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The use of double-faced, metallized, perforated PCB planes, segmented into strips for the anodic read-out of ionization signals in liquid argon TPCs, is emerging as a promising technology for charge readout in liquid argon TPCs used in large volume detectors.As a proof of concept, a prototype liquid Argon TPC hosting this new anode configuration based on single side perforated PCB planes has been constructed and exposed to cosmic rays at LNL in Italy. Tests were performed with both the metallized and insulating sides of the anode facing the drift volume, providing the first evidence of the focusing effect on drift electron trajectories through the PCB holes due to charge accumulation on the insulator surface., Comment: 12 pages, 13 figures

Recognizing daily activities with unobtrusive sensors in smart environments enables various healthcare applications. Monitoring how subjects perform activities at home and their changes over time can reveal early symptoms of health issues, such as cognitive decline. Most approaches in this field use deep learning models, which are often seen as black boxes mapping sensor data to activities. However, non-expert users like clinicians need to trust and understand these models' outputs. Thus, eXplainable AI (XAI) methods for Human Activity Recognition have emerged to provide intuitive natural language explanations from these models. Different XAI methods generate different explanations, and their effectiveness is typically evaluated through user surveys, that are often challenging in terms of costs and fairness. This paper proposes an automatic evaluation method using Large Language Models (LLMs) to identify, in a pool of candidates, the best XAI approach for non-expert users. Our preliminary results suggest that LLM evaluation aligns with user surveys., Comment: Accepted for publication at UbiComp / ISWC 2024's XAIforU workshop

The sensor-based recognition of Activities of Daily Living (ADLs) in smart home environments enables several applications in the areas of energy management, safety, well-being, and healthcare. ADLs recognition is typically based on deep learning methods requiring large datasets to be trained. Recently, several studies proved that Large Language Models (LLMs) effectively capture common-sense knowledge about human activities. However, the effectiveness of LLMs for ADLs recognition in smart home environments still deserves to be investigated. In this work, we propose ADL-LLM, a novel LLM-based ADLs recognition system. ADLLLM transforms raw sensor data into textual representations, that are processed by an LLM to perform zero-shot ADLs recognition. Moreover, in the scenario where a small labeled dataset is available, ADL-LLM can also be empowered with few-shot prompting. We evaluated ADL-LLM on two public datasets, showing its effectiveness in this domain., Comment: Currently under review

A search for full energy depositions from bosonic keV-scale dark matter candidates of masses between 65 keV and 1021 keV has been performed with data collected during Phase II of the GERmanium Detector Array (GERDA) experiment. Our analysis includes direct dark matter absorption as well as dark Compton scattering. With a total exposure of 105.5 kg yr, no evidence for a signal above the background has been observed. The resulting exclusion limits deduced with either Bayesian or Frequentist statistics are the most stringent direct constraints in the major part of the 140-1021 keV mass range. As an example, at a mass of 150 keV the dimensionless coupling of dark photons and axion-like particles to electrons has been constrained to $\alpha$'/$\alpha$ < 8.7x10$^{-24}$ and g$_{ae}$ < 3.3x10$^{-12}$ at 90% credible interval (CI), respectively. Additionally, a search for peak-like signals from beyond the Standard Model decays of nucleons and electrons is performed. We find for the inclusive decay of a single neutron in $^{76}$Ge a lower lifetime limit of $\tau_n$ > 1.5x10$^{24}$ yr and for a proton $\tau_p$ > 1.3x10$^{24}$ yr at 90% CI. For the electron decay e$^-\rightarrow\nu_e\gamma$ a lower limit of $\tau_e$ > 5.4x10$^{25}$ yr at 90% CI has been determined., Comment: 20 pages, 12 figures, 7 tables

The study of behavioral diversity in Multi-Agent Reinforcement Learning (MARL) is a nascent yet promising field. In this context, the present work deals with the question of how to control the diversity of a multi-agent system. With no existing approaches to control diversity to a set value, current solutions focus on blindly promoting it via intrinsic rewards or additional loss functions, effectively changing the learning objective and lacking a principled measure for it. To address this, we introduce Diversity Control (DiCo), a method able to control diversity to an exact value of a given metric by representing policies as the sum of a parameter-shared component and dynamically scaled per-agent components. By applying constraints directly to the policy architecture, DiCo leaves the learning objective unchanged, enabling its applicability to any actor-critic MARL algorithm. We theoretically prove that DiCo achieves the desired diversity, and we provide several experiments, both in cooperative and competitive tasks, that show how DiCo can be employed as a novel paradigm to increase performance and sample efficiency in MARL. Multimedia results are available on the paper's website: https://sites.google.com/view/dico-marl.

