Meta-learning accelerates detector design optimization

Maxim Borisyak; Nikita Gladin; Andrey Ustyuzhanin

physics.ins-detSep 22, 2026

Meta-learning accelerates detector design optimization

Authors: Maxim Borisyak, Nikita Gladin, Andrey Ustyuzhanin

Organizations: Constructor University Bremen · University of Bologna · Constructor Labs · National University of Singapore

Abstract

The quality of a detector design is ultimately determined by the quality of the inference it enables, that is, by the accuracy with which the quantities of interest are reconstructed from the raw detector response. For complex detectors, the inference is performed by machine learning models, and the relation between the design and the attainable inference performance is, in general, non-trivial. In this work, we consider the optimization of the inference performance with respect to the detector design. The conventional approach prescribes retraining the inference model at every candidate design, thus, treating the evaluations as independent tasks and discarding the shared structure of the optimal inference algorithms at different designs. We propose the meta-learned objective estimate (MLOE): instead of solving the inference problem anew at every candidate design, a single meta-inference model, conditioned on the design and trained continually along the optimization path, is shared across all of them. We test MLOE on three families of optimization problems, the last of which comprises two design spaces of the Spectrometer Straw Tracker of the Search for Hidden Particles (SHiP) experiment; under matched budgets of simulation calls, the meta-inference model evaluates a candidate design using fewer simulation calls than the baseline strategies and holds the better rank over the convergence curve in all examined cases.

Figures & tables

Panel of Figure 1 — Figure 1: Illustration of the meta-learning effect for an explicitly known design. In this illustration, the event is empty, $x\in\mathbb{R}^{0}$ , and the target is equal to the design itself. Conventional approaches estimate $f$ at each design individually (per-design). The proposed method amounts to a regression in the extended space $f(x,\theta)$ , here demonstrated with polynomial models of several degrees: under a suitable inductive bias, the estimate at one design benefits from the events simulated at all the others. The effect persists even when the inference model is more flexible than the ground truth, and becomes more pronounced as the number of designs grows.

Strategy	Network initialization	Training dataset
independent	random initialization	$x,y\mid\theta_{n}$
continual	$W^{*}_{n-1}$	$x,y\mid\theta_{n}$
MLOE	$W^{*}_{n-1}$	mixture of $x,y\mid\theta_{n}$ and accumulated $x,y\mid\theta_{1..n-1}$

Table 1: Training strategies.

\theta_{i},i=1,\dots,n

denote the trajectory of the optimizer in the design space. All strategies use the same architecture, optimizer, and stopping procedure.

Figure 2: Probes for linear model at four dimensionalities

m

, ten seeds each; the design has

m(m+1)

dimensions. Upper row: mean verified best objective against the cumulative number of simulation calls; average integral ranks are indicated in parentheses. Lower row: distribution of the simulation calls consumed per design; mean sample consumption per design is indicated in parentheses. The objective is verified by the exact expression for the Bayes risk ( 11 ). For illustration purposes, convergence curves for

m=1

and

m=2

are cut to

2^{17}

and

2^{18}

simulation calls.

	Strategy	Q1	Q2	Q3	Q4	overall
$m=1$	MLOE	$\bm{1.69\pm 0.05}$	$\bm{2.00\pm 0.00}$	$\bm{2.00\pm 0.00}$	$\bm{2.00\pm 0.00}$	$\bm{1.92\pm 0.01}$
	independent	$2.15\pm 0.05$	$\bm{2.00\pm 0.00}$	$\bm{2.00\pm 0.00}$	$\bm{2.00\pm 0.00}$	$2.04\pm 0.01$
	continual	$2.16\pm 0.04$	$\bm{2.00\pm 0.00}$	$\bm{2.00\pm 0.00}$	$\bm{2.00\pm 0.00}$	$2.04\pm 0.01$
$m=2$	MLOE	$\bm{1.35\pm 0.05}$	$\bm{1.36\pm 0.16}$	$\bm{1.61\pm 0.18}$	$\bm{1.79\pm 0.22}$	$\bm{1.53\pm 0.10}$
	independent	$2.42\pm 0.04$	$2.44\pm 0.17$	$2.25\pm 0.24$	1.95 $\pm$ 0.27	$2.26\pm 0.13$
	continual	$2.23\pm 0.04$	$2.20\pm 0.16$	2.14 $\pm$ 0.22	2.26 $\pm$ 0.25	$2.21\pm 0.12$

Table 2: Average integral rank on the probes for linear model, by quarter of the simulation budget, over the ten seeds of each dimensionality

m

. The objective is verified by the exact expression for the Bayes risk ( 11 ). The lowest values are marked in bold, the values within two combined standard errors — in italics. The values for

m=1

are degenerate after Q1, as all strategies converge on the exact solution within the first quarter.

