RESuM: A Rare Event Surrogate Model for  Physics Detector Design
Authors: Ann-Kathrin Schuetz
A.W.P. Poon
Aobo Li
Venue: ICLR 2025 Spotlight (2025)
Keywords: surrogate model
simulation
rare event search
AI4Sci
AI for physics
conditional neural process
Bayesian methods
emulator
Multi-Fidelity Gaussian Process
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: We developed a Rare Event Surrogate Model (RESuM) powered by Conditional Neural Process to optimize particle physics detector design under high-variance design metrics.

Abstract: The experimental discovery of neutrinoless double-beta decay (NLDBD) would answer one of the most important questions in physics: Why is there more matter than antimatter in our universe? To maximize the chances of discovery, NLDBD experiments must optimize their detector designs to minimize the probability of background events contaminating the detector. Given that this probability is inherently low, design optimization either requires extremely costly simulations to generate sufficient background counts or contending with significant variance. In this work, we formalize this dilemma as a Rare Event Design (RED) problem: identifying optimal design parameters when the design metric to be minimized is inherently small. We then designed the Rare Event Surrogate Model (RESuM) for physics detector design optimization under RED conditions. RESuM uses a pre-trained Conditional Neural Process (CNP) model to incorporate additional prior knowledge into a Multi-Fidelity Gaussian Process model. We applied RESuM to optimize neutron shielding designs for the LEGEND NLDBD experiment, identifying an optimal design that reduces the neutron background by $(66.5 \pm 3.5)$% while using only 3.3% of the computational resources compared to traditional methods. Given the prevalence of RED problems in other fields of physical sciences, especially in rare-event searches, the RESuM algorithm has broad potential for accelerating simulation-intensive applications.

Published as a conference paper at ICLR 2025
RESUM: RARE EVENT SURROGATE MODEL FOR PHYSICS DETECTOR DESIGN
Ann-Kathrin Schuetz' aschuetz 1bl.@llb.... lll.bo.
Alan W. P. Poon' awpoon 1
Aobo Li aol002@ucsd.eed
NNuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA "Halcıoğlu Data Science Institute, Department of Physics, UC San Diego, La Jolla, CA 92093,
USA
Corresponding Authors
ABSTRACT
The experimental discovery of neutrinoless double-beta decay (NLDBD) would answer one of the most important questions in physics: Why is there more matter than antimatter in our universe? To maximize the chances of detection, NLDBD experiments must optimize their detector designs to minimize the probability background events contaminating the detector. Given that this probability is in- herently low, design optimization either requires extremely costly simulations to generate sufficient background counts contending with significant variance. In this work, we formalize this dilemma as Rare Event Design (RED) problem: identifying optimal design parameters when the design metric to be minimized is inherently small. We then designed the Rare Event Surrogate Model (RESuM) for physies detector design optimization under RED conditions. RESuM uses pretrained Conditional Neural Process (CNP) model to incorporate additional prior knowledges into a Multi-Fidelity Gaussian Process (MFGP) model. We ap- plied RESuM to optimize neutron moderator designs for the LEGEND NLDBD experiment, identifying an optimal design that reduces neutron background by (66.5 + 3.5)% while using only 3.3% of the computational resources compared to traditional methods. Given the prevalence of RED problems in other fields of physical sciences, the RESuM algorithm has broad potential for simulation- intensive applications.
INTRODUCTION
Why is there more matter than antimatter in our universe? This question remains one of the most im- portant yet unsolved questions in physics. Several Nobel Prizes have been awarded for groundbreak- ing discoveries that have advanced our understanding of this questions, including the discovery of CP violation in kaons (Cronin and Fitch 1980), the detection of cosmic neutrinos (Koshiba) 2002). and the development of the Kobayashi-Maskawe 2008 Despite these monumental achievements, the reason for the dominance of matter over anti- matter remains unsolved. One of the most promising next steps toward answering this question is potential discovery of Neutrinoless Double-Bert Beta Decay (NLDBD) (Dolinski et al. 2019). Such discovery would represent major milestone in this direction and would undoubtedly be consid- cred a Nobel-Prize-
theory of CP violation (Kobayashi and Maskawa
breakthrough in physics. Due to its utmost importance, the entire U.S. nuclear physics community has gathered for year-long discussion in 2023 and recommended the experimental search for NLDBD as the second-hight -highst priority (Committee) 2023) for next 10 years.
The most challenging aspect of NLDBD search is dealing with background events: physical events that are not NLDBD, but are indistinguishable from it. Since NLDBD hypothesized to occur less than once every three years gle background event entering the detector could potentially ruin the entire detection effort. There- fore, designing ultra-pure NLDBD detectors with optimal parameters minimize the probability of background events entering the detector becomes the utmost goal of all NLDBD experiments.
et al.) 2021; Dolinski et al. 2201)),
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Traditionally, the detector design procedure is conducted with simulations: we first simulate detectors and N1 background events under a certain design parameter 01. then count the number of background events that eventually enter our detector, m,. We then repeat the simulation process with another design parameter 0₂ and count m2. If m1 m1/N1 < m2// N2, it suggests that the design 0, is better than Oz. This simulation process can be repeated multiple times until an optimal design is found. An obvious shortcoming of this traditional approach is the computational cost: due to the ulltr--pure nature of the NLDBD detector, NN needs to be very large for m to even be non-zero. This is amplified by the complexity of the design space, involving numerous and often non- inter- dependent parameters such as detector geometry, material properties, and environmental conditions.
An obvious solution to this problem is to build surrogate model that can significantly accelerate our simulations (Li et al., 2023 Ravi et al., 2024). However, due to the rare event nature, m either 0 or small, discrete integer, which leads to high variance in our design metric m/N. /N.. This variance renders training traditional continuous surrogate models extremely difficult. In this paper, we formulate this problem as a Rare Event Design (RED) problem and present RESuM-a Rare Event Surrogate model to solve this problem. RESuM navigate through a complex landscape and approximate the complex relationships between the design parameters @ and rare event design metric m/N. The benchmarking result shows that RESuM could reduce the LEGEND neutron background by (.5)5 using only 3.3% of the computational power compared traditional methods. Due to the broad presence of RED problems in physical sciences, RESuM has the potential to be applied to other domains, including Astronomy and Material Science.
RELATED WORKS
Due to the computational cost of particle physics simulations, generative models like VAE (Z. Fu eet al. 2024). GAN (Kansal et al., 2021 Vallecorsa| 2018 Hashemi et al. 2024). and diffusion models are widely used surrogate models for fast simulation (Kansal et al., 2023). Although these deep generative models, usually trained on large datasets, robustly reproduce enriched high- dimensional data, their black-box nature renders them non-interpretable and lacking clear statistical meaning. Meanwhile, the CNP model (Garnelo et al., 2018). as a probabilistic generative model, offers the distinct advantage of few-shot learning and provides clear statistical interpretation. It has demonstrated good performance in few-shot problems. including classification tasks (Requeimal let al. 2019p, statistical downscaling (Vaughan et al.) 2022), and hydrogeological modeling (Cui et al. 2022). In this study, we explore a novel surrogate modeling approach that focuses solely on key detector design metrics. The CNP model was not used as a generative model, but as a predictive model to smooth out discreteness of rare design metrics.
Another related field is rare event simulation and modeling in reliability engineering. The rare event problem here focuses on estimating extremely low failure probabilities Py. Since direct Monte Carlo simulation becomes intractable as P, approaches zero, specialized techniques, including adaptive sampling and FORM/SORM methods, have been developed. The development progressed from FORM by Hasofer (974) and its extension to non-normal distributions (Fiessler et al. 1979). com- prehensively reviewed in Der Kiureghian et al. (2005). Methodological advances include adaptive sampling (Bucher 1988), surrogate-baser methods (Li and Xiu 2010 Li et al. [001], sequential importance sampling (Papaioannou et al., 2016). and multi-fidelity approaches (Peherstorfer al., 2016 2018). Recent work has introduced multilevel sampling (Wagner et al.) 2020) and ensemble
Kalman filters (Wagner et al. (2022). While Adaptive Importance Sampling (AIS) can potentially solve the RED problem, its implementation to LEGEND simulation presents several significant chal- lenges, as discussed in Appendix [15] In contrast, the RESuM model proposed in this work provides a simple yet efficient solution to the RED problem in LEGEND.
3 RARE EVENT DESIGN PROBLEM
Definition Let 0 E O be the vector of design parameters, where θ represents the space of all pos- sible design parameters. Consider simulation involving N events, or data points, under design pa- rameter 0 each event can either trigger a signal or not. Define a stochastic process (X1,... XN
""trigger signal" could represent any event of interest depending on the task setup. In the case of the NLDBD background minimization task, it means background event successfully leaches into the detector
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where each random variable X; corresponds to the 1 event in the simulation and X, if the ith event triggers a signal and X, = if it doesn't t. Each random variable X, is statistically independent of all other X, for j * i.
Each simulated event is considered independent, and the outcome of X, depends on two sets of parameters: set of design parameters 0 which is universal across all events, and another sets of event-specific parameters Φ; E + where & represents the space of all possible event-specific parameters. The probability that the i-th event will trigger signal is thereby defined as a function of both 0 and Φi. which could be denoted as 1(0, Φz).
Let m represent the number of events that trigger signal. The design metric y is then defined as:
m
-==x, X,
(1)
N
Rare Event Assumption The number of triggered events m follows binomial distribution with the triggering probability t(0, Φt). Under the rare event assumption that m < N and the triggering probability for each event t(0, Φbs) is small, the number of triggered events m can be approximated by a Poisson distribution as m ~ Poisson (NE()). Where i(0) is the expected triggering probability over all simulated events when N goes to infinity:
t(O) / ()
(2)
The function (ΦΦ) denotes predefined probability density function (PDF) where o; could be sam- pled from during the simulation process. t(0) is obtained by marginalizing t(O, Φ) over g()). Therefore, the ultimate metric that we want to minimize f, which the expectation of y:
0 = arg min t(0) EE
(3)
Since f depends 0, minimizing i requires extensive sampling of different f values within the design space θ to identify the optimal parameter.
Large N Scenario Assuming that i(0) remains fixed. When N becomes large, according to the central limit theorem, m will tend to follow a normal distribution:
m ~~~N m~N((N())). NI(())
Since = m//, this means that y will also follow normal distribution with symmetric, well- defined statistical uncertainties E(0)/N:
y NO)))
As N - +00. y will asymptotically approximate f(0) with statistical uncertainties approaching 0.
