at the

1\%

allowance, selected under the same scientific-risk restriction and cross-evaluated using held-out Full action and truth-relative hidden-law squared sliced-Wasserstein distance. The two diagnostics induce different orderings; values are comparable within columns but not across columns.

Figure 1: Downstream consequences of baseline-preserving selection. A: At signal amplitude

\epsilon=0.10

and occupancy localization scale

\kappa=4

, we evaluate truth-relative local-occupancy RMSE on

16

smooth observables that are withheld from both the scientific risk and all secondary design criteria. Scientific risk is identical across the five candidate geometries. Full and static

W_{2}

select

\alpha=0

, while Tangent, residual

L^{2}

, and forward KL select

\alpha=0.75

. The Law bar is the expected RMSE under uniform tie-breaking over its five risk-minimizing geometries; open circles show the corresponding geometry-specific errors. B: The selected geometries are then frozen while a true departure of strength

\gamma

is introduced in the task-unprotected coordinate. The ordinate is occupancy RMSE minus the expected Law RMSE, so negative values indicate improvement over Law and positive values indicate harm. The Full/static-

W_{2}

geometry changes from lower to higher error than Law as misspecification grows, while the Tangent/residual-

L^{2}

/forward-KL geometry reverses in the opposite direction; both crossings occur between

\gamma=0.5

and

0.6

in this representative setting.

Panel of Figure 2 — Figure 2: Empirical overview. Left: Held-out scientific risk and Full action across the tested risk allowances in the prospective double-gyre study. Right: Interacting-skyrmion setup, where aggregate sensor views constrain a $32$ -dimensional configuration law without determining it.

Appendix figures & tables22 assets

Supplementary material from the paper’s appendix.

Appendix

Allowance	Full holdout risk	Risk increase (%)	Excess over allowance (pp)
$0.5\%$	0.038777	0.4404	$-0.0596$
$1\%$	0.039016	1.0598	$+0.0598$
$2\%$	0.039289	1.7686	$-0.2314$
$3\%$	0.039531	2.3945	$-0.6055$
$4\%$	0.039950	3.4794	$-0.5206$
$5\%$	0.040538	5.0023	$+0.0023$

Appendix

Table 2: Held-out scientific-risk transfer for the retrospective frontier. The holdout Law risk is

0.038607

. Negative excess indicates a point estimate below the nominal allowance. Values are descriptive because the retained summaries do not contain paired per-trial scientific losses needed for matching simultaneous risk-feasibility inference.

Allowance	Law	Tangent	GlobalReaction	Full	Tangent	GlobalReaction	Full	Full interval (%)
	Mean Full action				Reduction vs. Law (%)
$1\%$	0.77523	0.76973	0.76652	0.74994	0.71	1.12	3.26	$[1.50,\ 5.03]$
$2\%$	0.77523	0.76502	0.76205	0.73475	1.32	1.70	5.22	$[3.46,\ 6.99]$
$3\%$	0.77523	0.75927	0.75660	0.72193	2.06	2.40	6.88	$[5.11,\ 8.64]$
$4\%$	0.77523	0.75296	0.75043	0.71029	2.87	3.20	8.38	$[6.61,\ 10.14]$
$5\%$	0.77523	0.74635	0.74397	0.69936	3.72	4.03	9.79	$[8.02,\ 11.55]$

Appendix

Table 3: Fresh finite-measure comparison at

\kappa=1

. Each frozen selector is evaluated using the same numerical Full-action functional on

128

paired validation trials. Reductions are computed relative to the Law action at the same

\kappa

. Tangent and GlobalReaction are descriptive structural comparators; the final column reports the paired interval for the prespecified Full-versus-Law action reduction.