Figure 4: Enzyme inhibitor assay: 16 dimensions, twenty seeds at a budget of

2^{21}

simulation calls. Left: mean verified best objective against the cumulative number of simulation calls; average integral ranks are indicated in parentheses. Right: distribution of the simulation calls consumed per design; mean sample consumption per design is indicated in parentheses.

Figure 5: Simulation calls per design against the cumulative simulation calls (denoted as context size) on the enzyme inhibitor assay. Semi-transparent points: sample consumption on designs; lines: mean

\pm

standard error in bins of equal count, the first designs are a bin of their own. Twenty seeds; twelve of the 1349 designs lie above the frame.

Strategy	Q1	Q2	Q3	Q4	overall
MLOE	2.08 $\pm$ 0.09	$\bm{1.78\pm 0.11}$	$\bm{1.80\pm 0.15}$	$\bm{1.71\pm 0.17}$	$\bm{1.83\pm 0.08}$
independent	$\bm{1.84\pm 0.09}$	2.07 $\pm$ 0.16	$2.38\pm 0.12$	$2.33\pm 0.16$	$2.16\pm 0.08$
continual	2.08 $\pm$ 0.09	2.15 $\pm$ 0.16	1.83 $\pm$ 0.15	1.97 $\pm$ 0.15	2.01 $\pm$ 0.10

Table 3: Average integral rank on the enzyme inhibitor assay, by quarter of the simulation budget, computed on the verified objective values over twenty seeds. The lowest values are marked in bold, the values within two combined standard errors — in italics.

Figure 6: A typical event in the Spectrometer Straw Tracker at the nominal design, viewed from upstream. The layers are drawn as semi-transparent tilted parallelograms, triggered straw tubes are highlighted in red, and the particle trajectories are shown in green and blue. All layers are perpendicular to the $z$ axis.

Figure 7: The two design spaces of the SHiP SST. SST angle: the stations are frozen and the stereo angle alone is free, 1 dimension, ten seeds. SST layout: the station positions and the stereo angle are free and overlapping stations are priced, 5 dimensions, ten seeds. Upper row: mean verified best objective against the cumulative number of simulation calls; average integral ranks are indicated in parentheses. Lower row: distribution of the simulation calls consumed per design; mean sample consumption per design is indicated in parentheses.

Figure 8: Simulation calls per design against the cumulative simulation calls (denoted as context size). Semi-transparent points: sample consumption on designs; lines: mean

\pm

standard error in bins of equal count, the first designs are a bin of their own. Ten seeds per strategy per task.

	Strategy	Q1	Q2	Q3	Q4	overall
SST angle	MLOE	$\bm{1.66\pm 0.18}$	$\bm{1.17\pm 0.09}$	$\bm{1.26\pm 0.10}$	$\bm{1.26\pm 0.13}$	$\bm{1.34\pm 0.09}$
	independent	$2.23\pm 0.11$	$2.50\pm 0.15$	$2.80\pm 0.10$	$2.80\pm 0.13$	$2.58\pm 0.10$
	continual	2.11 $\pm$ 0.21	$2.33\pm 0.12$	$1.95\pm 0.14$	$1.94\pm 0.22$	$2.07\pm 0.13$
SST layout	MLOE	$\bm{1.08\pm 0.08}$	$\bm{1.19\pm 0.10}$	$\bm{1.15\pm 0.11}$	$\bm{1.13\pm 0.10}$	$\bm{1.14\pm 0.06}$
	independent	$2.49\pm 0.04$	$2.62\pm 0.07$	$2.90\pm 0.10$	$2.87\pm 0.10$	$2.72\pm 0.06$
	continual	$2.42\pm 0.11$	$2.19\pm 0.14$	$1.95\pm 0.16$	$2.00\pm 0.15$	$2.14\pm 0.08$

Table 4: Average integral rank on the two design spaces of the SHiP SST, by quarter of the simulation budget, computed on the verified objective values over ten seeds per task. The lowest values are marked in bold, the values within two combined standard errors — in italics.