Small N Scenario When N becomes small, the total number of instances m that trigger a signal has higher variance, as each individual instance has a significant impact on m. The accuracy measure y N can no longer be approximated with normal distribution. This makes y more sensitive statistical fluctuations of a few simulated events. Furthermore, there a neg-negli probability that no event will trigger signal, meaning that m = 0 and y ~ T = 0. In summary, the small N scenario, the design metric y of interests will only takes on a discrete set of values, E { (N+ I N. N
4 RARE EVENT SURROGATE MODEL
The Rare Event Surrogate Model (RESuM) aims to solve the RED problem under the constraint of limited access to large N simulations and an unknown triggering probability 1(0, Φi). Consider a scenario where we run K simulation trials with different design parameter 0, indexed by k; each simulation trial contains N events indexed by i. The RESuM model includes three components: Conditional Neural Process (CNP) (Garnelo et al.] 2018) model that is trained event level; a Multi- Fidelity Gaussian Process (MFGP) (Kennedy and O O'Hagan 2000; Qian and WuI 2008) model that trains simulation trial level; and active learning techniques to sequentially sample the parameter space after training. The conceptual framework and details of our model design are outlined in the following subsections.
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4.1 BAYESIAN PRIOR KNOWLEDGE WITH CONDITIONAL NEURAL PROCESS
The random variable X5= represents whether the 4th event triggered a signal or not. In traditional particle physics, the value of Xk: is determined through a Monte Carlo simulation process: first, parameter Φ& is sampled from the distribution g(Φ) to generate the event. This event then propa- gates through the detector, characterized by the design parameter 0z. The outcome of the simulation, which implicitly involves the joint distribution t(x,, Φki), is only observed as Xki. As discussed before, X1z can only be or 1. In the small N scenario, the root cause of the discreteness of y is this binary nature: 1 if a signal is triggered or 0 if not. This produces significant statistical variance in y. Suppose we want to model this simulation process with a Bernoulli distribution:
X Bernoulli((( sbx))
The goal of incorporating prior knowledge is to smooth out the binary XRs into continuous, floating-point score 33 between 0 and 1. The score Ser should approximate 1(0x, Φki) given de-
sign parameter 0& and event-specific parameter Φki-
This work provides an alternative solution by adopting a similar idea to the CNP model. CNP works by learning a representation of input-output relationships from context data to predict outputs for new inputs (Garnelo et al. 2018). In our case, the input is 0₂ and Φkz. and the output is the random variable X Xki- The random process that generates X&s based on the inputs is the Bernoulli process controlled by t(0, Φ). We then adopt the same representation learning idea used in the CNP, which involves approximating the random process by sampling from a Gaussian distribution conditioned at observed data. The mean and variance are modeled with neural networks:
Bernoulli(t(O.. Φ)) R Bernoulli(3)
(5)
3 3~ 3~ NNNNN ob, u) N. (0, Φ,, x
(6)
Where 3 is the CNP- event (ith event in the tthh simulation trial). The nuisance parameters represent the trainable parameters of the neural network (Garnelo et al., 2018), including the weights and biases, that are optimized during training by minimizing the likel of the observed data. Importantly, the neural networks are not trained to predict the binary observable X, but rather to estimate the continuous floating-poine score A comprehensive description of the CNP model, along with the interpretation of the score 3 and the associated loss function (likelihood), is provided in Appendix [3 The score for each simulated event serves as prior information that incorporated into the MFGP surrogate model.
score in general and 3%s Blea. is the CNP score of a specific
4.2 MODEL DESCRIPTION
Building on the conceptual framework described in 4.1 we will provide an end-to-end overview of RESuM as shown in Figure [ We generate two types of simulations: low-fidelity (LF) and high- fidelity (HF). Detailed descriptions of these simulations can be found in Section [5.1 The primary distinction between them lies in the number of simulated events N, where Nur > NLr- Another key difference is the distribution g(Φ) from which the parameter Φb of each event is sampled, where HF simulation contains a more complicated, physics-oriented g()). The low computational cost of LF simulation allows us to simulate more trials thereby exploring a broader range of 0. The first step is to train the CNP model. The CNP comprises three primary components: an encoder, an aggregator, and decoder. The parameters O₂. Φki. and X of each simulated event are first concatenated into a context point. The encoder, implemented a Multi-Layer Perceptron (MLP), transforms each context point into a low-dimensional representation. These representations are then aggregated through averaging to form a unified representation that represents I(0). The decoder uses t(O) and the du of new data to output parameters μ4s and of for each event i. We then use μą: and of to form a normal distribution and sample a CNP score B&i from it. The scores 3x are chosen to naturally fit normal-like distribution but bounded between O and 1. Since the CNP is trained at event level, 3kg will be the same regardless of whether the event is generated in HF or LF simulation.
Based on the trained CNP model, the next step involves in calculating three design metries at dif- ferent fidelities. The first one is ynow m/N from HF simulations, which is the ultimate design
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Siudation
Conditional Neural Process
Multi-fidelity Gaussian
7OAP
PY
Model Training --> Model Inference
Figure 1: Overview the RESuM framework for solving RED problems. The left side illustrates the CNP used for modeling both LF and HF simulation data. The CNP aggregates event-specific parameters Φ; and design parameters 0 from LF and HF simulations to produce VENP and SCNP. which, together with HF simulation output vkaw serve as inputs to the surrogate model. The right side shows the multi-MFGP that combines predictions ÓCNP from LF and HF to estimate the HF design metric all
metric we want our surrogate model to emulate. The second metric also derived from HF simula- tions but is defined as the average CNP score of all simulated events:
N
JCNP
The third metric is JONN calculated over LF simulations. These three design metrics are then incorporated into MFGP model to train the surrogate model. Co-kriging was used account for correlations among different design metrics. The mathematical detail of MFGP can be found in Appendix [I]
After training the MFGP model, we adopt active learning to select new sampling points Onew to generate Yfow with HF simulations. Since HF simulation is expensive, determine which point to collect next, we use gradient-brrs based optimizer to find 0x+1 arg maXøex I(0) (Paleyes et al., 2023 The acquisition function I(0) determines the next data point to explore by balancing ex- ploration (high variance) and exploitation (high mean). We chose the integrated variance reduction method, where the next point, 02+1, is selected to maximally reduce the total variance the model Sacks et al. 1989), More detail about the active learning method can be found in Appendix 12
EXPERIMENT AND RESULT
The Large Enriched Germanium Experiment for Neutrinoless Double-Beta Decay (LEGEND) is a next-generation pioneering experiment in the search for NLDBD, with over 300 international col- laborators. One of the major background event type in LEGEND are 77(m) Ge, which are produced through a three-step physics process: (1) Cosmic Muons are high-energy particles that constantly shower down from the sky. (2) When cosmic muons enter the LEGEND outer detector, they in- teract with materials in the outer detector, which generates a lot of neutrons.
(3) Neutrons then propagate through the LEGEND detector system. If neutron enters the inner detector, it has chance to produce 77(m) by neutron capture, which is the primary background of concern. 77(m) Ge, once produced, will be particularly challenging, because it could mimic NLDBD events, making it nearly impossible to distinguish and reject once produced The most viable solution to mitigate this background is through the design of neutron moderator- neutron shield that slows down the neutrons and reduce the neutron flux entering the inner detector system between step (2) and (3). Figure 2 provides overview of the LEGEND detector and proposed neutron moderator design. Our goal with the RESuM is to optimize the geometric design of the neutron moderator to prevent most neutrons from leaching into the inner detector. The optimization process consists of two key steps: generating simulations under different design parameters, and adopting the RESuM model surrogate and identify the optimal design of neutron moderators.
Currently, there are no efficient methods eliminate 77(m) Ge once created, aside from employing complex active tagging algorithms (Neuherger 202T) that introduce additional dead time.
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Figure 2: Left: Rendering of the LEGEND- 1000 experiment with the hundreds of detectors. The inner detector region is shown in green while the rest are outer detectors. Right: Illustration of the neutron moderator/shield design with the 5 design parameters (viewed from top).
5.1 NEUTRON MODERATOR SIMULATIONS
Two neutron moderator design schemes were proposed: (1) continuous, cylindrical-shaped shield surrounding the detector array, and (2) a turbine-like shield composed of several panels. As il- lustrated in Figure 2 (Right), we define five design parameters to include both geometries under continuous transition: the radius of the cylindrical layer or the distance of the panels, the shield thickness d, the number of panels n, the angle of the panels şp, and the panel length L. These five parameters constitute the design parameter space θ where each parameter is allowed to vary within a predefined range.
A specific design 0 can be sampled from + to perform simulations and obtain the corresponding design metric FRaw- For a given O, utilized a Monte Carlo (MC) simulation package based on the GEANT-4 toolkit (Agostinelli et al.] [2033] Allison et (al,)2000). integrated with the existing LEG- END software frameworks (Neuberger) Ramachers and Morgan). We first simulate the LEGEND detector with neutron moderator configuration according to the sampled design parameter 0, as shown in shown in Figure 2 Second, we simulate neutrons which are allowed to propagate through the detector within the simulation system, and the number of 77(m) produced, m, is counted within the inner detector. The design metric VRaw is then computed as YRow = m/N.
We implemented two levels of simulation fidelity: high-fidelity (HF) and low-fidelity (LF). Since neutrons are primarily produced by cosmic muons showering down from the atmosphere, the HF simulation starts by generating muons outside the LEGEND detector using physics information provided by the MUSUN muon simulation software (Kudryavtsev 2009p. The event-specific pa- rameter du of each muon is sampled from a carefulll-ddsigneed distribution ( These muons, with predefined energies and directions, will propagate through the detector geometry, leading to the production of neutrons with their associated 6in assigning an initial position (æ, y. 2). initial momentum (Pz. Py: P=), and initial energy E to the created neutrons. The total number of simulated muons NFF for each design parameter O& is typically very large, the order of 107. The LF simu- lation, on the other hand, skipped the muon simulation step. Neutrons are directly simulated within the detector, and the epent-specific parameter dé is randomly sampled from a uniform distribution gime (pins) in a predefined range. The total number of neutrons simulated, NLF is on the order of 103, which is also significantly smaller than in the HF simulation. To fully assess the probability of a neutron in step (3) producing 77(m) background event, we must also consider the neu- tron's 's end position, the time from its creation to termination, and its energy transfer at termination. Consequently, parameters form the event-specific parameter & ( .gfin) obtained from (fh
predefined distribution g(Φ) = ( (b! ( Φn) 9u ())))) dep,, (as discussed in Section 3)
While the LF simulation is based on simplified distribution gf (gin)) (instead of g. ( (w)), it offers significant computational advantages, with cost of only 0.15 CPU hours per run-about 1 130 times faster than the HF simulation. This allows a broader exploration in the design param- eter space O. Conversely, the HF simulations are crucial for providing rigorous estimates of the 77(m) Ge background event production rate, ensuring that the optimized designs meet the stringent

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background requirements for the LEGEND experiment. In total, 4 HF and 304 LF simulations were generated to form the training dataset for the surrogate model, with the four HF samples being reused from previous simulation study, suggested in Neuberger et al. (202T).