Frequency $n$	Moment lower bound	Static I-projection	Restricted dynamic	Reduction (%)
$1$	$3.295850$	$6.521546$	$5.890663$	$9.67$
$2$	$13.183401$	$26.086185$	$20.867915$	$20.00$
$4$	$52.733603$	$104.344741$	$77.063457$	$26.15$

Appendix

Table 4: Effect of the completion rule at fixed measured moments. The static column evaluates Full on the independent per-time information projections in equation 254 . The restricted dynamic column optimizes one additional unmeasured Fourier coordinate over time while preserving exactly the same trajectory

c_{n}(t)

. “Reduction” is

100(\mathcal{A}_{\mathrm{IP}}-\mathcal{A}_{\mathrm{dyn}})/\mathcal{A}_{\mathrm{IP}}

. The moment lower bound is the time integral of equation 258 and is not asserted to be attainable.

Figure 3: Candidate-family occupancy error under baseline correctness. At

\epsilon=0.10

and

\gamma=0

, truth-relative local-occupancy RMSE is shown for all five frozen geometries and all three prespecified localization scales

\kappa\in\{2,4,8\}

. Scientific risk is identical across the five designs. Full and static

W_{2}

select

\alpha=0

, Tangent, residual

L^{2}

, and forward KL select

\alpha=0.75

, and Law is indifferent among all five geometries. The occupancy loss is withheld from design and is not used by any selector.

Figure 4: Candidate-family occupancy error under misspecification. At

\epsilon=0.10

, truth-relative local-occupancy RMSE is shown across all five frozen geometries as the unprotected departure strength

\gamma

increases. Blue denotes

\alpha=0

, orange denotes

\alpha=1

, and gray denotes

\alpha\in\{0.25,0.5,0.75\}

. The baseline-preserving

\alpha=0

geometry is best near

\gamma=0

but eventually becomes worse than more departure-sensitive geometries. Dashed vertical lines mark the independently predicted small-amplitude

\alpha=0

versus

\alpha=1

occupancy crossovers from equation 268 :

\gamma_{\star}=0.5632

0.5314

, and

0.4482

for

\kappa=2,4,8

, respectively. No geometry is reselected as

\gamma

varies; Figure 1 gives the selector-level comparison emphasized in the main text.

Figure 5: Do the held-out designs satisfy their scientific-risk allowances under sampling uncertainty? Descriptive held-out relative risk increase divided by the nominal allowance for the Gaussian-mixture, prospective double-gyre, and skyrmion validation studies. Gaussian and prospective double-gyre use their respective

64

-trial independent holdouts, while skyrmion uses

32

fresh paired validation trials. The dashed line at one marks equality with the nominal allowance. Filled markers denote operating points whose simultaneous one-sided

95\%

upper bound for

\mathbb{E}[Z_{i}(p)]

is below zero; open markers are unresolved. The plotted ratio is descriptive: inferential status is determined by the paired excess-loss statistic in equation 269 , not by the ratio alone.

Study	Allowance	Risk increase (%)	Simultaneous upper	Conclusion
Gaussian	$0.5\%$	0.516	$1.61\times 10^{-4}$	Unresolved
Gaussian	$1\%$	0.656	$-7.58\times 10^{-5}$	Below cap
Gaussian	$2\%$	0.656	$-7.33\times 10^{-4}$	Below cap
Gaussian	$3\%$	0.656	$-1.39\times 10^{-3}$	Below cap
Gaussian	$4\%$	0.656	$-2.05\times 10^{-3}$	Below cap
Gaussian	$5\%$	0.656	$-2.70\times 10^{-3}$	Below cap

Appendix

Table 5: Held-out risk-feasibility inference. Relative risk increase is

100(\overline{\ell}_{\mathrm{Full}}/\overline{\ell}_{\mathrm{Law}}-1)

. The fourth column gives the simultaneous one-sided

95\%

upper bound for the paired excess-loss statistic

Z_{i}(p)

within the corresponding benchmark family. A point is labeled below cap only when this bound is negative.

