Finding and using interpretable latents in a neutrino foundation model with sparse autoencoders

Raphaël Bonnet-Guerrini; Johann Ioannou-Nikolaides; Inar Timiryasov; Vincenzo Piuri

astro-ph.HEAug 26, 2026

Finding and using interpretable latents in a neutrino foundation model with sparse autoencoders

Authors: Raphaël Bonnet-Guerrini, Johann Ioannou-Nikolaides, Inar Timiryasov, Vincenzo Piuri

Organizations: Computer Science Department, Università degli Studi di Milano, Milano, Italy · INFN Sezione di Milano, Milano, Italy · Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

Abstract

We present a first application of sparse-autoencoder-based mechanistic interpretability to particle physics. Studying a neutrino foundation model pretrained on IceCube data and fine-tuned for direction reconstruction, we identify a validated atlas of physical concepts in the model representation, using a strict validation protocol consisting of held-out tests, matched nuisance controls, and replication across independent dictionary trainings. Causal interventions show that the direction head barely draws on this atlas. Motivated by this underused information, we train an uncertainty head on the same event-level representation to predict the model's angular reconstruction error. Unlike the direction head, it depends causally on quality and brightness features from the atlas. At $20\%$ selection efficiency, this interpretable estimator improves the median angular resolution from $20.2^\circ$ to $3.2^\circ$ . These results suggest that mechanistic interpretability can reveal learned latent physics encoded within a model's internal representation and help design downstream tasks that exploit it.

Figures & tables

Split name	Role	Fraction
sae_train	SAE weight training	$75\%$
sae_reconstruction_val	Reconstruction and fidelity validation	$10\%$
concept_dev	Candidate latent discovery	$7\%$
concept_test	Independent concept validation	$6\%$
causal_test	Independent causal interventions	$2\%$

Table 1: Disjoint event splits used for SAE training and downstream validation. Fractions refer to the nominal one-million-event partition.

Choice	Values tested	Selected	Basis
Expansion $m/d$	$\{2,4,8\}$	$4$	intrinsic diagnostics
BatchTopK budget $k$	$\{8,16,32\}$	$16$	fidelity–sparsity tradeoff
AuxK coefficient $\lambda_{\mathrm{aux}}$	$\{0,2^{-8},2^{-6},2^{-5},2^{-4},2^{-3},2^{-2}\}$	$2^{-5}$	zero dead latents
AuxK budget $k_{\mathrm{aux}}$	$\{64,256,512\}$	$512$	validation diagnostics
Dead-step threshold $\tau_{\mathrm{dead}}$	$\{2^{8},2^{9},2^{10},2^{12},2^{13}\}$	$2^{8}$	validation diagnostics

Table 2: Concept-independent SAE selection. We select the configuration using intrinsic reconstruction and sparsity diagnostics on sae_reconstruction_val set, before any concept validation or causal analysis.

Panel of Figure 1 — Figure 1: Concept-validation protocol. A candidate SAE read-out is first identified by association, then tested for held-out selectivity, and finally intervened on to establish whether a particular downstream head uses it. Robustness checks are applied across the validation chain. Exact numerical criteria are given in Table 5 .

Figure 2: Illustration of five events ordered by increasing activation of

z_{\mathrm{bc}}

on causal_test (

z=0.35

0.45

0.58

1.03

2.67

). Panels share camera, detector frame and charge scale.

Feature	Role	Validation / response	Min. control AUROC	Response analog
$z_{\mathrm{bc}}$	bright-clean events	$0.91$	$0.79$	$4/4$
$z_{\mathrm{bc}}^{\,\prime}$	clean events, secondary	$0.69$	$0.84$	$4/4$
$z_{\mathrm{aux}}$	auxiliary activity	$0.84$	$0.85$	$3/4$
$z_{\mathrm{bs}}$	brightness-associated support (dense)	charge monot. $+0.92$	non-selective	$4/4$
$z_{\mathrm{depth}}$	detector depth (layer 2)	$0.986$	$0.984$	layer bank

Table 3: Named sparse directions identified in the paper. The final column reports whether an analogous response appears across dictionary seeds; it does not report replication of the complete validation criteria. Section 3.3 gives the corresponding seed-by-seed verdicts.

Figure 3: Response profiles of the main SAE latents along the auxiliary fraction and the total charge, on concept_dev set events. Bands show the min-max spread across dictionary seeds.