5.2 CONDITIONAL NEURAL PROCESS RESULT
Training is performed using supervised learning, where signal label (1) is assigned to neutrons that successfully produce 77(m) Ge background, and background label (0) is assigned to neutrons that do not. The network input parameter x here consists the design parameters 0g. as well as the event-specific parameters a and & A major challenge in training the CNP is the severe imbalance between signal and background, with ratio of approximately 1:55 10 This consistent with the rare event assump- tion, where < N. To address this imbalance, we apply mixup (Zhang et al.) (018 techniques which generates new training samples & by forming linear combinations of existing signal samples 2y and background samples Ij, along with their corresponding labels yt and yy:
The network structure and motivation of the CNP are discussed in Section 4.2
â = λzy + (1 (1- and ÿ = Ayı + (1 X)yj
where À is randomly drawn from a beta distribution B(0.1, 0.1). This process introduces a weighted blend of signal and background, helping to alleviate the imbalance in the data and improve the model's generalization and robustness. To demonstrate the effectiveness of the CNP in reducing
2
27 K000
Figure 3: Comparison of raw data (red points) and CNP predictions (green points with error bars) across different design parameters. a) CNP result on the training dataset with mixup. b) predicted scores for validation dataset. c) to g) provides scatter plots of the raw metric yRaw (red points) and YCNP (green points with error bars), plotted against each design parameter in θ The CNP model offers smoother, more refined predictions compared to VRaw-
statistical variance in the design metric, calculated two different metrics for each LF simulation trial: the raw design metric yRaw (in [nuclei/ (r)))))) and the averaged CNP score SCNP. The results of both metrics are plotted against the five design parameters in Figure As anticipated, the raw metric yRaw exhibits significant statistical fluctuations that overshadows any correlations with respect to each design parameter. In contrast, the CNP score JENP reveals clear dependencies on the radius and number of panels (see Figure 3 Boottom. This indicates that CNP effectively reveal additional prior information into the MFGP.
5.3 RESUM RESULT
With -NN-ggeerated scores, the full RESuM model was trained with three design metrics at differ- ent fidelities: VRaw for HF simulations, yčKp for HF simulations, and yCMP for LF simulations. The ultimate goal is to emulate YRaw for HF simulations which provides more accurate representation of 77(m) Ge production rates under design parameter O. The MFGP model was carried out by using the Emukit python library (Paleyes et al. 2023). Figure 4 (Left Bottom) illustrates the active learning process, including the HF model prediction (Top) and the acquisition function (Bottom). The surro- gate model provides an estimate of the design metric of interest (YRow from HF simulations) along with its associated uncertainty each point in the input space. The acquisition function evolves as new data points refine the objective function approximation, balancing exploitation of known op- tima with exploration of uncertain regions. As notable in the lower panel of Figure 4 (Left), the
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03
04
31
02
116
189
Thakees
2315
0.4
03-
5333
2.2
2 3.5
34
50
200
250
20
4adi5mmm
Figure 4: Left: One-dimensional CNP-LF (cyan), CNP-HF (dark cyan) and HF (orange) model pre- dictions (dashed line) with uncertainty band (shaded area) as a function of the radius т. The lower panel shows the acquisition function as a function of the radius after each iteration. It guides the selection of future evaluation points in the input space to efficiently search for the optimal solution. Right: One dimensional model predictions over various design parameters. illustrates the sequen- tial model prediction update by adding new sampling points in each iteration.
acquisition function initially explores regions with high uncertainty, especially medium distances where the optimum is likely to be found. The active learning procedure can be iterated as long as computation resources allow. In this work, six active learning iterations performed to obtain the final result. The trends shown in Figure Feffeect well-known detector physics: r represents the moderator-detector distance, which determines the moderator's coverage volume, while denotes the number of moderator panels, where inter-panel gaps allow neutrons to escape. The dependence on other parameters are less obvious due to the schocastic nature of the system.
The HF model predictions for the 77(m) Ge production rate shown in Figure which displays one-dimensional projections LFF. from LF simulations (cyan), VCNP from HF simulations (dark cyan), and yRaw from HF simulations (orange) model predictions (dashed line) along with associ- ated uncertainties (shaded area), as functions the model's uncertainty decreased with each new sampling iteration. After six active learning it- erations, the model predictions converged on several optimal designs, as shown in Table I These designs exhibit range of configurations, with most favoring smaller panel angles &p and higher number of shorter panels, while one achieve optimal performance with fewer but significantly longer panels. Additionally, the optimal designs tend to cluster around two radii ranges, approximately 165 and 200 cm, suggesting that positioning the neutron moderator near these distances provides the best balance between effective neutron capture and material efficiency. This positioning allows for sufficient moderator mass trap neutrons while maintaining appropriate distance to minimize neutron escape. These designs outperformed those with larger gaps between panels, primarily due the increased thickness of neutron moderator shielding. The optimal design reduces the 77(m)) Ge production rate from 0.238 nuclei/ (kg yr) to 0.0798 nuclei/ (kg yr). leading to (66.5 + 3.5)% reduction in ncutron-induced background in LEGEND. Notably, RESuM identified the optimal de-
design parameters. These figures illustrate how
Table 1: Optimal neutron moderator design parameters identified by the RESuM model.
Radius [em]
Thickness d [cm]
NPanels n
Angle +p [deg]
Length T [em]
canin
RRa
[nuclei/ /kg yr)]
[nuclei /kg yr)
165.6
3.3
188
19.3
7.5
0.0798
0.0483
207.3
2.8
3.5
3.2
0.0786
0.0494
202.2
2.4
153
9.1
3.0
0.0787
0.0489
164.3
4.2
192
15.4
3.1
0.0784
0.0485
sign parameters with drastically reduced computational costs. Each HF simulation required 170 CPU hours, while each LF simulation needed just 0.15 CPU hour. If space with only HF simulations, traditional methods would have required 52,700 CPU hours to ex- plore all 310 design parameter sets. In contrast, RESuM used 310 LFF simulations and 10 HF (4 for
to explore the design
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MFGP training and 6 for active learning) simulations, totaling 1746.5 CPU hours only 3.3% of the computational resources required by conventional approaches.
MODEL VALIDATION AND BENCHMARKING
The ultimate goal of this manuscript is to learn the functional mapping f - y. Each trained model should be able to take 0 as input and predict jRaw as the predicted mean and oRaw as the prediction uncertainty. For this purpose, additional independent, out-of-sample HF simulations generated as validation dataset, and the ground truth VRaw for each HF simulation is calculated by counting the number of background events m. Given the high computational demands of HF sim- ulations and our limited resources, we generated 100 out-of-sample HF simulations at randomly sampled 0 values. We benchmarked the RESuM (MFGP+CNP) model against two baseline models: a Multi- Multi-Fidelity Bayesian Neural Network (MF-BNN,Binghard et al.] (2019); /uoo et (2022). and a standard MFGP model Paleyes et al. (2023). Due to convergence speed difference, each model was trained on different dataset variants and validated the same validation dataset. The compar- ison utilized five benchmarking metries: the Mean Square Error (MSE) between VRaw and ORaw. statistical coverage at 10, 20, and 30 coverages, and computational cost to generate training data expressed as percentage relative to traditional methods. Model selections and additional details discussed in Appendix 14 The benchmarking result is shown in Table 2
For each trial, the
Table 2: Benchmarking Result of RESuM model with respect to 3 different baseline models.
Trial
Model
Dataset (HHF
MSE
1ō [%]
2ö [%]
[%]
Cost
[%]
MF-BNN
(305,5)
0.471
100
100
100
1.7
2
MF-BNN
(307,7)
0.439
100
100
100
2.3
3
MF-BNN
(310,10)
0.021
100
100
100
3.3
MFGP
(305,5)
0.025
100
100
1.7
5
MFGP
(307,7)
0.007
31
56
79
2.3
6
MFGP
(310,10)
0.015
17
33
48
3.3
AIS-MFGP
(305,5)
0.002
39
66
80
1.7
8
AIS-MFGP
(307,7)
0.023
33
2.3
9
AIS-MFGP
(310,10)
0.001
39
64
86
3.3
10
RESuM
(305,5)
0.002
67
96
99
1.7
11
RESuM
(307,7)
0.003
92
99
2.3
12
RESuM
(310,10)
0.002
95
100
3.3
13
RESuM (100x)
(310,10)
0.003
62.4
92.2
99.6
3.3
Proper Statistical Coverage
68.27
95.45
99.73
coverage is computed by counting the percentage of ground truth yRaw that falls within the 1/2/3 Ô Raw band of URaw- Assuming standard normal distribution, the expected 2, and 36 coverages are 68. 68.27%, 95.45%, and 99.73% respectively. Among all benchmarked models, only the RE- SuM model (Trial 10-13) achieves both high prediction accuracy (low MSE) and proper statistical coverage. In comparison, the MFGP (Trial 4-6) and AIS-MFGP (Trial 7-9) models exhibit worse prediction accuracy and significant undercoverage. The MF-BNN model (Trial 1-3) yields much worse prediction accuracy with significant overcoverage. We also trained the RESuM model 100 times (Trial 13) with different randomly sampled datasets and validation sets. It shows that RE- SuM's prediction and coverage are consistently good. The coverage results, illustrated in Figure underscore RESuM's capability effectively surrogate complex detector design simulations.
LIMITATIONS AND APPLICATIONS
The primary limitation of this work is the restricted computational resources avail- able for thoroughly validating the model's performance. Due to these constraints, we were limited to generating only 100 HF simulations to assess coverage. Ideally, with unlimited computational re- sources, a comprehensive grid search across the 5-dimensional design parameter space would allow for more robust validation. Additionally, the active learning strategy employed in RESuM is rela- tively simplistic. Future work will focus on exploring more sophisticated active learning algorithms more efficiently identify the optimal design.
Limitations:

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0.5
HF Vildetion Data
RESUM =
:2c
#30
9.
02-
3.0
100
Figure 5: Coverage plot of the RESuM model predictions (Trial 9 in Table 5). The shaded regions represent the uncertainty bands different confidence levels: =1ō (green), =26 (yellow), and =: (red). The RESuM model captures the overall trend of the HF validation data with proper coverage.