Compact robotic platforms with powerful compute and actuation capabilities are key enablers for practical, real-world deployments of multi-agent research. This article introduces a tightly integrated hardware, control, and simulation software stack on a fleet of holonomic ground robot platforms designed with this motivation. Our robots, a fleet of customised DJI Robomaster S1 vehicles, offer a balance between small robots that do not possess sufficient compute or actuation capabilities and larger robots that are unsuitable for indoor multi-robot tests. They run a modular ROS2-based optimal estimation and control stack for full onboard autonomy, contain ad-hoc peer-to-peer communication infrastructure, and can zero-shot run multi-agent reinforcement learning (MARL) policies trained in our vectorized multi-agent simulation framework. We present an in-depth review of other platforms currently available, showcase new experimental validation of our system's capabilities, and introduce case studies that highlight the versatility and reliability of our system as a testbed for a wide range of research demonstrations. Our system as well as supplementary material is available online. https://proroklab.github.io/cambridge-robomaster

The real-time monitoring of the structural displacement of the Vacuum Vessel (VV) of thermonuclear fusion devices caused by electromagnetic (EM) loads is of great interest. In this paper, Model Order Reduction (MOR) is applied to the Integral Equation Methods (IEM) and the Finite Elements Method (FEM) to develop Electromagnetic and Structural Reduced Order Models (ROMs) compatible with real-time execution which allows for the real-time monitoring of strain and displacement in critical positions of Tokamaks machines. Low-rank compression techniques based on hierarchical matrices are applied to reduce the computational cost during the offline stage when the ROMs are constructed. Numerical results show the accuracy of the approach and demonstrate the compatibility with real-time execution in standard hardware.

Human Activity Recognition (HAR) based on the sensors of mobile/wearable devices aims to detect the physical activities performed by humans in their daily lives. Although supervised learning methods are the most effective in this task, their effectiveness is constrained to using a large amount of labeled data during training. While collecting raw unlabeled data can be relatively easy, annotating data is challenging due to costs, intrusiveness, and time constraints. To address these challenges, this paper explores alternative approaches for accurate HAR using a limited amount of labeled data. In particular, we have adapted recent Self-Supervised Learning (SSL) algorithms to the HAR domain and compared their effectiveness. We investigate three state-of-the-art SSL techniques of different families: contrastive, generative, and predictive. Additionally, we evaluate the impact of the underlying neural network on the recognition rate by comparing state-of-the-art CNN and transformer architectures. Our results show that a Masked Auto Encoder (MAE) approach significantly outperforms other SSL approaches, including SimCLR, commonly considered one of the best-performing SSL methods in the HAR domain. The code and the pre-trained SSL models are publicly available for further research and development.

Context-aware Human Activity Recognition (HAR) is a hot research area in mobile computing, and the most effective solutions in the literature are based on supervised deep learning models. However, the actual deployment of these systems is limited by the scarcity of labeled data that is required for training. Neuro-Symbolic AI (NeSy) provides an interesting research direction to mitigate this issue, by infusing common-sense knowledge about human activities and the contexts in which they can be performed into HAR deep learning classifiers. Existing NeSy methods for context-aware HAR rely on knowledge encoded in logic-based models (e.g., ontologies) whose design, implementation, and maintenance to capture new activities and contexts require significant human engineering efforts, technical knowledge, and domain expertise. Recent works show that pre-trained Large Language Models (LLMs) effectively encode common-sense knowledge about human activities. In this work, we propose ContextGPT: a novel prompt engineering approach to retrieve from LLMs common-sense knowledge about the relationship between human activities and the context in which they are performed. Unlike ontologies, ContextGPT requires limited human effort and expertise. An extensive evaluation carried out on two public datasets shows how a NeSy model obtained by infusing common-sense knowledge from ContextGPT is effective in data scarcity scenarios, leading to similar (and sometimes better) recognition rates than logic-based approaches with a fraction of the effort.