Figure 4: Highest-activating event of each latent on causal_test . Markers are DOMs (charge summed per DOM), area grows with $\sqrt{\text{charge}}$ , color encodes hit time within the event, and filled versus hollow marks non-auxiliary versus auxiliary hits. Panels share axes and charge scale.

Figure 5: Dose response of the brightness-support latent

z_{\mathrm{bs}}

(latent

973

) on causal_test : this single coordinate is rescaled as

z\to s\,z

before decoding. In both panels the two curves are clean events (

f_{\mathrm{aux}}\leq 0.05

) split by brightness alone: bright (

Q_{\mathrm{tot}}\geq 160

p.e.) and dim (

Q_{\mathrm{tot}}\leq 122

p.e.). Left: the resulting change in the true angular error of the direction head. Right: the resulting shift

M_{1}

in the log-error predicted by the uncertainty head (Section 4.3 ).

Figure 6: Sensitivity structure of the direction and uncertainty heads. Left: normalized singular-value spectra of the aggregate head Jacobians on the concept_dev and concept_test splits. Right: absolute cosine similarity between each head’s leading sensitivity axis and selected SAE decoder directions, physics-probe directions, and the first two principal components. The dashed vertical line marks

|\cos|=0.14

	Direction head			Uncertainty head
Intervention	effect	verdict	seeds	$M_{1}$	verdict	seeds
remove $z_{\mathrm{bc}}$	$+0.04^{\circ}$	null	$4/4$	$+0.31$	causal	$4/4$
remove $z_{\mathrm{aux}}$	$+0.00^{\circ}$	null	$3/4$	$-0.059$	null, sign only	$3/4$
remove clean pair	$+0.03^{\circ}$	null	$4/4$	$+0.32$	causal	$4/4$
remove clean family	$-0.02^{\circ}$	null	$3/4$	$+0.78$	causal	$4/4$
remove clean core	$+0.02^{\circ}$	null	$3/4$	$+0.37$	causal	$4/4$

Table 4: Intervention effects on the final layer for the direction head and the uncertainty head. The “seeds” columns report replication of the corresponding verdict across the four independently trained SAE dictionaries. Direction-head effects are measured as shifts of angular error in degrees. Uncertainty-head effects are measured as shifts of predicted log error,

M_{1}

in Eq. ( 21 ).

Figure 7: Left: median angular error of the retained sample as a function of selection efficiency for detector-level observables, distributed representation read-outs, the sparse feature

z_{\mathrm{bc}}

, and the uncertainty head. The horizontal line gives the unselected resolution, and bands show bootstrap intervals. Right: median true angular error in bins of the error predicted by

U_{\mathrm{lin}}

and

U_{\mathrm{MLP}}

. The diagonal indicates perfect calibration, while the shaded regions show the within-bin spread of true angular errors.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 8: Held-out mean angular error of a linear CLS probe at each layer. Lower values are better; the frozen direction head is shown as a reference.

Stage	Fixed rule
Primary binary candidate	Discovery AUROC $\geq 0.75$ and survival of every evaluable matched-control scheme
Secondary read-out	Reported separately when it survives the nuisance controls but remains below the primary AUROC bar
Independent confirmation	Re-evaluate the fixed candidate on concept_test , without reselection or threshold tuning
Family membership	Auxiliary-monotonicity score $\mathrm{ams}\leq-0.85$ for the clean side or $\mathrm{ams}\geq+0.85$ for the auxiliary side
Direction-head causality	$\Delta_{+}\geq 0.05^{\circ}$ , $z(\Delta_{+})\geq 2$ , and $z(\Delta_{+}-\Delta_{-})\geq 2$
Uncertainty-head causality	$\|M_{1}\|\geq 0.02$ , $z(M_{1})\geq 2$ , and the pre-specified sign

Appendix

Table 5: Fixed discovery, validation, and intervention rules. Each row gives the condition used at that stage.

Latent	AUROC	Top- $1\%$ enrich.	Firing rate	Min. ctrl AUROC	Status
Positive class: clean-trigger events
$\mathbf{640}$	$\mathbf{0.913}$	$15.40\times$	$0.139$	$0.818$	primary
$1006$	$0.698$	$14.94\times$	$0.087$	$0.699$	secondary
$751$	$0.631$	$6.13\times$	$0.134$	$0.550$	rejected
$946$	$0.601$	$15.18\times$	$0.014$	$0.510$	rejected
$418$	$0.591$	$3.89\times$	$0.126$	$0.590$	rejected

Appendix

Table 6: Discovery-split rankings for the clean-trigger and auxiliary-dominated contrasts. AUROC measures event-level discrimination, top-

1\%

enrichment is the prevalence of the positive class in the highest-activation tail relative to its overall prevalence, firing rate is

P(z_{j}>0)

, and “Min. ctrl AUROC” is the lowest AUROC across the matched-control schemes. Status records the role assigned after applying the fixed discovery criteria.