Generalizability: Although this work focuses on specific detector design scenario within the LEGEND experiment, we believe that the mathematical formulation of RED problem, as outlined in Section 3 is applicable to wide range of simulation and optimization challenges in the physical sciences. A few examples are provided below:
Astromony: In computational astronomy, emerging involves simulating binary black hole (BBH) mergers to match the gravitational wave (GW) signals detected by the LIGO experiments (Fishbach and Holz (017). This process involves highly complex and computationally expensive many-body simulations (Kruckow et al. 2018). In this context, cosmological constants, such as the Hubble Constant and Dark Energy Density, can be treated as design parameters 0, while the initial position, mass, and spin of each black hole are considered event-specific parameters Φ. The design metric m is defined the number of BBH mergers occuring over given time period, with N representing the black holes simulated. The ground truth provided by the GW catalog from I.GO. Some related work in this direction include (Lin et al. 2021 Brockgaarden et al.) 2019).
First-principles simulations using Density Functional Theory (DFT) are fundamental in condensed matter research for material property prediction and design optimization (Dovesi et al. 2018 Hutter et al.) 2014, Kang et al., 2019). While DFT simulations involve multiple parameters (temperature, pressure, doping concentration) and produce specific outputs like electronic band structures, their computational expense often limits thorough parameter space exploration. This limitation drives the need for efficient surrogate models that can rapidly approximate DFT results while maintaining accuracy, thereby accelerating materials discovery.
Material Science:
8 CONCLUSION AND OUTLOOK
In this work, we presented RESuM, a rare event surrogate model designed for detector design op- timization problems in physics. We began by statistically define the RED problem and proposed CNP-enhanced surrogate model to solve it. We demonstrated the effectiveness of RESuM on opti- mizing the neutron moderator design for the LEGEND NLDBD experiment. Our results show that RESuM successfully identified an optimal design, reducing neutron background by 66.5% while uti- lizing only 3.3% of the computational resources required by traditional methods. The accuracy and coverage of the trained RESuM model were successfully validated with independently simulated HF datasets, showing that RESuM successfully emulate physical simulations with proper coverage and statistical robustness. Since all events (neutrons) are generated by traditional, first-principle-basee physics simulation framework GEANT4, we believe that the RESuM model is more interpretable compared to other black- surrogate models, such as those based on VAE or GANs (de Oliveira al. 2017 Z.. Fu et al. 2024). Although this work focuses on a specific detector design problem in physics, we believe that RED problems are prevalent in many other domains, as discussed in Sec- tion 7 Our future work will focus on two key directions: first, we want to further refine the RESuM model by validating with more HF simulations and improving the active learning algorithm; second, we intend to explore additional application of the RESuM model, especially in simulating Binary Black Hole mergers in Astronomy, as outlined in Section 7 This effort seeks to foster greater col- laboration across the machine learning, physics, and astronomy communities, ultimately benefiting all fields involved.
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9 ACKNOWLEDGEMENT
A. L. thanks the University of California, San Diego and San Diego Supercomputer Center for providing resources to this work. The work at Lawrence Berkeley National Laboratory (LBNL), including computational resources provided by the National Energy Research Scientific Comput- ing Center (NERSC), is supported by the U.S. Department Agreement AE-2200205CH 1231. The authors thank Yue Ma for her significant contribution to this work. The authors also thank Simon Mak, Yian Ma, the MODE (Machine Learning Optimized Design of Experiment) and LEGEND collaborations for valuable discussions.
Energy (DOE) under Federal Prime
10 REPRODUCIBILITY STATEMENT
To ensure that the RESuM model can be reliably reproduced, we have carefully documented all aspects of methodology and experiment. The simulation tool and package used in our work explicitly referenced in Section 5.1 The dataset, preprocessing steps, and model architecture are described in Section 4.2 Section 5.1 and Section
5.2 Model parameters and evaluation metrics are clearly defined in Section 4.2 and Section 5.2 The code of this work is anonymized and released as the supplementary material of this submission. All scripts for data handling, model training, and evaluation are included in the supplementary material, along with environment specifications and fixed random seeds to minimize variability. The training data this work too large as it involves in expensive simulations. The authors plan release training data in the camera-ready version.
REFERENCES
S. Agostinelli et al. GEANT4-a simulation toolkit. Nucl. Instrum. Meth. A, 506:250-303, 2003. doi: 10.1016/50168 (0006//0166--900000001111511 9002(00))
Allison et al. Geant4 developments and applications. IEEE Transactions on Nuclear Science, 53
(1):270-278, 2006. doi: 10.1 2TT.TNSS.200686990
Eli Bingham, Jonathan P. Chen, Martin Jankowiak, Fritz Obermeyer, Neeraj Pradhan, Theofanis Karaletsos, Rohit Singh, Paul Szerlip, Paul Horsfall, and Noah D. Goodman. Pyro: Deep universal probabilistic programming. Journal of Machine Learning Research, 20(28):1-6, 2019. URL http:://mmm www.iirrrrrppppppppee /vpeers/v20// /v20 403. html
S. Broekgaarden, S. Justham, S. de Mink, Gair, Mandel. S. Stevenson. W. Barrett, A. Gómnz, and C. J. Neijssel. stroopwafel: simulating rare outcomes from astrophysical populations, with application to gravitational wave sources. Monthly Notices of the Royal Astro- nomical Sociery, 490(4):5228 5248 09 2019. ISSN 0035-8711.
doi: 00.1099/mmrrs//t/255.88
URL /ts/htttps////..o///000.
Christian Bucher. Adaptive sampling an iterative fast monte carlo procedure. Structural Safety, ((2):119-126, 1988.
Yilun Chen, Ami Wiesel, and Alfred O. Hero. Robust shrinkage estimation of high-dimensional covariance matrices. IEEE Transactions on Signal Processing. 58(10):5016-502, 2010.
Nuclear Science Advisory Committee. A earse nuclear science, 2023. htt.::/:eemmieeeeeeeeeeeeeeeeerr
of discovery: The 2023 long range plan for
uploads/2024/002
606476 2024 LRP 11x1111111-111 www.111pp--1.22.11 1101:=---11111111111121101111111
J. Cronin and V. Fitch. The nobel prize in physies 1980: Discovery of violation in the decay of neutral kaons, 1980. UtL.bttps://www. summary/ Accessed: 2024-09-11.
Z. Cui, Q. Chen, and G. Liu. Characterization of subsurface hydrogeological structures with convo- lutional conditional neural processes on limited training data. Water Resources Research, 58(12): e2022WR033161, 2022.
11

---

Published as a conference paper at ICLR 2025
L. de Oliveira, M. Paganini, and B. Nachman. Learning particle physics by example: Location- aware generative adversarial networks for physics synthesis. Computing and Software for Big Science, 1, 09 2017. doi: 10. 00.1007/s41881-07-000044
Armen Der Kiureghian, Terje Haukaas, and Koji Fujimura. First-and second-order reliability meth- ods. In E. Nikolaidis, D. M. Ghiocel, and S. Singhal, editors, Engineering Design Reliability Handbook, chapter 14. CRC Press, 2005.
M. J. Dolinski, A.W.P. Poon, and W. Rodejohann.
Neutrinoless double-beta de- cay: Status and prospects. Annual Review of Nuclear and Particle Science, 69 (Volume 69, 2019):219 2-1, 2019. ISSN 1545-4134. doi: http://doi.org/10011466
annurev-nucl- nuc--101118-0240077
URL https: /tt///wwww www. www.aaaeeewew..orgooooooooooooee
journa 8 8/10. 110...14655 1146/anuuree
023407
R. Dovesi, A. Erba, R. Orlando, C. M. Zicovich-Wiin Wilson, Civalleri, L. Maschio, M. Rérat, S. Casassa, J. Baima, S. Salustro, et al. Quantum-mechanical condensed matter simulations with crystal. Wiley Interdisciplinary Reviews: Computational Molecular Science, 8(4),e1360, 2018.
B. Fiessler, H.-J. Neumann, and R. Rackwitz. Quadratic limit states in structural reliability. Journal of the Engineering Mechanics Division, 105(4):6611674) 1979.
M. Fishbach and D. E. Holz. Where are ligo's big black holes? The Astrophysical Journal Letters, (51,2): dec 2017. doi: 000444/20041-811//aaaa URL httthtttt:/// http:///dxxdoo.ogg//11... 006772041-8213//1
M. Garnelo, D. Rosenbaum, C. J. Maddison, T. Ramalho, D. Saxton, M. Shanahan, Y. Whye Teh,
D. J. Rezende, and S. M. Ali Eslami. Conditional neural processes, 2018. URL https://
arxiv. abg/abs//r//b//../. 1807. 01613
Michael Gelbart, Jasper Snoek, and Ryan P. Adams. Constrained bayesian optimization with noisy experiments. In Proceedings of the 27th International Conference on Neural Information Processing Systems (NeurIPS), pages 117-125. MIT Press, 2014.
Mengwu Guo, Andrea Manzoni, Maurice Amendt, Paolo Conti, and Jan S. Hesthaven. Multi-fidelity regression using artificial neural networks: Efficient approximation of parameter-dependent out- put quantities. Computer Methods in Applied Mechanics and Engineering, 389: 14378, 2022.
Baran Hashemi, Nikolai Hartmann, Sahand Sharifzadeh, James Kahn, and Thomas Kuhr. Ultra-
high-granularity detector simulation with intra-event aware generative adversarial network and self-supervised relational reasoning. Nature Commun., 15(1):4916, 2024. doi: 10.1038/
181467-024-4910442
[Erratum: Nature Commun. 1 115, 5825 (2024)).
Abraham M. Hasofer. An exact and invariant first order reliability format. Journal of Engineering Mechanics Division, ASCE, 100(1):111-21) 1974.
J. Hutter, M. lannuzzi, F. Schiffmann, and J. Vandele cp2k: atomistic simulations of con- densed matter systems. Wiley Interdisciplinary Reviews: Computational Molecular Science, 4(1): 15-25, 2014.
L. Kang, F. Liang, X. Jiang, Z. Lin, and C. Chen. First-principles design and simulations promote the development nonlinear optical crystals. Accounts of Chemical Research, 53(1):209-217, 2019.
R. Kansal, J. Duarte, H. Su, B. Orzari, T. Tomei, M. Pierini, M. Touranakou, D. Gunopulos, et al. Particle cloud generation with message passing generative adversarial networks. Advances in Neural Information Processing Systems, 14423858--23877 2021.
R. Kansal, A. Li, J. Duarte, N. Chernyavskaya, M. Pierini, B. Orzari, and T. Tomei. Evaluating generative models in high energy physics. Physical Review D, 107(7):076017, 2023.
M. C. Kennedy and A. O' O'Hag... Predicting the output from a complex computer code when fast approximations available. Biometrika, vol. 87(no. 1):1-13, 2000.
D. P. Kingma. Auto-encoding variational bayes. arXiv preprint arXiv: 2013.
12

---

Published as a conference paper at ICLR 2025
M. Kobayashi and T. Maskawa. The nobel prize in physics 2008: Discovery of the origin of cp violation, 2008. URL https:: www. http://wppp.........o./aaaaaaaaa
summary/ Accessed: 2024-09-11.