We study the impact of a bonus policy implemented by Hawai'i Public Schools starting in fall 2020 that raised the salaries of all special education teachers in the state by $10,000. We estimate that the introduction of this policy reduced the proportion of vacant special education teaching positions by 32%, or 1.2 percentage points, and the proportion of special education positions that were vacant or filled by an unlicensed teacher by 35%, or 4.0 percentage points. The bonus policy did not have significant impacts on special education teacher retention; instead, the impacts of the policy were driven almost entirely by an increase in the number of general education teachers in the state who moved into open special education teaching positions. The effects of the bonus policy were also largest in historically hard-to-staff schools in which all teachers also received "tiered school" bonuses of up to $8,000. Hawai'i therefore represents a unique but instructive case of how strategic financial incentives can help address special education teacher shortages.

We use over 25 years of longitudinal data from Washington state to provide a descriptive portrait of the paraeducator workforce in the state. Paraeducators are more racially and ethnically diverse than special education teachers, particularly in the last decade, and tend to be less experienced. They also have full-time salaries that are about half of the average special education teacher salary. Paraeducator-to-student ratios have decreased over time in the state, but they are higher in schools serving more students of color. Finally, and perhaps most importantly, paraeducator attrition rates from the state's public school workforce have increased dramatically over time; for example, the paraeducator attrition rate after the 2021-22 school year (23%) was over twice as high as the attrition rate after the 2008-09 school year (8%). These findings have implications for how policymakers and school leaders should approach decision-making related to the paraeducator workforce, as well as how researchers might approach further research with this group of educators.

We develop a model reduction technique for non-smooth dynamical systems using spectral submanifolds. Specifically, we construct low-dimensional, sparse, nonlinear and non-smooth models on unions of slow and attracting spectral submanifolds (SSMs) for each smooth subregion of the phase space and then properly match them. We apply this methodology to both equation-driven and data-driven problems, with and without external forcing., Comment: Submitted to International Journal of Non-Linear Mechanics

The field of Multi-Agent Reinforcement Learning (MARL) is currently facing a reproducibility crisis. While solutions for standardized reporting have been proposed to address the issue, we still lack a benchmarking tool that enables standardization and reproducibility, while leveraging cutting-edge Reinforcement Learning (RL) implementations. In this paper, we introduce BenchMARL, the first MARL training library created to enable standardized benchmarking across different algorithms, models, and environments. BenchMARL uses TorchRL as its backend, granting it high performance and maintained state-of-the-art implementations while addressing the broad community of MARL PyTorch users. Its design enables systematic configuration and reporting, thus allowing users to create and run complex benchmarks from simple one-line inputs. BenchMARL is open-sourced on GitHub: https://github.com/facebookresearch/BenchMARL

The catalytic sites of MoS$_{2}$ monolayers towards hydrogen evolution are well known to be vacancies and edge-like defects. However, it is still very challenging to control the position, size, and defective areas on the basal plane of Mo$S_{2}$ monolayers by most of defect-engineering routes. In this work, the fabrication of etched arrays on ultra-large supported and free-standing MoS$_{2}$ monolayers using focused ion beam (FIB) is reported for the first time. By tuning the Ga+ ion dose, it is possible to confine defects near the etched edges or spread them over ultra-large areas on the basal plane. The electrocatalytic activity of the arrays toward hydrogen evolution reaction (HER) was measured by fabricating microelectrodes using a new method that preserves the catalytic sites. We demonstrate that the overpotential can be decreased up to 290 mV by assessing electrochemical activity only at the basal plane. High-resolution transmission electron microscopy images obtained on FIB patterned freestanding MoS$_{2}$ monolayers reveal the presence of amorphous regions and X-ray photoelectron spectroscopy indicates sulfur excess in these regions. Density-functional theory calculations provide identification of catalytic defect sites. Our results demonstrate a new rational control of amorphous-crystalline surface boundaries and future insight for defect optimization in MoS$_{2}$ monolayers., Comment: 10 pages, 5 figures

The GERmanium Detector Array (GERDA) experiment operated enriched high-purity germanium detectors in a liquid argon cryostat, which contains 0.33% of $^{36}$Ar, a candidate isotope for the two-neutrino double-electron capture (2$\nu$ECEC) and therefore for the neutrinoless double-electron capture (0$\nu$ECEC). If detected, this process would give evidence of lepton number violation and the Majorana nature of neutrinos. In the radiative 0$\nu$ECEC of $^{36}$Ar, a monochromatic photon is emitted with an energy of 429.88 keV, which may be detected by the GERDA germanium detectors. We searched for the $^{36}$Ar 0$\nu$ECEC with GERDA data, with a total live time of 4.34 yr (3.08 yr accumulated during GERDA Phase II and 1.26 yr during GERDA Phase I). No signal was found and a 90% C.L. lower limit on the half-life of this process was established T$_{1/2}$ > 1.5x10$^{22}$ yr, Comment: 10 pages, 5 figures, 1 table