Family	Definition	$n$
clean_pair	$\{640,\,1006\}$	$2$
clean_family	All $\mathrm{ams}\leq-0.85$ , excluding the brightness-support coordinate $z_{\mathrm{bs}}$	$194$
clean_core	Members of clean_family with firing rate $\geq 0.05$	$5$
aux_family	All $\mathrm{ams}\geq+0.85$	$135$

Appendix

Table 7: Reference-dictionary families. The final column gives the number of included latents.

Positive class	Score	Test AUROC	Control AUROC
clean	PC $10$	$0.846$	$0.727$
clean	$z_{\mathrm{bc}}$	$0.911$	$0.787$
auxiliary	PC $10$	$0.847$	$0.845$
auxiliary	$z_{\mathrm{aux}}$	$0.840$	$0.854$
high charge	PC $10$	$0.848$ † † † High/low charge is a binary contrast between the top and bottom quintiles of total charge.	$0.750$
high charge	$z_{\mathrm{bs}}$ ( $973$ )	$0.532$ ‡ ‡ ‡ $z_{\mathrm{bs}}$ is strongly charge-monotonic in population means but has near-chance per-event high/low-charge discrimination. This is why we describe it as brightness-associated rather than as a validated brightness read-out.	$0.516$

Appendix

Table 8: PCA and SAE scores evaluated on the independent concept_test split. Control AUROC is the strictest matched-control value for each target.

	Layer 8				Layer 3
Target	CLS	SAE	$k{=}1$	$k_{90\%}$	CLS	SAE	$k{=}1$	$k_{90\%}$
linefit speed	$0.48$	$0.42$	$0.16$	$128$	$0.61$	$0.53$	$0.14$	$64$
linearity	$0.52$	$0.53$	$0.12$	$128$	$0.49$	$0.52$	$0.10$	$128$
time extent	$0.78$	$0.78$	$0.36$	$32$	$0.88$	$0.88$	$0.43$	$16$
top-5 charge frac.	$0.60$	$0.51$	$0.05$	$256$	$0.73$	$0.64$	$0.08$	$64$
$z$ RMS	$0.67$	$0.67$	$0.10$	$256$	$0.74$	$0.71$	$0.19$	$64$

Appendix

Table 9: Held-out morphology-probe results at layers 8 and 3. Within each layer, CLS and SAE are the

R^{2}

values of dense ridge probes using the raw CLS representation and all SAE latents, respectively. The

k{=}1

column gives the best single-latent

R^{2}

, while

k_{90\%}

is the smallest number of selected latents needed to reach

90\%

of the dense-SAE

R^{2}

Layer	Latent	Depth AUROC	Min. ctrl AUROC	Direction-head $\Delta_{+}$
$2$	$171$	$0.986$	$0.984$	$+0.0064^{\circ}$
$3$	$191$	$0.925$	$0.899$	$-0.009^{\circ}$
$8$	none retained	$0.598$	—	—

Appendix

Table 10: Depth isolation across layers. “Min. ctrl AUROC” is the lowest AUROC across the matched-control schemes, and

\Delta_{+}

is the direction-head angular-error shift after removing the selected latent. At layer 8,

0.598

is the best raw AUROC, but no latent passed the isolation criteria.

Role	Quantity	Seed 1	Seed 2	Seed 3	Seed 4
Clean primary	latent	$640$	$754$	$617$	$947$
	test AUROC	$0.911$	$0.874$	$0.912$	$0.996$
	min. ctrl AUROC	$0.787$	$0.740$	$0.781$	$0.987$
	all ctrl schemes	Yes	No	Yes	Yes
Clean secondary	latent	$1006$	$662$	$918$	$808$
	test AUROC	$0.694$	$0.744$	$0.691$	$0.679$

Appendix

Table 11: Functional read-out analogues across SAE seeds. Each block reports the coordinate and its held-out validation quantities.