M. Koshiba. The nobel prize in physies 2002: Detection of cosmic neutrinos, 2002. URL https:
// ////w. www.:.oobllpriei... htt.://w...............eooaas//a http://///2///////////
Accessed: 2024-09.
11.
M. U. Kruckow, T. M. Tauris, N. Langer, M. Kramer, and R. G. Izzard. Progenitors of gravitational wave mergers: binary evolution with the stellar grid-based code ComBinE. Monthly Notices of the Royal Astronomical Sociery, 481 1949, 08 2018. ISSN 0035-8711. doi: 10.1093/ mnras/sty2110. URL http:::::/a...o////ppa.oe
mnras sttpaassssyy1g)
V.A. Kudryavtsev. Muon simulation codes music and musun for underground physies. Comput. Phys. Commun., 180(3):339-346, 2009, ISSN 0010-4655. doi: http::/doo.orr/10..11166. 2008.10.013.
LEGEND-CCllaboration et al. Legend-1000 preconceptual design report, 2021. URL https:
/aaxii.org//bb//1100 11462
Jing Li and Dongbin Xiu. Evaluation of failure probability via surrogate models. Journal of Com-
putational Physics, 229((3) 8980, 2010.
Jing Li, Jinglai Li, and Dongbin Xiu. An efficient surrogate-based method for computing rare failure probability. Journal of Computational Physics, 230(24)88683-86)) -697, 2011.
K. Li, S. Mak, J. Paquet, and S. Bass. Additive multi-index gaussian process modeling, with appli- cation multi physiis surrogate modeling of the quark-gluon plasma, 06 2023.
L. Lin, D. Bingham, F. Broekgaarden, and Mandel. Uncertainty quantification of computer model for binary black hole formation, 06 2021.
M. Neuberger.
warwick-legend.
httpss//gghhub..ccm////riixeeeer
warwick-legend
M. Neuberger et al. Phys.: Conf. Ser. 2156(012216), 2021.
A. Paleyes, M. Mahsereci, and N. D. Lawrence. Emukit tutorial: Experimental design introduction.
2019.
A. Paleyes, M. Mahsereci, and N. D. Lawrence. Emukit: A Python toolkit for decision making under uncertainty. Proc. Python Sei. Conf., 2023.
lason Papaioannou, Costas Papadimitriou, and Daniel Straub. Sequential importance sampling for structural reliability analysis. Structural Safery. 62:66-75 75,66-75,226 2016.
Benjamin Peherstorfer, Tiangang Cui, Youssef Marzouk, and Karen Willcox. Multifidelity impor- tance sampling. Computer Methods in Applied Mechanics and Engineering, 300:490-509, 2016.
Benjamin Peherstorfer, Boris Kramer, and Karen Willcox. Multifidelity preconditioning of the cross- entropy method for rare event simulation and failure probability estimation. SIAM/ASA Journal on Uncertainty Quantification, 6(2):737-761, 2018.
P.Z.G. Qian and C.F.J. Wu. Bayesian hierarchical modeling for integrating low-accuracy and high- accuracy experiments. Technometrics, 50(2):192-204, 2008. URL httpps::///oo..o///111
1198/0040170000000000
Y. Ramachers and Morgan. werwick-legend. wa warwick-legent
stttps///gittubb.cm//bbeeee
K. Ravi, V. Fediukov, F. Dietrich, T. Neckel, F. Buse, M. Bergmann, and H.-J. Bungartz. Multi- fidelity gaussian process surrogate modeling for regression problems in physics, 2024. URL
https::/arxiv.cot http:///wwiivv.orr////////////// .1966 11965
13

---

Published as a conference paper at ICLR 2025
J. Requeima, J. Gordon, J. Bronskill, S. Nowozin, and R.E. Turner Fast and flexible multi-tass clas- sification using conditional neural adaptive processes. Advances in neural information processing systems, 32, 2019.
Peter J. Rousseeuw. Silhouettes: A graphical aid to the interpretation and validation of cluster analysis. Journal of Computational and Applied Mathematics, 20(1):53-65, 1987.
Sacks, W. J. Welch, T. Mitchell, and Wynn. Design and Analysis of Computer Ex- periments. Statistical Science. 4(4):409 - 423, 1989. doi: 10.2214//11170122113 URL
Vallecorsa. Generative models for fast simulation. Journal of Physics: Conference Series, volume 1085, page 022005. IOP Publishing, 2018 8.
A. Vaughan, W. Tebbutt, S. Hosking, and R. E. Turner. Convolutional conditional neural processes for local climate downscaling. Geoscientific Model Development, 15(1):251-205 2022.
Felix Wagner, Jonas Latz, lason Papaioannou, and Elisabeth Ullmann. Multilevel sequential impor- tance sampling for rare event estimation. SIAM Journal on Scientific Computing, 42(4):A2062- A2087, 2020.
Felix Wagner, lason Papaioannou, and Elisabeth Ullmann. The ensemble kalman filter for event estimation. SIAM/ASA Journal on Uncertainty Quantification, 10(1):317-349, 2022.
Z. Fu, Ch. Grant, D. M. Krawiec, A. Li, and L. A. Winslow. Generative models for simulation of kamland-zen. The European Physical Journal C, 84(6):651, 2024.
H. Zhang, M. Cisse, N. Dauphin, and D) Lopez-Paz. mixup: Beyond empirical risk minimization, 2018.
APPENDIX
11 MULTI-FIDELITY GAUSSIAN PROCESS
Our implementation builds upon the architecture described Ravi al. (2024). For completeness, we provide a succinet overview of the essential components below.
In Gaussian process (GP) a function f(0) is modeled as:
f(0) ~ (P(,(0),(0000))
where k(0 0') is the covariance function.
Given data Dx {0,y), the posterior mean and variance at a new point O, are:
H. K(O,, (),
o² = 0.) - K(O.,
where K is constructed by evaluating the covariance function A at the corresponding input points. This model represents the joint distribution of multiple fidelities as a multivariate GP with specified covariance structure. The covariance matrix in this model includes both correlation terms between fidelities and discrepancy terms within fidelities.
We assume additive relationship between the High and Low fidelities, that the HF model f (0) could be expressed in terms of the LF model (0) with discrepancy 8(0):
fn (0) (f) 5(0)
where p scaling factor and 5(0) another random variable modeled as a GP. According this relationship, the joint distribution of the two random variables f2(0) and fu (0) could be calculated
~N( ~N(1, (H()))
f2((0)
(0.(Kr KLL. (KIL
K + Ks )
PKLI.
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Here KLL and K5 are the covariance of fL (0) and 8(0). For more fidelity levels, the method recursively applies:
- + (((()))))))
12 ACTIVE LEARNING STRATEGY
We use integrated variance reduction to quantify and minimize the expected posterior variance over the input space 0. The goal of integrated variance reduction is minimize the total variance across the design space:
I(O / [ (ppiior(0)) (0') posse (0' 0)] d0',
where oprior (f') denotes the prior variance of the GP at point O' and of post (0' 0) is the posterior variance 0' given observation 0, The new sampling point Onew is selected by maximizing this integrated variance:
θ new arg max I(0") f'
In the case of a Gaussian Process (GP), the acquisition function I(0) defined by integrated variance reduction simplifies to the Lower Confidence Bound (LCB) acquisition function under the assump- tion that the variance reduction directly translates to balancing the predicted mean and uncertainty:
#samples
k0,
(0)
#samples
o²(0)
with o²) representing the predictive variance at the observed point and k is the Radial Basis Function (RBF) kernel with 0; sampled randomly (Paleyes et al. 2019). Furthermore, we aim to optimize the acquisition function under parameter constraints-whic limit the allowable values of the design parameters- using gradient descent algorithm to locate the global maximum. These constraints ensure that the parameters remain within feasible ranges, reflecting the practical and structural requirements necessary for maintaining the integrity and functionality of the overall de- sign. The constraints are incorporated into the acquisition function by adding a mulitiplicative factor nullifying the acquisition function's 's values when constraints violated and thereby discouraging the algorithm from exploring infeasible regions (Gelbart et al. 2014).
13 CONDITIONAL NEURAL PROCESS
The core principle of the Conditional Neural Process (CNP) framework is to approximate arbitrary random processes using Gaussian sampling, where the mean μ and variance neural networks. In this section, we show that the CNP score / can be viewed as an estimate of t(0,)) Φ). with the Gaussian distribution representing the posterior of i(0, Φ). Furthermore, the RED problem can be aligned with the thcoretical framework of the Variational Autoencoder (VAE), as detailed in Kingma (2013), with E () interpreted as the latent vector.
parameterized by
We begin by formulating the RED problem within a Bayesian framework: The training data consists of finite set of X values generated through simulation, and the objective is construct en- coder that approximates the posterior distribution of t(0, Φ), conditioned on the observed dataset {Xas, Φki, xx).
q(t(O, x)) X Φki, 0x) p(((0, x) Φki, 0k.
(8)
According to Bayes' theorem, the conditioned posterior in Eq. hood of the observed dataset and the prior of (0,
3 could calculated with the likeli-
((t(0, Xxis Pkis 0x) x L(N Φ OR, 1(0, (t())))( Φ))
The prior p( x)) is conventionally set constant. Let LA represents the k-th simulation, the combined likelihood of the full dataset is therefore:
K
K No
L(Xxd Pes 0 t ())
II
c.
Bernouli(x = XxslP = t(0 obaa)) (10)
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Note that the ground truth of t(0,, Φ) is unknown. In the Bayesian framework, can only estimate it with probability density function, which is ((t(O, Φ)). The estimation of the likelihood is therefore:
M No.
C L(Xillkii Rn, (())
Bernoulli(:: Xx: 1(0k,, ( 4))dt
(11)
k=1
It is important to note that the quantity t here is not variable but a function, and the integration is performed in the Hilbert space. The problem, therefore, becomes to find the function q" (t(O, Φ)) so that:
max q' = arg (EI(0,0)
L(Xxi) On: (()))
(12)
Here f0, Φ) represents the set of arbitrary normalized functions of Φ). While this optimization problem is mathematically solvable, the integration computation in Hilbert space is mathematically non-trivial and computationally expensive, rendering this solution impractical.
Then, here comes the CNP model, which simplifies and tackles this optimization problem in the following steps:
First, approximate with parameterized Gaussian, which is natural choice if we regard the task as a statistical parameter estimation for ((,, s):
qN (t(O, Φ)) (NNN o, w) NN (0, ())) &u..0
(13)
Then, the neural network is trained minimize the likelihood deseribed in Eq. [T] Therefore, the CNP model actually performs statistical estimation for I(0,, Φ) by approximating the posterior with Gaussians.
If we consider 1(0, Φ)) as the latent vector in the VAE model, the pre-defined Bernoulli process acts as probabilistic "decoder", which generates data given the latent vector.