Special educators serving students with emotional/behavioral disorders (EBD) depend on their principals' support, yet prior research has not explored principals' roles in supporting these programs. Using constructivist grounded theory methods, we analyzed interviews with five elementary school principals about their roles in supporting self-contained programs for students with EBD. Principals held widely varying "conceptions of students with EBD, visions" for their program, and understandings of their own "roles and responsibilities" for their programs; collectively, interactions among these gave rise to principals' "senses of agency," both for the program's capacity to be a causal agent in promoting student growth and their own capacity, as leaders, to improve the program. Our findings indicate the centrality of principals' conceptions of EBD for how they support self-contained programs for students with EBD, which has important implications for future research and practice regarding principal leadership for special education. [This is the online first version of an article published in "Remedial and Special Education."]

We present the measurement of the two-neutrino double-$\beta$ decay rate of $^{76}$Ge performed with the GERDA Phase II experiment. With a subset of the entire GERDA exposure, 11.8 kg$\cdot$yr, the half-life of the process has been determined: $T^{2\nu}_{1/2} = (2.022 \pm 0.018_{stat} \pm 0.038_{sys})\times10^{21}$ yr. This is the most precise determination of the $^{76}$Ge two-neutrino double-$\beta$ decay half-life and one of the most precise measurements of a double-$\beta$ decay process. The relevant nuclear matrix element can be extracted: $M^{2\nu}_{\text{eff}} = (0.101\pm0.001).$, Comment: 7 pages, 4 figures, 2 tables

We search for tri-nucleon decays of $^{76}$Ge in the dataset from the GERmanium Detector Array (GERDA) experiment. Decays that populate excited levels of the daughter nucleus above the threshold for particle emission lead to disintegration and are not considered. The ppp-, ppn-, and pnn-decays lead to $^{73}$Cu, $^{73}$Zn, and $^{73}$Ga nuclei, respectively. These nuclei are unstable and eventually proceed by the beta decay of $^{73}$Ga to $^{73}$Ge (stable). We search for the $^{73}$Ga decay exploiting the fact that it dominantly populates the 66.7 keV $^{73m}$Ga state with half-life of 0.5 s. The nnn-decays of $^{76}$Ge that proceed via $^{73m}$Ge are also included in our analysis. We find no signal candidate and place a limit on the sum of the decay widths of the inclusive tri-nucleon decays that corresponds to a lower lifetime limit of 1.2x10$^{26}$ yr (90% credible interval). This result improves previous limits for tri-nucleon decays by one to three orders of magnitude., Comment: 8 pages, 9 figures, 3 tables

We present an environmentally benign methodology for the covalent functionalization (arylation) of reduced graphene oxide (rGO) nanosheets with arylazo sulfones. A variety of tagged aryl units were conveniently accommodated at the rGO surface via visible light irradiation of suspensions of carbon nanostructured materials in aqueous media. Mild reaction conditions, absence of photosensitizers, functional group tolerance and high atomic fractions (XPS analysis) represent some of the salient features characterizing the present methodology. Control experiments for the mechanistic elucidation (Raman analysis) and chemical nanomanipulation of the tagged rGO surfaces are also reported.

The use of supervised learning for Human Activity Recognition (HAR) on mobile devices leads to strong classification performances. Such an approach, however, requires large amounts of labeled data, both for the initial training of the models and for their customization on specific clients (whose data often differ greatly from the training data). This is actually impractical to obtain due to the costs, intrusiveness, and time-consuming nature of data annotation. Moreover, even with the help of a significant amount of labeled data, model deployment on heterogeneous clients faces difficulties in generalizing well on unseen data. Other domains, like Computer Vision or Natural Language Processing, have proposed the notion of pre-trained models, leveraging large corpora, to reduce the need for annotated data and better manage heterogeneity. This promising approach has not been implemented in the HAR domain so far because of the lack of public datasets of sufficient size. In this paper, we propose a novel strategy to combine publicly available datasets with the goal of learning a generalized HAR model that can be fine-tuned using a limited amount of labeled data on an unseen target domain. Our experimental evaluation, which includes experimenting with different state-of-the-art neural network architectures, shows that combining public datasets can significantly reduce the number of labeled samples required to achieve satisfactory performance on an unseen target domain., Comment: IEEE SMARTCOMP 2023