Criterion	Scientific risk	Own objective	Full action
Law	0.066186	0.066186	30.2016
Tangent	0.066511	2.10920	38.0653
Full	0.066586	19.71935	19.7193
Residual $L^{2}$	0.066478	52.50258	36.5877
Forward KL	0.066387	0.235235	33.1209
Static $W_{2}$	0.066387	0.134317	33.1209

Appendix

Table 6: Gaussian criterion-specific selections on a shared promoted candidate pool. Each row gives the minimizer of one criterion over the same set of

17

scientifically feasible designs. The “own objective” column is reported on each criterion’s native scale and is therefore not comparable across rows; the final column cross-evaluates every selected design using the common Full-action functional. Forward KL and static

W_{2}

select the same candidate, whereas Law, Tangent, Full, and residual

L^{2}

select distinct candidates.

Panel of Figure 6 — Figure 6: Gaussian criterion-specific selections on the common promoted candidate pool. Gray points denote the remaining promoted candidates and labeled markers denote the six criterion-specific minimizers obtained by reselecting each objective over the shared pool. Left: scientific risks occupy a narrow range while Full action varies substantially across candidate designs; the dashed line marks the frozen $1\%$ operational risk ceiling. Forward KL and static $W_{2}$ select the same candidate and therefore overlap. Right: the corresponding sensor geometries occupy distinct regions of the design space, again with the forward-KL and static- $W_{2}$ selections coincident.

Figure 7: Controlled Gaussian reference misspecification on the fixed

1\%

admissible pool. Left: covariance-normalized detectability of the discrepancy between the analytic target and the perturbed reference; larger values mean that the selected sensors expose the reference error more strongly. Right: truth-relative hidden-law sliced-Wasserstein error of the corresponding information-projected completion; smaller values mean closer agreement with the analytic intermediate law. The two diagnostics need not induce the same ordering.

Evaluator	Law action	Full action	Reduction (%)	Paired Full $-$ Law interval
$128\times 64$	1.50697	1.16113	22.95	$[-0.388,-0.307]$
$192\times 96$	1.50973	1.16290	22.97	$[-0.389,-0.308]$
$256\times 128$	1.51070	1.16352	22.98	$[-0.389,-0.308]$
Bandwidth $0.8\times$	1.63096	1.24506	23.66	$[-0.430,-0.344]$
Bandwidth $1.2\times$	1.38646	1.08121	22.02	$[-0.344,-0.269]$
PCG tolerance $10^{-10}$	1.50697	1.16113	22.95	$[-0.388,-0.307]$

Appendix

Table 7: Is the prospective Full-action reduction stable to the numerical evaluator? Mean numerical Full actions for the frozen Law and

5\%

Full geometries under spatial refinement, density–source bandwidth perturbation, and solver-tolerance perturbation. The reported paired intervals are for the Full-minus-Law action difference. No setting produces a numerical failure, and the action reduction remains between

22.02\%

and

23.66\%

Frequency $k$	Relative error at $N=512$	Empirical spatial order
$1$	$1.156\times 10^{-5}$	$2.00004$
$2$	$4.623\times 10^{-5}$	$2.00015$
$4$	$1.850\times 10^{-4}$	$2.00061$

Appendix

Table 8: Manufactured-solution convergence of the periodic Full evaluator. Relative errors compare the discrete time-integrated action with an independently quadrature-evaluated continuum benchmark. The empirical order is computed from the final two spatial refinements. At fixed

N

, the larger relative error at higher frequency is consistent with the expected dependence of a second-order spatial discretization on the dimensionless resolution

kh

Predictor condition	Feasible	Risk-rank $\rho$	Held-out risk change (%)	Held-out action reduction (%)
Unperturbed	246	1.000	2.030	24.324
Mean bias, $0.25$ local SD	16	$-0.294$	$-0.141$	0.327
Mean bias, $0.50$ local SD	1	$-0.355$	0.000	0.000
Sampling covariance $\times 0.75$	246	1.000	2.030	24.324
Sampling covariance $\times 1.25$	246	1.000	2.279	23.133
QoI-target bias, $0.25$ scale	224	0.800	$-1.749$	26.747