With this framework, our task can be generally described as follows: Assume we have predefined probabilistic decoder, p(æ c). where represents the latent vector and c the condition. Ad- ditionally, have simulation informed by domain knowledge that generates data based the nominal latent vector =", though its exact value is unknown. Our objective is to develop a surrogate model that performs statistical estimation, represented ((2(ar) to infer the nominal latent vector from the simulated data æ. The posterior distribution q(= x)) serves the same role as the proba- bilistic encoder in VAE model. We can then sample the latent vector z from g(2(ar), which can subsequently be used to generate the "reconstructed" (surrogate) data æ'. The illustration of this architecture shown in Figure
Conditions
Simuluted Data
Conditions
Surrogata Data
Surogate
Probabiliatic
2
Decoder
[x xx)
Z
X'
p(x|2,c)
Nominal Latent (unknown)) function
Sompled
Figure 6: Architecture of Surrogate Model for the RED Problem
In this work, as the primary objective is to optimize the design parameters rather than generate ad- ditional simulated datasets, we utilize only the averaged CNP score 3 as input to the subsequent MFGP model. Nevertheless, it is important to highlight that the trained CNP model has the poten- tial to generate surrogate data as fast simulator. This capability can be especially beneficial in other studies where the detailed distribution of the latent vector parameters is of interest. enabling more efficient exploration of the parameter space and supporting applications such as uncertainty quantification and model validation.
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For the RED problem, specifically, t(0, Φ) corresponds to the latent vector 2, but with infinite dimensions, as represents a continuous function over (0, Φ). (0, Φ) acts as the condition c, and the fusion of latent vector and condition is performed by plugging 0, Φ into the function t to get t(6,00 0)
It is worth noting that the latent vector, denoted by 2, in our model, is not conventional finite- dimensional vector but rather a vector situated in Hilbert space. Dimensionality reduction can be achieved through quantization, which involves discretizing the space by creating a grid over (0, Φ) and computing values only at the selected grid points. An alternative approach is to project the latent vector onto a set of basis functions and impose a cutoff. For instance, with Fourier decomposition, the vector's 's coefficients can be retained up to certain frequency limit, effectively serving as a high- frequeney cutoff. In this work, we adopt the CNP model to represent with a fixed dimensionality.
14 MODEL BENCHMARKING DETAILS
Training and Validation Data Additional independent, out-of-sample HF simulations are generated as validation dataset. Given the high computational demands of HF simulations and our limited resources, generated 100 out-of-sample HF simulations at randomly sampled O values. The training dataset is identical to the RESuM training data described in Section S comprising 310 LF and 10 HF simulations. We further divided this dataset into three subsets: Small (305 LF + 5 HF), medium (307 LF + 7 HF), and full (310 LF + 10 HF). The small dataset establishes a threshold, meaning our RED problem at hand necessitates a minimum of 305 LF data samples to adequately capture the mapping of the 5-dimensional design space, as subsets with fewer samples would lack sufficient coverage. Additionally, some experiments reveal that at least 5 HF samples are required to establish a reliable correlation between LF and HF, enabling meaningful integration of the two fidelities.
Baseline Models There are two requirements for selecting baseline models: (1) the model must be multi-fidelity, and (2) it must be able to produce uncertainty estimates. Given these constraints, we selected two baseline models for benchmarking against the RESuM model (which combines MFGP and CNP): the Multi-Fidelity Gaussian Process (MFGP), and the Multi-Fidelity Bayesian Neural Network (MF-BNN). Leveraging several information sources, the MFGP and the RESuM frameworks employ a co-kriging approach such that the high-fidelity output is represented as: Yht = p Yhr + 6, where p is scaling factor and ó a discrepancy term -both inferred from the data. The MFGP and the RESuM model follow the architecture described in Chapter[[] In case of the Bayesian Neutral Network (BNN) (Bingham et al., (201)), we implemented Multi-hdelity net- work by adopting a hierarchical architecture inspired by Guo et al. (2022). The MF-BNN integrates low- and high-fidelity data using two interconnected BNNs.
Each model, including RESuM, was trained on small, medium, and full training datasets and val- idated on out-of-sample validation dataset. The benchmarking metrics, described in the next paragraph, were evaluated for each model. Additionally, to further demonstrate the robustness of the RESuM model, we performed 100 iterations, where each iteration involved randomly splitting the 410 LF and 1 110 HF samples into a full training dataset and validation set (100 HF samples). The benchmarking metrics were then computed as averages across these 100 iterations to ensure statistical reliability and consistency.
Benchmarking Metric The ultimate goal of this manuscript is learn the functional mapping f: 0 ->>>> y. Each trained model should be able to take O as input and predict jRaw as the predicted and yRew as the prediction uncertainty. Meanwhile, the ground truth yRaw for each HF simulation in the validation dataset is determined by counting the number of background events m. Based these variables, we propose five validation metrics to assess the accuracy and coverage of each model:
MSE: the mean square error (gRaw - )Raw)? summed over 100 HF validation datasets.
1/2/3al %1: J: The 1/2/3a coverage of model prediction. This metric calculates the percent- age of validation dataset for which yRaw falls within ynaw + 1/2/3ōRmw Assuming standard normal distribution, the expected coverage at #lōraw +26 and +3ō are 68.27%. 95.45%. and 99.73% respectively.
Cost: The computational cost of generating the training data is shown as a percentage relative the cost of traditional method (52,700 CPU hours). This metric depends only on
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the size of the training dataset, with values of 1.1%, 2.3%, and 3.3% for the small, medium, and full datasets, respectively.
Benchmarking Results To provide comprehensive understanding of the benchmarking results in Table 2 we include detailed plots in Figure / illustrating the statistical coverage for each baseline model for the full dataset (Trial Trial and Trial Table 2). The plots in Figure 7 visualize the percentage of ground truth VRaw (black marker) for all trials of the validation dataset that falls within the 1 Raw Raw and Raw bands (shaded area) of the model prediction under study.
RESuM
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Figure 7: Validation comparison of RESuM (Upper) with the simplified MFGP (Upper Middle) AIS- MFGP (Lower Middle) and MF-BNN (Bottom) models using 310 LF and 10 HF data samples during training. Both comparison models fail to adequately describe ÿRaw. as demonstrated by their overly narrow (in case of MFGP) or wide (in case of MF-BNN) prediction bands and poor alignment with the HF validation data. Despite the narrow (wide) uncertainty bands, the predictions lack physical relevance, highlighting the models' inability to capture the complexity of the design space without the CNP model.
When comparing RESuM (Trial 12, Figure 7 (Upper)) to simpler MFGP model (Trial 6, Figure (Upper Middle) that only utilized the LF VRow and HF Yıaw of the exact same LF and HF data (excluding the CNP outputs JCNP), the contrast in outcomes were striking. The simplified MFGP model not only failed to capture the complex dependencies between design parameters, but its pre- dictions were also physically inconsistent. As shown in Figure 7 (Upper Middle), the prediction
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bands for the simplified MFGP model are excessively narrow and do not capture the actual vari- ability of the HF validation data. The model's inability to describe jRaw is evident, as does not adequately reflect the complex interactions within the design space. The model's Raw Rō and 3oRaw confidence intervals are unrealistically tight, providing little insight into the true uncertainties of the system. With that, the coverage R/raw 20 and Rew the simplified model was only 17%. 33% and 48%, a notably poor result compared to the RESuM model, which achieved a Lōraw, Row. and Raw coverage of 69%, 95% and 100% with much more realistic uncertainty predictions. Similarly, the MF-BNN model (Trial 3, see Figure 7 Bottom) failed to adequately represent the data, resulting in predictions that lack physical interpretability and deviated from the system's 'ss expected behavior. The poor performance of the MF-BNN model can probably be attributed to the insuffi- cient amount of training data, particularly the limited number of HF samples. The extremely wide uncertainty bands produced by the MF-BNN highlight the high degree of uncertainty in its predic- tions, further emphasizing its inability to effectively learn from the sparse training data. Mcanwhile, an Adaptive Importance Sampling (AIS) approach combined with MFGP achieved 1ō Rōw and Rr coverage rates of 39%, 64%, and 86%, respectively. These results indicate that the AIS approach manages to sample from relevant regions and produce physically interpretable predictions, however, its coverage and overall performance remain significantly below RESuM's ' levels.
These findings underscore the complexity of the design space and the limitations of traditional Gaus- sian regression or neural network approaches with limited HF data. In contrast, the CNP-enabled RESuM model effectively reduced statistical variance and provided meaningful physical insights by capturing complex relationships between design parameters O and event-specific parameters Φ.
15 ADAPTIVE IMPORTANCE SAMPLING
major challenge in the Rare Event Detection (RED) problem is the limited number of signal events generated by low-fidelity (LF) simulations. To address this, we implemented Adaptive Importance Sampling (AIS), which strategically concentrates sampling efforts in regions more likely to produce signal events. In our standard LF simulations using uniform sampling, only 15-30 signal events occur per 50,000 neutron simulations.
We aim to conduct benchmark comparison where we fed the AlS-enhanced L.F simulations into the Multi-Fidelity Gaussian Process (MFGP) algorithm to compare its performance against the RESuM model. The AIS+MFGP benchmark can be considered as an ablation test of the CNP model. In this section, we focus on explaining the technical details of our AIS algorithm implementation.
Building on the rare event problem setting outlined in Section B and using the adaptive sampling framework, the expected probability t(Ox) of a signal event for a fixed design parameter 0% is calculated as the expectation value of 1(xx: Φ) over the target distribution f(Ø 0z), where [(Φ | 0x) replaces g(b) in Equation 2 It is closely related to g(h 0k), the distribution of all event- specific parameters in the simulation. Specifically, ((b (%) can be expressed as:
f(b 0x) = g( Ok) tig (kk Φ).
where taig (Ok. b) represents the probability of triggering signal given the detector design 0₂ and the event-specific parameters Φ. The trigger probability faig (fx), representing the overall probability of the Bernoulli process for signal events given Oz, defined the expectation value of the indicator function laig (c)), which evaluates to 1 if a signal event occurs and 0 otherwise. Mathematically, is expressed as:
1sig (0k) = Tas (
0g) de.