Appendix figures & tables23 assets

Supplementary material from the paper’s appendix.

Appendix

Task	$n_{0}$	increment	$\varepsilon$	$\tau$	warm-up	budget
probes for linear model, $m=1$	2048	1024	$2\times 10^{-2}$	16	16	$2^{19}$
probes for linear model, $m=2,3,4$	4096	2048	$2\times 10^{-2}$	16	16	$2^{19}$
enzyme inhibitor assay	16384	8192	$10^{-2}$	16	16	$2^{21}$
SST angle	32768	16384	$10^{-2}$	16	2	$2^{21}$
SST layout	32768	16384	$10^{-2}$	16	2	$2^{21}$

Appendix

Table 5: Parameters of the training procedure for each task: the initial training dataset size

n_{0}

, the increment added at every data addition, the target precision

\varepsilon

, the patience

\tau

, the unconditional warm-up in epochs and the total budget of simulation calls. A quarter of every acquisition is held out for validation.

Task	Warm-start training regime	MLOE	independent	continual
probes for linear model	mix, $\rho=1$ (none)	$4.53\pm 0.08$	$4.65\pm 0.06$	$4.60\pm 0.07$
	mix, $\rho=0.9$	$4.08\pm 0.09$	$4.09\pm 0.05$	$4.10\pm 0.09$
	mix, $\rho=0.75$	$3.20\pm 0.06$	$3.11\pm 0.03$	$3.17\pm 0.05$
	shrink-and-perturb, $\alpha=0.3$	$\bm{1.23\pm 0.12}$	$\bm{1.05\pm 0.05}$	$\bm{1.04\pm 0.04}$
	shrink-and-perturb, $\alpha=0.6$	$1.96\pm 0.10$	$2.10\pm 0.03$	$2.09\pm 0.05$
enzyme inhibitor assay	mix, $\rho=1$ (none)	3.17 $\pm$ 0.61	2.99 $\pm$ 0.33	$3.23\pm 0.19$

Appendix

Table 6: Results of hyperparameter optimization: average integral rank of each warm-start training regime for each training strategy, computed on the verified objective values over the three selection runs, mean

\pm

standard error; on the probes for linear model, one selection is pooled over the four dimensionalities, twelve runs per strategy. Lower is better; 1 is the best attainable and 5 the worst. The lowest values are marked in bold, the values within two combined standard errors — in italics.

Task	Strategy	$\rho=1$	$\rho=0.9$	$\rho=0.75$	$\alpha=0.3$	$\alpha=0.6$
probes for linear model	MLOE	$\bm{1.17\pm 0.10}$	$\bm{1.27\pm 0.12}$	$\bm{1.23\pm 0.09}$	$\bm{1.63\pm 0.17}$	$\bm{1.08\pm 0.05}$
	independent	$2.26\pm 0.12$	$2.04\pm 0.10$	$1.91\pm 0.09$	$2.11\pm 0.16$	$2.37\pm 0.10$
	continual	$2.57\pm 0.10$	$2.69\pm 0.08$	$2.86\pm 0.02$	$2.26\pm 0.11$	$2.56\pm 0.09$
enzyme inhibitor assay	MLOE	$\bm{1.56\pm 0.12}$	$\bm{1.74\pm 0.25}$	$\bm{1.69\pm 0.11}$	$\bm{1.82\pm 0.51}$	$\bm{1.72\pm 0.34}$
	independent	$2.15\pm 0.15$	2.13 $\pm$ 0.21	2.10 $\pm$ 0.42	2.12 $\pm$ 0.33	1.97 $\pm$ 0.31
	continual	$2.29\pm 0.20$	2.13 $\pm$ 0.10	2.21 $\pm$ 0.31	2.06 $\pm$ 0.31	2.31 $\pm$ 0.09

Appendix

Table 7: Average integral rank of each training strategy under each warm-start training regime (

\rho

: mixing,

\alpha

: shrink-and-perturb), on the verified objective values over three seeds, mean

\pm

standard error; on the probes for linear model, over the three seeds of each of the four dimensionalities. Lower is better; 1 is the best attainable and 3 the worst. The lowest values are marked in bold, the values within two combined standard errors — in italics.