Appendix

Table 9: Sensitivity of prospective selection to the aggregate predictive interface. All conditions rank the same fixed pool of

541

candidate geometries. “Feasible” is the number of candidates satisfying the original numerical/support requirements and the condition-specific

5\%

Law-relative scientific-risk band. Risk-rank

\rho

is the Spearman correlation with the unperturbed condition over the

406

statically admissible geometries. Held-out quantities compare the Full and Law designs selected under the same predictor condition on the common post-freeze validation experiment.

Figure 8: Predictor perturbations reshape the prospective design-time risk landscape in different ways. Each panel shows the same fixed pool of

541

candidate sensor geometries, with minimum pairwise sensor separation on the horizontal axis and scientific-risk margin

R_{\mathrm{sel}}/R_{\mathrm{Law}}^{\mathcal{C}}-1

relative to the condition-specific candidate-pool Law anchor on the vertical axis. The vertical dashed line marks the minimum allowed separation

0.24

, and the horizontal dashed line marks the

5\%

Law-relative risk allowance. Highlighted markers denote the Law and Full designs selected from the fixed pool. The tested covariance rescalings leave the landscape nearly unchanged, whereas sensor-mean perturbations substantially reorganize the relative risk ordering and move an increasing fraction of the below-cap candidates into the separation-invalid region. QoI-target perturbations alter the scientific-risk landscape without changing the underlying reconstructed trajectories. Because each panel is normalized by its own Law anchor, vertical positions describe within-condition relative risk rather than absolute risk across perturbations. The two dashed thresholds alone do not define final feasibility, which also requires the condition-dependent numerical and support criteria.

Risk allowance	Base	Fourier-400	Fourier-560	Interaction-416
1%	3.51	3.51	3.52	3.53
2%	6.91	6.92	6.93	6.95
3%	9.38	9.40	9.41	9.43
4%	9.38	9.40	9.41	9.43
5%	12.22	12.25	12.28	12.31

Appendix

Table 10: Does the action reduction persist under a richer potential space? Percentage reductions in mean fitted action relative to Law under each evaluator. All 32 trials are retained, including failed-certification cases; the 3%/4% row represents one shared geometry. Fourier-400 and Fourier-560 refine the Fourier dictionary, while Interaction-416 adds interaction features. Fourier-560 has six held-out energy-check failures, and ESS support failures remain under every evaluator.

Design	Base ( $K=280$ )	Fourier-560 ( $K=560$ )	Increase (%)
Law	$0.024433$	$0.024581$	$0.61$
Full ( $5\%$ allowance)	$0.021449$	$0.021564$	$0.54$

Appendix

Table 11: Absolute fitted action under nested Galerkin enrichment. Mean fitted actions over the same

32

validation trials. In agreement with the monotonicity of the underlying restricted variational problem in Proposition 6 , enlarging the Fourier potential space raises the fitted action for both frozen designs. The corresponding Full-versus-Law reduction changes only from

12.22\%

12.28\%

Allowance	$\kappa=0.25$	$\kappa=1$	$\kappa=4$
$1\%$	3.02	3.26	2.97
$2\%$	4.79	5.22	4.95
$3\%$	6.27	6.88	6.59
$4\%$	7.61	8.38	8.09
$5\%$	8.85	9.79	9.49

Appendix

Table 12: Sensitivity of the Full-versus-Law contrast to the transport–reaction metric. Entries are percentage reductions in mean Full action relative to Law, computed separately within each value of

\kappa

. Because

\kappa

changes the correction metric, absolute action values are not compared across columns.