This integral represents the wenshty fraction of the parameter space that contributes to signal events. In practice, {(0x) is approximated using Monte Carlo sampling as the weighted fraction of signal-producing samples from the target distribution. Specifically, adaptive sampling begins with a broad initial proposal distribution q( O2, 0), chosen to cover the entire event-specific parameter space & for the given design 4₁. In each iteration I, samples Φ; are drawn from the current proposal distribution q();; 0x. l). These samples are evaluated under f( 0x) to compute importance weights:
f(b. 0%) u4
a( On. 1)
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The proposal distribution is updated iteratively to better approximate f(ø 0k)- For Gaussian proposals, the weighted samples are used to recompute the mean and covariance matrix:
Σ Mi -= wiP: EE Wi
No wi 1 ()= M1+1)
14+1
E
W4
This iterative process continues until Φb; 0k. 1) sufficiently concentrates the high-probabiln
signal regions in & for the fixed design 2z. By iteratively refining the sampling distribution, adaptive sampling reduces variance in the estimation of t(Ok))
We apply the Adaptive Importance Sampling (AIS) framework to each LF simulation with a fixed de- sign parameter OR. allowing efficient approximation of the conditional signal distribution f(Ø 0k) and increasing neutron capture occurrences. Initially, a proposal distribution q(Φ:; tu. 0) is con- structed using k-means clustering on event-specific input parameters (d [d=33 2), such as position, mo- mentum, and energy. Clusters emphasize regions where the moderator strongly influences neutron interactions. In each iteration 4, the AIS algorithm draws samples Φi from the current proposal and assigns importance weights w;, reflecting the ratio of the target density to the proposal density. These weighted samples drive updates to the Gaussian mixture components of q, recalculating their means and covariances to better approximate f | 0%) Covariance matrices are regularly checked and "regularized" for numerical stability, while an optional Principal Component Analysis (PCA) step may reduce dimensionality to further stabilize updates. By iteratively adding and adjusting Gaussian components based on where weighted samples cluster, the proposal distribution adapts toward regions deemed most significant by the target density.
In each iteration, once the samples are drawn and their importance weights are computed, a simula- tion run is performed with these sampled points and the results are added to the existing sample pool, forming an aggregated distribution. This aggregated distribution is then included the denominator of the weight calculation. This helps maintain broader coverage, prevents mode collapse, and lever- ages prior knowledge. As the algorithm progresses over multiple iterations, this aggregation acts as "safety net" distribution, maintaining exploration in under-sampled regions while still allowing the main proposal components to adapt toward regions of high target density. Diagnostics such as Effective Sample Size (ESS) are monitored gauge performance and convergence. with higher ESS indicating that the proposal distribution is capturing the target distribution more effectively.
Challenges arise from the limited signal-to-backee backggood ratio (2 10²) in LF simulations, which impedes identification of statistically meaningful clusters. To ensure statistically meaningful covari- ance matrix estimation, a heuristic suggests that each cluster must contain at least d'd + 1) samples (Chen et al., 2010). The initial proposal is derived from 40% of the available event samples for RESuM analysis. Using 2 clusters and with fewer than 15 signal events on average per initial proposal, this heuristic is not satisfied in early stages, shown in Figure g Nevertheless, posely do not merge data across different LF simulations for the AIS k-means clustering because each simulation uses distinct, fixed design parameter 0. The merges of all data shown in Figure g are for demonstration purposes only. For evaluation of the quality of the clustering, we show the silhouette score, which measures how similar data point is its assigned cluster compared to other clusters. A higher silhouette score indicates better-defined and well-separated clusters. The incorporated constraints focus sampling on regions of relevance while maintaining well-separated clusters with silhouette score (Roussecuw) [1987) exceeding 0.5 (see Figure 8 Bottom). This bal- ance between exploration and exploitation ensures robust initialization and refinement of the AIS proposal distribution.
pur-
The AIS framework in combination with a MFGP achieved coverage rates of 39%, 64%, and 86% for 10 Rā and 3/ respectively. While the AIS approach has demonstrated its ability to improve the efficiency of LF simulations by increasing the sonnal-t-backkrouudd ratio and di- recting sampling toward relevant regions, it still does not achieve the performance level of RESuM within the same 50,000-sample budget (see Figure (Lower Middle)); additional samples would be required to ensure more robust coverage and convergence.
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030
CM0
030
02.
Momber Cuaters
Figure 8: Visualization of signal samples and K-means cluster centers (top row) and corresponding silhouette scores (bottom row) for different data conditions. The first column shows the clustering of all signal samples using 100% of the available data, while the second column shows clustering with only 40% of the available data. The third column shows the signal distributions of one LF simulation using 40% of the available event samples. The bottom row illustrates the silhouette score as a function of the number of clusters for each corresponding scenario, highlighting the clustering quality under different data availability and conditions.
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Reviews:
[{"review_id": "eeq6vdGzdC", "reviewer": "Reviewer_BhVu", "summary": "The paper presents a surrogate model for rare events in physics.  The model is a hybrid model combining a conditional neural process (CNP) model and a multi-fidelity Gaussian process (MFGP).  The CNP is pre-trained on both low-fidelity (LF) and high-fidelity (HF) simulation data.  Subsequently the CNP model is used to calculate additional design data (averaged CNP scores for LF and HF data).  The MFGP model is initially trained using HF, HF CNP and LF CNP data, and further tuned using Bayesian Optimization (BO) equipped with the integrated variance reduction acquisition function to minimize the total variance of the model.", "questions": "In addition to the above:\n\n- I am a little unclear regarding the role of CNP here.  Is it primarily a means to include the LF simulations in the overall model in a way that takes advantage of the physical insight in the combined LF/HF data?\n- in the active learning / Bayesian optimization phase you only run HF simulations.  Have you considered using multi-fidelity BO [1,2] to take advantage of the cheaper LF simulations as well?\n- As a minor note, on line 53 you say that N is $\\mathcal{O} (10^4)$.  Do you mean $N$ is of order $10^4$?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The RESuM model is well-motivated and the application in physics is quite interesting.  While I am not a statistician nor familiar with the CNP model, I am very familiar with GP models and BO design, and can see no obvious mistakes in the paper.  Moreover the results are thorough and I can see a clear application for this model outside of that considered here.", "weaknesses": "It appears that the real goal of this paper is to optimise particle detector design.  As such, it strikes me that the surrogate model need only be accurate in those regions of design space that are \"good\" - ie minimise background noise.  However the RESuM model is designed to model the entire design space, including \"bad\" designs (eg it may request HF simulations for regions whose prior confidence bounds are such that we know whp will not be appropriate in practice).\n\nAs such perhaps a more efficient approach would be to replace the integrated variance reduction acquisition function used in the BO portion of the design with a more goal-oriented acquisition function like expected improvement (EI) or GP-UCB, which would naturally focus computational effort on optimizing the detector design (ie the real underlying goal) rather than modeling all possible detectors (essentially the first step in the current design procedure).\n\n(Note I intend this more as a point for discussion rather than a major criticism - the current approach is valid, but perhaps the strong focus on modeling is obscuring the real goal).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "OUFITB0bFa", "reviewer": "Reviewer_Y2Co", "summary": "The article presents a study on the optimal design of a neutrinoless double-beta decay detector, a Rare Event Design. The authors introduce a surrogate model, RESuM, to solve this optimization problem and gain computational cost. This algorithm is based on Conditional Neural Process, incorporating prior information. The authors applied their work to the LEGEND experiment, a physics detector for neutrinoless double-beta decay.\n\nThe Rare event Design problem is mainly well stated and very clear.  The Large and Small N scenarios provide a very good understanding of the problem.\n\nBayesian prior knowledge with Conditional Neural Process (CNP) is clear in the text. CNP is explained in Appendix 11. The details shows the CNP is rich : by the use of surrogate modelling it generates data as fast simulator. It is more efficient for exploration of parameter space, uncertainty quantification and model validation.\n\nThe simulations, results and validation of RESuM are well explained.\n\nRESuM reduces LEGEND neutron background by (66.5 \\pm 3.5)% using only 3.3% of the computational power of traditional methods. This is a very good achievement.\n\nAll in all, this is a very good paper.", "questions": "Line 53, this is interesting, where does this order of magnitude 'N needs to be extremely large (O(10^4))'come from? Could you give a quick explanation or a reference, please?\n\nIn part 5.0.2. why are relationship dependencies clear for R and N and not for the thickness and the angle? What does this implies on the model's interpretation or performance?\n\nLine 171, the nuisance parameters are introduced quickly. Can you provide a brief explanation and give a reference, please?", "limitations": "", "rating": 10, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "The study is well conducted, very clear and concise, very interesting. The stated results are also very impressive and I hope this will lead to a larger breakthrough regarding neutrinoless double-beta decay community. The code is well presented and commented.\n\nThe work can be applied to Astronomy or Material Science. There are many other application possibilities in physics.\n\nThe work is motivated by the experimental discovery of neutrinoless double-beta decay (NLDBD), leading to answering the question ‘Why is there more matter than antimatter in our universe?’. Which makes a link with one of the opening questions of Javier Duarte talks at ICML24 ‘What is our universe made of?'. Thus, this work is very relevant for the community, and source of enthusiasm since the potential discovery of NLDBD would  lead to a Nobel-Prize-level breakthrough in physics.", "weaknesses": "The authors indicates in the limitations and applications part that due to limitation of ressources available they computed only 100 High Fidelity simulation, maybe the publication of the article will help to convince to gain more ressources.\n\nThe authors also add that the active learning functions used un RESuM are simplistic, they will consider more sophisticated one in future work.\n\nMinor comments that do not impact the score :\n\nThey are many inconsistencies in the notations, here are what I noted while reading, some might be redundant\n\nLine 95, independent of the other events\n\nLine 97, \\mathbf{Phi} is not defined. Is it for conciseness?\n\nLine 97,  dollar i dollar-th\n\nLine 133, dollar { y \\in  blablabla } dollar should make it appear on the same line\n\nLine 153-154, Why highlight +1 and +0? Should be in dollar 1 dollar, same for dollar 0 dollar\n\nLine 186, \\mathbf{\\phi}\n\nLine 187, the same\n\nLine 188,  \\mathbf for theta\n\nLine 172, The use of the nuisance parameters is unclear to me. The vertical bar followed by X_ki, phi_ki, theta_k means conditionally, if possible this should be improved.