Deep Learning models are a standard solution for sensor-based Human Activity Recognition (HAR), but their deployment is often limited by labeled data scarcity and models' opacity. Neuro-Symbolic AI (NeSy) provides an interesting research direction to mitigate these issues by infusing knowledge about context information into HAR deep learning classifiers. However, existing NeSy methods for context-aware HAR require computationally expensive symbolic reasoners during classification, making them less suitable for deployment on resource-constrained devices (e.g., mobile devices). Additionally, NeSy approaches for context-aware HAR have never been evaluated on in-the-wild datasets, and their generalization capabilities in real-world scenarios are questionable. In this work, we propose a novel approach based on a semantic loss function that infuses knowledge constraints in the HAR model during the training phase, avoiding symbolic reasoning during classification. Our results on scripted and in-the-wild datasets show the impact of different semantic loss functions in outperforming a purely data-driven model. We also compare our solution with existing NeSy methods and analyze each approach's strengths and weaknesses. Our semantic loss remains the only NeSy solution that can be deployed as a single DNN without the need for symbolic reasoning modules, reaching recognition rates close (and better in some cases) to existing approaches.

Striking a balance between integration and modularity is crucial for a machine learning library to be versatile and user-friendly, especially in handling decision and control tasks that involve large development teams and complex, real-world data, and environments. To address this issue, we propose TorchRL, a generalistic control library for PyTorch that provides well-integrated, yet standalone components. With a versatile and robust primitive design, TorchRL facilitates streamlined algorithm development across the many branches of Reinforcement Learning (RL) and control. We introduce a new PyTorch primitive, TensorDict, as a flexible data carrier that empowers the integration of the library's components while preserving their modularity. Hence replay buffers, datasets, distributed data collectors, environments, transforms and objectives can be effortlessly used in isolation or combined. We provide a detailed description of the building blocks, supporting code examples and an extensive overview of the library across domains and tasks. Finally, we show comparative benchmarks to demonstrate its computational efficiency. TorchRL fosters long-term support and is publicly available on GitHub for greater reproducibility and collaboration within the research community. The code is opensourced on https://github.com/pytorch/rl.

Evolutionary science provides evidence that diversity confers resilience in natural systems. Yet, traditional multi-agent reinforcement learning techniques commonly enforce homogeneity to increase training sample efficiency. When a system of learning agents is not constrained to homogeneous policies, individuals may develop diverse behaviors, resulting in emergent complementarity that benefits the system. Despite this, there is a surprising lack of tools that quantify behavioral diversity. Such techniques would pave the way towards understanding the impact of diversity in collective artificial intelligence and enabling its control. In this paper, we introduce System Neural Diversity (SND): a measure of behavioral heterogeneity in multi-agent systems. We discuss and prove its theoretical properties, and compare it with alternate, state-of-the-art behavioral diversity metrics used in the robotics domain. Through simulations of a variety of cooperative multi-robot tasks, we show how our metric constitutes an important tool that enables measurement and control of behavioral heterogeneity. In dynamic tasks, where the problem is affected by repeated disturbances during training, we show that SND allows us to measure latent resilience skills acquired by the agents, while other proxies, such as task performance (reward), fail to. Finally, we show how the metric can be employed to control diversity, allowing us to enforce a desired heterogeneity set-point or range. We demonstrate how this paradigm can be used to bootstrap the exploration phase, finding optimal policies faster, thus enabling novel and more efficient MARL paradigms.

Supervised Deep Learning (DL) models are currently the leading approach for sensor-based Human Activity Recognition (HAR) on wearable and mobile devices. However, training them requires large amounts of labeled data whose collection is often time-consuming, expensive, and error-prone. At the same time, due to the intra- and inter-variability of activity execution, activity models should be personalized for each user. In this work, we propose SelfAct: a novel framework for HAR combining self-supervised and active learning to mitigate these problems. SelfAct leverages a large pool of unlabeled data collected from many users to pre-train through self-supervision a DL model, with the goal of learning a meaningful and efficient latent representation of sensor data. The resulting pre-trained model can be locally used by new users, which will fine-tune it thanks to a novel unsupervised active learning strategy. Our experiments on two publicly available HAR datasets demonstrate that SelfAct achieves results that are close to or even better than the ones of fully supervised approaches with a small number of active learning queries.