Family	$n$	Status	$\Delta_{+}$	$\Delta_{-}$	Select.	$z_{+}$	Ctrl med.	Ctrl mean
clean_pair	$2$	null	$+0.033^{\circ}$	$-0.007^{\circ}$	$+0.039^{\circ}$	$-0.33$	$+0.024^{\circ}$	$+0.844^{\circ}$
clean_family	$194$	null	$-0.020^{\circ}$	$-0.016^{\circ}$	$-0.005^{\circ}$	$-1.25$	$+7.651^{\circ}$	$+9.239^{\circ}$
$z_{\mathrm{aux}}$	$1$	null	$+0.004^{\circ}$	$+0.033^{\circ}$	$-0.029^{\circ}$	$-0.22$	$-0.006^{\circ}$	$+0.247^{\circ}$
aux_family	$135$	null	$-0.008^{\circ}$	$+0.031^{\circ}$	$-0.039^{\circ}$	$-0.75$	$+0.025^{\circ}$	$+1.305^{\circ}$
clean_core	$5$	null	$+0.018^{\circ}$	$-0.010^{\circ}$	$+0.028^{\circ}$	$-0.63$	$+0.033^{\circ}$	$+2.966^{\circ}$

Appendix

Table 12: Reference-dictionary removals on causal_test . “Select.” is

\Delta_{+}-\Delta_{-}

; the last columns summarize matched controls.

Role	Seed 1	Seed 2	Seed 3	Seed 4
Clean primary	$+0.057^{\circ}$ (null)	$+0.098^{\circ}$ (null)	$+0.031^{\circ}$ (null)	$-0.033^{\circ}$ (null)
Clean secondary	$+0.082^{\circ}$ (pass)	$+0.113^{\circ}$ (null)	$+0.042^{\circ}$ (null)	$-0.019^{\circ}$ (null)
Auxiliary	$+0.156^{\circ}$ (null)	$-0.117^{\circ}$ (null)	—	$-0.040^{\circ}$ (null)
Brightness support	$+13.51^{\circ}$ (V)	$+14.81^{\circ}$ (V)	$+11.61^{\circ}$ (V)	$+14.59^{\circ}$ (V)

Appendix

Table 13: Direction-head shifts across SAE seeds. Parentheses give the fixed-rule verdict or dose-response shape.

Experiment	$\Delta$ recip.	$\Delta$ donor	$z(\Delta_{+})$	$z(\mathrm{sel.})$	Verdict
write $640$ into aux	$-0.012^{\circ}$	$+0.025^{\circ}$	$-0.92$	$-0.71$	null
full swap ( $+$ zero $195$ )	$-0.002^{\circ}$	$+0.034^{\circ}$	$-0.01$	$-0.51$	null
write $\{640,1006\}$ into aux	$+0.003^{\circ}$	$+0.045^{\circ}$	$-0.21$	$+0.03$	null
full swap	$+0.013^{\circ}$	$+0.053^{\circ}$	$+0.38$	$-0.22$	null
write clean core ( $5$ lat.)	$+0.006^{\circ}$	$+0.044^{\circ}$	$-0.06$	$+0.25$	null
full swap	$+0.016^{\circ}$	$+0.051^{\circ}$	$+0.28$	$+0.56$	null

Appendix

Table 14: Direction-head writing tests. Recipient and donor columns give the induced angular-error shifts.

Dictionary	Objective	Bright-clean AUROC	Causal set	Best concept AUROC	Validated pairs
One-shot	pure functional	$0.53$	$16$	—	$0$
R1	pure functional	$0.53$	$11$	$0.59$	$0$
R9	reconstruction anchored	$0.87$	$6$	$0.69$	$0$
R10	reconstruction anchored	$0.878$	$6$	$<0.75$	$0$

Appendix

Table 15: Results for the functionally trained dictionaries. ‘Bright-clean AUROC” gives the best bright-clean read-out, ‘Causal set” counts latents that affect the direction head, and ‘Best concept AUROC” tests those latents against the full concept vocabulary. ‘Validated pairs” counts the pairs that pass the complete validation procedure.

Intervention	Seed 1	Seed 2	Seed 3	Seed 4
remove $z_{\mathrm{bc}}$	$+0.310$	$+0.193$	$+0.295$	$+0.612$
remove clean pair	$+0.321$	$+0.374$	$+0.307$	$+0.612$
remove clean family	$+0.775$	$+0.837$	$+1.012$	$+0.805$
remove clean core	$+0.373$	$+0.493$	$+0.681$	$+0.617$
clean core after support exclusion	$+0.373$	$+0.493$	$+0.302$	$+0.617$

Appendix

Table 16: Cross-seed uncertainty-head removals. Entries are shifts

M_{1}

in predicted log error.

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