\n\nLine 183, Figure 1 title is ‘Overview of the RESuM framework’ which is not the case of Figure 3. I assume the authors mean Figure 1 then.\n\nIn Figure 1, if the MFGP combines prediction from LF and HF simulations, why are the arrows pointing at them and not starting from them?\n\nIn the section 3, the vector of design parameters is \\mathbf{\\theta} thus it should be the same in all the study.\n\nIn the introduction, line 49, 51, 52. According to section 4 it should be in \\mathbf?\n\nIn the section 3,\n\nLine 92-93, is it in \\mathbf?\n\nLine 96, in \\mathbf\n\nLine 108-109, the same\n\nLine 113-114, the same for \\theta and \\phi\n\nLine 120,  the same for \\theta, N should be in  dollar N dollar\n\nLine 124-125, the same for \\theta\n\nLine 126, the same for \\theta twice in the eq\n\nLine 126-127, \\infty should be used and same for \\theta\n\nIn the section 4,\n\nLine 140, mathbf for \\theta\n\nLine 151, the same for \\theta_k the k-th simulation trial\n\nLine 152, the same for \\theta_k and \\phi_ki\n\nLine 153, it should be  dollar N dollar,  dollar y dollar also\n\nLine 160-161, \\mathbf for \\theta_k and \\phi_ki\n\nLine 187-188, \\mathbf for \\phi and \\theta\n\nLine 204, the same for \\theta\n\nLine 209, the same for \\theta_k and \\phi_ki\n\nLine 212-213, the same for \\theta and \\phi_ki\n\nIn the section 5,\n\nLine 261-263, the parameters of design should be the same as in the figure 2 (\\varphi in particular). Figure 2 represents explicitly 4 parameters, n is missing\n\nLine 265, \\mathbf for the design parameters and the corresponding space\n\nLine 289-290, \\mathbf for \\phi\n\nLine 298, the same for \\phi\n\nLine 360, I suggest ‘plotted against the five parameters in Fig3.’ Since you only present five.\n\nLine 370-371, the same for \\theta\n\nLine 395-400, parameters should be consistent with line 261-263\n\nLine 404-405, parameters should be consistent with line 261-263\n\nLine 409, also\n\nFigure 4 and Table 1, should be consistent with line 261-263\n\nLine 441-446, the same for \\theta and the corresponding space\n\nIn section 6,\n\nLine 491 and 492, the same for \\theta and \\phi\n\nIn section 2,\n\nLine 93-93, I would have introduced the stochastic process X_1, …, X_N and specify after that\n\nLine 213, \\mathbf for theta and phi\n\nLine 222, ^{HF} is necessary for y_CNP\n\nLine 224, ^{LF} is necessary for y_CNP\n\nIn the supplementary materials\n\nLine 738, \\mathcal{L}_k appears two times\n\nA.9 must be improved:\n\nAre \\theta and \\theta’ vectors as in the manuscript?\n\nWhat kind of matrix is K? What are K_LL, K_\\delta?\n\nCould you give a reference or improve the theory, please?\n\nA.10 \\Theta must be in bold\n\nIs \\theta a scalar or a vector?\n\nCould you give a ref or a proof for the approximation of I(\\theta) for GP model? It seems simple, but I would like to be convinced this is right.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ntZ38JK4I9", "reviewer": "Reviewer_aEWx", "summary": "This paper addresses a key question in physics: the matter-antimatter asymmetry in the universe, focusing on neutrinoless double-beta decay (NLDBD). The authors tackle the challenge of optimizing detector designs to minimize background event contamination, framing it as a Rare Event Design (RED) problem.\nThey introduce the Rare Event Surrogate Model (RESuM), which combines a pre-trained Conditional Neural Process with a Multi-Fidelity Gaussian Process to optimize detector designs. Applied to neutron moderator designs for the LEGEND NLDBD experiment, RESuM achieves a 66.5±3.5% reduction in neutron background while using only 3.3% of the computational resources of traditional methods. This innovative approach has broad implications for similar challenges in physical sciences.", "questions": "Are there no other works attempting to solve the same problem that we can use to benchmark your work against?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The authors did a great job narrating each step of the procedure and model description. However, since I am not in the field, I got a little lost in the setup of the Experiment section, which involves applications. The results are convincing, though.", "weaknesses": "- It would be better if the authors provided more detailed explanations in the Experiment section.\n- Several mathematical symbols are not in math mode.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "rn7s5plZlC", "reviewer": "Reviewer_KsYG", "summary": "This work presents a generative approach for optimizing physics detector design in rare event scenarios by leveraging a Conditional Neural Process (CNP). By incorporating techniques like simple co-kriging, data augmentation, and multi-fidelity modeling, the authors demonstrate that their surrogate model effectively addresses the computational demands of rare event detection while maintaining the accuracy.", "questions": "- This works uses CNP as the major tool for rare event modeling. Is it model originally proposed by the authors? If not, please provide the corresponding citations. Also, please give the motivation here. Why CNP is more advantageous than VAE/GAN in this setting? \n\n- In line 51, the authors said \"If $m_1/N_1 < m_2/N_2$, it suggests that the design $\\theta_1$ is better than $\\theta_2$.\" This statement holds only if $N_1, N_2$ are sufficiently large, right? Otherwise, since it is in a rare event scenario, a bad $\\theta$ may also lead to zero values when $N$ is not large enough. \n\n- It looks like $X_{ki}$ is equivalent with $t(\\theta_k, \\phi_{ki})$, why use two separate notations? Please correct me if I am wrong.\n\n- Formula (4): I don't understand what does Bernoulli($p=t(\\theta_k, \\phi_{ki})$) mean here. Do you mean Bernoulli($p=\\bar{t}(\\theta_k)$)? Also, the letter $p$ has been used as density function. \n\n- The variable $\\beta$ is applied in different versions without clear definitions. In line 160, the authors introduce $\\beta_{ki}$, while $\\beta$ appears in formula (5) without clearly showing its dependency. My guess is that $\\beta_{ki} = \\beta(\\theta_k,\\phi_{ki})=\\beta\\vert\\theta_k, \\phi_{ki}=\\beta$. Please clarify. \n\n- Line 182, typo: \"RESuM as shown in Figure 3.\" Here it is supposed to be Figure 1.\n\n- Line 230: you don't need to give the fourmulation of the acquisition function, since $\\boldsymbol{x}$ is not introduced here.\n\n- Line 299 and line 308: The density $g(\\phi)$ for LF and HF models are different, right? If so, please use different notations for them.\n\n- Line 517-520: \"Based on the statistical formulation and ... for accelerating simulations. \" I don't see the comparison between the proposed method with VAE/GANs, could you please clarify how you draw this conclusion?", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The authors provide a solid background introduction to the NLDBD problem and offer clear, detailed explanations of the technical methods used in this work, including co-kriging and Conditional Neural Processes (CNP), in the appendix.", "weaknesses": "Some of the critical weaknesses: \n\n- While the authors provide a solid introduction to NLDBD and its motivation, the paper lacks a literature review on rare event simulation/modeling. Rare event simulation is a well-established field in engineering, including techniques such as the first/second-order reliability methods (FORM/SORM) [1, 2, 3], polynomial-based surrogate modeling [4, 5], adaptive and sequential importance sampling [6, 7], and ensemble Kalman filters [8]. These approaches have also been extended to multi-fidelity settings, as seen in [9, 10, 11]. Although it’s not necessary to cite all related works, the authors should clarify why they focus on deep generative modeling, highlighting its advantages over traditional models.\n\n- The computational results of the proposed method appear to lack a baseline, making it challenging to assess performance without a fair comparison.\n\n- Some notations are not statistically rigorous. Specific comments on notations are provided in the questions section. Also, the variables $\\phi$ and $\\theta$ are inconsistently bolded, which can cause confusion. \n\nOverall, we find that this work falls below the acceptance threshold for ICLR. We encourage the authors to: (1) expand the literature review to include rare event simulation/probability estimation as outlined here, (2) refine the statistical notations, and (3) add baselines for experiments. With these adjustments, this work would be in a better shape.\n\n[1] A. M. Hasofer, An exact and invarient first order reliability format, J. Eng. Mech. Div., Proc. ASCE 100 (1974) 111–121.\n\n[2] A. Der Kiureghian, et al., First-and second-order reliability methods, Engineering design reliability handbook 14 (2005).\n\n[3] B. Fiessler, H.-J. Neumann, R. Rackwitz, Quadratic limit states in structural reliability, Journal of the Engineering Mechanics Division 105 (1979) 661–676.\n\n[4] J. Li, D. Xiu, Evaluation of failure probability via surrogate models, Journal of Computational Physics 229 (2010) 8966–8980.\n\n[5] J. Li, J. Li, D. Xiu, An efficient surrogate-based method for computing rare failure probability, Journal of Computational Physics 230 (2011) 8683–8697.\n\n[6] C. G. Bucher, Adaptive sampling—an iterative fast monte carlo procedure, Structural safety 5 (1988) 119–126.\n\n[7] I. Papaioannou, C. Papadimitriou, D. Straub, Sequential importance sampling for structural reliability analysis, Structural safety 62 (2016) 66–75.\n\n[8] F. Wagner, I. Papaioannou, E. Ullmann, The ensemble kalman filter for rare event estimation, SIAM/ASA Journal on Uncertainty Quantification 10 (2022) 317–349.\n\n[9] B. Peherstorfer, T. Cui, Y. Marzouk, K. Willcox, Multifidelity importance sampling, Computer Methods in Applied Mechanics and Engineering 300 (2016) 490–509.\n\n[10] B. Peherstorfer, B. Kramer, K. Willcox, Multifidelity preconditioning of the crossentropy method for rare event simulation and failure probability estimation, SIAM/ASA Journal on Uncertainty Quantification 6 (2018) 737–761.\n\n[11] F. Wagner, J. Latz, I. Papaioannou, E. Ullmann, Multilevel sequential importance sampling for rare event estimation, SIAM Journal on Scientific Computing 42 (2020) A2062–A2087.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "5WQJqGBRJS", "reviewer": "Reviewer_TJgd", "summary": "The paper introduces RESuM (Rare Event Surrogate Model), designed to optimize physics detector design, specifically for reducing background events in neutrinoless double-beta decay (NLDBD) detection. The authors frame the challenge as a Rare Event Design (RED) problem, where background events are so rare that traditional simulation methods become computationally prohibitive. RESuM employs a multi-fidelity approach, using a Conditional Neural Process (CNP) model to learn from limited data and a Gaussian Process (GP) model to efficiently explore the design space. The authors apply RESuM to the LEGEND experiment’s neutron moderator design and achieve an  efficient performance.", "questions": "See Weaknesses for questions.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The model targets a critical challenge in physic.\n2. The integration of CNP and MFGP is a novel approach for handling rare event problems.\n3. The framework’s formulation makes it generalizable to other simulation-heavy domains.", "weaknesses": "I do not find major concerns.\nMinor issues:\n1. The success of RESuM’s multi-fidelity modeling approach relies on access to both high-fidelity and low-fidelity simulation data. In situations where these data sources are unavailable or highly dissimilar, would the model's effectiveness be reduced?\n2. It seems to me that the RESuM does not embed physics constraints or validation check to the model, thus it might be possible that RESuM would suggest some invalid designs. In practice, do we need to first define the valid range of parameters for RESuM? And thus for different application contexts, we need to modify the model accordingly?\n3. Section 5 uses subsubsections without subsections.\n4. Can RESuM scale to applications with larger or more complex detector designs? How to estimate the relationship between the dimension of design space and model performance? Also, can RESuM incorporate expert knowledge on the design space to make it more efficient?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Spotlight)