Real world applications of Reinforcement Learning (RL) are often partially observable, thus requiring memory. Despite this, partial observability is still largely ignored by contemporary RL benchmarks and libraries. We introduce Partially Observable Process Gym (POPGym), a two-part library containing (1) a diverse collection of 15 partially observable environments, each with multiple difficulties and (2) implementations of 13 memory model baselines -- the most in a single RL library. Existing partially observable benchmarks tend to fixate on 3D visual navigation, which is computationally expensive and only one type of POMDP. In contrast, POPGym environments are diverse, produce smaller observations, use less memory, and often converge within two hours of training on a consumer-grade GPU. We implement our high-level memory API and memory baselines on top of the popular RLlib framework, providing plug-and-play compatibility with various training algorithms, exploration strategies, and distributed training paradigms. Using POPGym, we execute the largest comparison across RL memory models to date. POPGym is available at https://github.com/proroklab/popgym.

Cooperative multi-robot tasks can benefit from heterogeneity in the robots' physical and behavioral traits. In spite of this, traditional Multi-Agent Reinforcement Learning (MARL) frameworks lack the ability to explicitly accommodate policy heterogeneity, and typically constrain agents to share neural network parameters. This enforced homogeneity limits application in cases where the tasks benefit from heterogeneous behaviors. In this paper, we crystallize the role of heterogeneity in MARL policies. Towards this end, we introduce Heterogeneous Graph Neural Network Proximal Policy Optimization (HetGPPO), a paradigm for training heterogeneous MARL policies that leverages a Graph Neural Network for differentiable inter-agent communication. HetGPPO allows communicating agents to learn heterogeneous behaviors while enabling fully decentralized training in partially observable environments. We complement this with a taxonomical overview that exposes more heterogeneity classes than previously identified. To motivate the need for our model, we present a characterization of techniques that homogeneous models can leverage to emulate heterogeneous behavior, and show how this "apparent heterogeneity" is brittle in real-world conditions. Through simulations and real-world experiments, we show that: (i) when homogeneous methods fail due to strong heterogeneous requirements, HetGPPO succeeds, and, (ii) when homogeneous methods are able to learn apparently heterogeneous behaviors, HetGPPO achieves higher resilience to both training and deployment noise.

The ability to detect liquid argon scintillation light from within a densely packed high-purity germanium detector array allowed the GERDA experiment to reach an exceptionally low background rate in the search for neutrinoless double beta decay of $^{76}$Ge. Proper modeling of the light propagation throughout the experimental setup, from any origin in the liquid argon volume to its eventual detection by the novel light read-out system, provides insight into the rejection capability and is a necessary ingredient to obtain robust background predictions. In this paper, we present a model of the GERDA liquid argon veto, as obtained by Monte Carlo simulations and constrained by calibration data, and highlight its application for background decomposition.

Joint bleeding is a common condition for people with hemophilia and, if untreated, can result in hemophilic arthropathy. Ultrasound imaging has recently emerged as an effective tool to diagnose joint recess distension caused by joint bleeding. However, no computer-aided diagnosis tool exists to support the practitioner in the diagnosis process. This paper addresses the problem of automatically detecting the recess and assessing whether it is distended in knee ultrasound images collected in patients with hemophilia. After framing the problem, we propose two different approaches: the first one adopts a one-stage object detection algorithm, while the second one is a multi-task approach with a classification and a detection branch. The experimental evaluation, conducted with $483$ annotated images, shows that the solution based on object detection alone has a balanced accuracy score of $0.74$ with a mean IoU value of $0.66$, while the multi-task approach has a higher balanced accuracy value ($0.78$) at the cost of a slightly lower mean IoU value.

A search for Beyond the Standard Model double-$\beta$ decay modes of $^{76}$Ge has been performed with data collected during the Phase II of the GERmanium Detector Array (GERDA) experiment, located at Laboratori Nazionali del Gran Sasso of INFN (Italy). Improved limits on the decays involving Majorons have been obtained, compared to previous experiments with $^{76}$Ge, with half-life values on the order of 10$^{23}$ yr. For the first time with $^{76}$Ge, limits on Lorentz invariance violation effects in double-$\beta$ decay have been obtained. The isotropic coefficient $\mathring{a}_\text{of}^{(3)}$, which embeds Lorentz violation in double-$\beta$ decay, has been constrained at the order of $10^{-6}$ GeV. We also set the first experimental limits on the search for light exotic fermions in double-$\beta$ decay, including sterile neutrinos.