Figure 9: Architecture for the inference model used in all experiments. The fully connected (FC) layers act on every element; each Bayesian aggregation (BA) combines the per-element means

\mu_{i}

and precisions

\tau_{i}

across elements; the circled C denotes concatenation of that representation with the mean of every element.

Task	Input (MLOE / baseline)	Block 1	Block 2	Target	Parameters
probes for linear model, $m$	$1+m$ / $2+m$	$24\to 32$	$32\to 24$	$m+1$	5.4–5.6 $\times 10^{3}$
enzyme inhibitor assay	$36$ / $36$	$24\to 16$	$16\to 24$	$2$	$3.1\times 10^{3}$
SST angle, SST layout	$4$ / $5$	$32\to 48$	$48\to 32$	$9$	$1.14\times 10^{4}$

Appendix

Table 8: Parameters of the inference model: per-element input width for MLOE and for the baselines, hidden and output widths of the two blocks, target width, and parameter count of one network. Ensembles have four identical networks.

		$m=1$		$m=2$		$m=3$		$m=4$
	Strategy	rank	calls, $10^{3}$	rank	calls, $10^{3}$	rank	calls, $10^{3}$	rank	calls, $10^{3}$
$\lambda=1$	MLOE	$\bm{1.92\pm 0.01}$	$\bm{3.9\pm 0.1}$	$\bm{1.53\pm 0.10}$	$\bm{7.4\pm 0.2}$	$\bm{1.32\pm 0.06}$	$\bm{8.2\pm 0.2}$	$\bm{1.54\pm 0.13}$	$\bm{7.5\pm 0.2}$
	independent	$2.04\pm 0.01$	$5.5\pm 0.1$	$2.26\pm 0.13$	$10.5\pm 0.3$	$2.46\pm 0.09$	$13.0\pm 0.6$	$2.19\pm 0.10$	$13.4\pm 0.5$
	continual	$2.04\pm 0.01$	$5.5\pm 0.2$	$2.21\pm 0.12$	$10.4\pm 0.1$	$2.22\pm 0.11$	$11.6\pm 0.4$	$2.28\pm 0.15$	$11.3\pm 0.5$
$\lambda=1/4$	MLOE	$\bm{1.95\pm 0.02}$	$\bm{4.9\pm 0.1}$	$\bm{1.69\pm 0.16}$	$\bm{8.2\pm 0.1}$	$\bm{1.45\pm 0.13}$	$\bm{9.1\pm 0.2}$	$\bm{1.66\pm 0.10}$	$\bm{9.3\pm 0.4}$
	independent	$2.03\pm 0.02$	$5.5\pm 0.1$	$2.21\pm 0.13$	$10.5\pm 0.3$	$2.42\pm 0.11$	$13.0\pm 0.6$	$2.13\pm 0.12$	$13.4\pm 0.5$

Appendix

Table 9: Average integral rank and mean simulation calls per design on the ten seeds of the test runs of the probes for linear model, mean

\pm

standard error. First two blocks: MLOE with the given weight

\lambda

against the baseline strategies; last block: the two values of

\lambda

ranked between themselves. Lower is better; 1 is the best attainable, 3 the worst in the first two blocks and 2 in the last. The lowest values are marked in bold, the values within two combined standard errors — in italics.