Figure 9: The within-metric Full-versus-Law action reduction persists across reaction penalties. Each curve reports the percentage reduction in mean Full action relative to Law at a fixed

\kappa

. Scientific observations are shared across the sweep, while

\kappa

changes only the transport–reaction correction metric; vertical comparisons of absolute action across

\kappa

are therefore not meaningful.

$\kappa$	Move / Full (%)	Reaction / Full (%)	Hidden / Full (%)	Full vs. Tangent (%)	Full vs. GlobalReaction (%)
0.25	21.99–22.65	77.35–78.01	43.15–43.80	1.95–6.12	2.23–6.28
1	23.97–24.21	75.79–76.03	64.58–65.51	2.57–6.30	2.16–6.00
4	38.79–39.33	60.67–61.21	73.51–74.62	2.27–5.62	1.76–5.20

Appendix

Table 13: Unbalanced mechanism summary across the

1\%

–

5\%

allowances. The first three quantities describe the Full-selected geometry and are reported as ranges over the five allowances. Move and Reaction form one decomposition of Full, while Tangent and Hidden form a separate orthogonal decomposition. The final two columns report the range of reductions in the common Full cross-score obtained by Full relative to the Tangent- and GlobalReaction-selected geometries.

Figure 10: Mechanism decomposition of the unbalanced Full correction. Left: fraction of Full action lying in the Tangent-invisible Hidden component. Right: fraction of Full action allocated to reaction rather than spatial movement. Fractions are computed within each fixed transport–reaction metric; the figure does not compare absolute action values across

\kappa

Study	Information available during selection	Information introduced after design freeze
Gaussian mixture	Analytic target law, target-derived moments and covariances, noisy finite-sample observations, population-risk screen, and one frozen endpoint-trained reference.	Fresh target draws, particle sampling, and detector noise from the same analytic family. The reference is unchanged.
Retrospective double-gyre	Frozen microscopic target simulation bank and three endpoint-trained references; candidate observations and scientific-risk quantities are evaluated directly from the target bank.	Fresh particle resampling and detector noise, conditional on the same target bank and reference ensemble.
Prospective double-gyre	Endpoint ensembles, one frozen endpoint-trained reference, and fixed aggregate sensor-response means, cross-sensor covariances, and scientific-QoI predictions. These aggregate fields are precomputed from a separate microscopic simulation bank and its intermediate particles are not exposed to the optimizer.	A separately seeded evaluation reference trained from the same endpoint dataset, together with new microscopic validation trajectories, sampling, and detector noise for action and scientific-risk validation.
Skyrmions	A fixed development set of interacting configurations, one frozen endpoint-trained permutation-equivariant reference, aggregate sensor observations, and the nine permutation-invariant scientific features used to evaluate development risk.	A separate set of $32$ fresh paired measurement trials for scientific-risk and action validation. The reference model and selected sensor geometries remain frozen. Richer potential dictionaries, leave-one-pair-out analyses, and larger reference banks are introduced only as post-selection numerical sensitivity checks.
Unbalanced Gaussian	A fixed analytical reference and finite-measure simulator, together with development observations and simulator truth used for scientific-risk evaluation.	A fresh bank of $128$ paired trials with new physical nuisances, particle observations, detector noise, and log-mass noise. The same frozen bank is reused across the $\kappa$ sensitivity analysis.

Appendix

Table 14: Information boundaries of the reported studies. “After design freeze” denotes information unavailable when optimizing the reported sensor geometries. Fresh evaluation is conditional on the quantities explicitly retained in the corresponding row.

Component	Configuration
Operating system	Ubuntu 24.04.3 LTS under WSL2
Linux kernel	6.6.87.2-microsoft-standard-WSL2
Processor	Intel Core Ultra 9 275HX (24 cores, up to 5.4 GHz)
GPU	NVIDIA GeForce RTX 5090 Laptop GPU
GPU memory	24 GB GDDR7
NVIDIA driver	581.57

Appendix

Table 15: Hardware and software environment used for the experiments.

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