		enzyme inhibitor assay		SST angle		SST layout
	Strategy	rank	calls, $10^{3}$	rank	calls, $10^{3}$	rank	calls, $10^{3}$
$\lambda=1$	MLOE	$\bm{1.83\pm 0.08}$	$\bm{81.2\pm 1.8}$	$\bm{1.34\pm 0.09}$	$\bm{165.6\pm 5.7}$	$\bm{1.14\pm 0.06}$	$\bm{157.9\pm 4.4}$
	independent	$2.16\pm 0.08$	$105.9\pm 2.6$	$2.58\pm 0.10$	$317.6\pm 7.3$	$2.72\pm 0.06$	$324.5\pm 6.4$
	continual	2.01 $\pm$ 0.10	$90.0\pm 2.3$	$2.07\pm 0.13$	$342.6\pm 9.8$	$2.14\pm 0.08$	$360.8\pm 8.9$
$\lambda=1/4$	MLOE	2.10 $\pm$ 0.08	$104.8\pm 3.1$	$\bm{1.30\pm 0.11}$	$\bm{236.0\pm 7.7}$	$\bm{1.25\pm 0.09}$	$\bm{227.6\pm 4.6}$
	independent	2.02 $\pm$ 0.10	$105.9\pm 2.6$	$2.61\pm 0.09$	$317.6\pm 7.3$	$2.62\pm 0.06$	$324.5\pm 6.4$

Appendix

Table 10: Average integral rank and mean simulation calls per design on the ten seeds of the test runs of the SST tasks and the twenty seeds of the test runs of the enzyme inhibitor assay, mean

\pm

standard error. First two blocks: MLOE with the given weight

\lambda

against the baseline strategies; last block: the two values of

\lambda

	SST angle		SST layout
	rank	calls per design, $10^{3}$	rank	calls per design, $10^{3}$
$\lambda=1$	$2.76\pm 0.14$	$\bm{182.3\pm 7.4}$	$\bm{2.16\pm 0.44}$	$\bm{178.9\pm 17.2}$
$\lambda=1/2$	2.05 $\pm$ 0.21	$210.2\pm 0.5$	2.57 $\pm$ 0.41	226.0 $\pm$ 16.7
$\lambda=1/4$	$\bm{1.81\pm 0.16}$	$247.0\pm 7.7$	2.87 $\pm$ 0.66	$246.4\pm 12.8$
$\lambda=1/8$	$3.38\pm 0.09$	$283.8\pm 17.9$	2.40 $\pm$ 0.26	$280.9\pm 7.9$

Appendix

Table 11: Average integral rank and mean simulation calls per design of MLOE at each weight

\lambda

, on the three seeds of the selection runs of the SST tasks, mean

\pm

standard error. All four weights are taken at the same warm-start training regime, mixing with

\rho=0.9

on the SST angle task and no mixing (

\rho=1

) on the SST layout task. Lower is better; 1 is the best attainable and 4 the worst. The lowest values are marked in bold, the values within two combined standard errors — in italics.

	SST angle		SST layout
	rank	calls per design, $10^{3}$	rank	calls per design, $10^{3}$
MLOE	$\bm{1.72\pm 0.11}$	$\bm{165.6\pm 5.7}$	$\bm{1.47\pm 0.11}$	$\bm{157.9\pm 4.4}$
MLOE without context	1.85 $\pm$ 0.15	$327.2\pm 11.5$	$1.80\pm 0.08$	$331.9\pm 9.8$
continual	$2.43\pm 0.16$	$342.6\pm 9.8$	$2.74\pm 0.06$	$360.8\pm 8.9$

Appendix

Table 12: Average integral rank and mean simulation calls per design of MLOE, MLOE without the context and the continual strategy, on the verified objective values over ten seeds, mean

\pm

standard error. Lower is better; 1 is the best attainable and 3 the worst. The lowest values are marked in bold, the values within two combined standard errors — in italics.

Task	Strategy	calls per design, $10^{3}$	time per design, h	sampling, core-hours
			simplified	high-fidelity
SST angle	MLOE	$166\pm 6$	$0.200\pm 0.007$	92–138
	independent	$318\pm 7$	$0.337\pm 0.011$	176–265
	continual	$343\pm 10$	$0.317\pm 0.014$	190–286
SST layout	MLOE	$158\pm 4$	$0.189\pm 0.007$	88–132
	independent	$324\pm 6$	$0.323\pm 0.010$	180–270

Appendix

Table 13: Cost of a single optimization step on the SST tasks using simplified simulations compared to estimated cost of generating the same number of samples using high-fidelity simulation.

Figure 10: Warm-start training regimes on the probes for linear model,

m=2

, one column per strategy. Top: verified best objective against cumulative simulation calls; average integral ranks are indicated in parentheses. Bottom: simulation calls per design; mean sample consumption per design is indicated in parentheses. Each warm-start training regime was evaluated on 3 independent runs.

Figure 11: Warm-start training regimes on the enzyme inhibitor assay, one column per strategy. Top: verified best objective against cumulative simulation calls; average integral ranks are indicated in parentheses. Bottom: simulation calls per design; mean sample consumption per design is indicated in parentheses. Each warm-start training regime was evaluated on 3 independent runs.

Figure 12: Warm-start training regimes on the SST angle task, one column per strategy. Top: verified best objective against cumulative simulation calls; average integral ranks are indicated in parentheses. Bottom: simulation calls per design; mean sample consumption per design is indicated in parentheses. Each warm-start training regime was evaluated on 3 independent runs.

Figure 13: Warm-start training regimes on the SST layout task, one column per strategy. Top: verified best objective against cumulative simulation calls; average integral ranks are indicated in parentheses. Bottom: simulation calls per design; mean sample consumption per design is indicated in parentheses. Each warm-start training regime was evaluated on 3 independent runs.

Figure 14: Training strategies under each warm-start training regime on the probes for linear model,

m=2

, one panel per warm-start training regime; the last panel shows each strategy under the warm-start training regime selected for it. Verified best objective against cumulative simulation calls; average integral ranks are indicated in parentheses. Each strategy was evaluated on 3 independent runs.

Figure 15: Training strategies under each warm-start training regime on the enzyme inhibitor assay, one panel per warm-start training regime; the last panel shows each strategy under the warm-start training regime selected for it. Verified best objective against cumulative simulation calls; average integral ranks are indicated in parentheses. Each strategy was evaluated on 3 independent runs.

Figure 16: Training strategies under each warm-start training regime on the SST angle task, one panel per warm-start training regime; the last panel shows each strategy under the warm-start training regime selected for it. Verified best objective against cumulative simulation calls; average integral ranks are indicated in parentheses. Each strategy was evaluated on 3 independent runs.

Figure 17: Training strategies under each warm-start training regime on the SST layout task, one panel per warm-start training regime; the last panel shows each strategy under the warm-start training regime selected for it. Verified best objective against cumulative simulation calls; average integral ranks are indicated in parentheses. Each strategy was evaluated on 3 independent runs.

Figure 18: MLOE with the weight

\lambda

of the context samples varied, on the probes for linear model, one column per dimensionality

m

. Top: verified best objective against cumulative simulation calls; average integral ranks are indicated in parentheses. Bottom: simulation calls per design; mean sample consumption per design is indicated in parentheses. Ten seeds. For illustration purposes, convergence curves for

m=1

and

m=2

are cut to

2^{17}

and

2^{18}

simulation calls.

Figure 19: MLOE with the weight

\lambda

of the context samples varied, on the enzyme inhibitor assay (left), SST angle (middle) and SST layout (right) tasks. Top: verified best objective against cumulative simulation calls; average integral ranks are indicated in parentheses. Bottom: simulation calls per design; mean sample consumption per design is indicated in parentheses. Ten seeds of the test runs per SST task, twenty on the enzyme inhibitor assay.

Figure 20: MLOE with the weight

\lambda

of the context samples varied over

1

1/2

1/4

and

1/8

, on the SST angle (left) and SST layout (right) tasks, at a common warm-start training regime . Top: verified best objective against cumulative simulation calls; average integral ranks are indicated in parentheses. Bottom: simulation calls per design; mean sample consumption per design is indicated in parentheses. Three seeds of the selection runs.

Figure 21: MLOE without the context, against MLOE and the continual strategy, on the SST angle (left) and SST layout (right) tasks. Top: verified best objective against cumulative simulation calls; average integral ranks are indicated in parentheses. Bottom: simulation calls per design; mean sample consumption per design is indicated in parentheses. Ten seeds.

Figure 22: Mean verified best objective against the optimization iteration. Top row: probes for linear model, one panel per dimensionality

m

, ten seeds per dimensionality. Bottom row: enzyme inhibitor assay, twenty seeds; SST angle and SST layout, ten seeds per task. A curve is drawn while every seed has a design at that iteration.

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