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Many approaches have been used to model chordae tendineae geometries in finite element simulations of atrioventricular heart valves. Unfortunately, current "functional" chordae tendineae geometries lack fidelity that would be helpful when informing clinical decisions. The objectives of this work are (i) to improve synthetic chordae tendineae geometry fidelity to consider branching and (ii) to define how the chordae tendineae geometry affects finite element simulations of valve closure. In this work, we develop an open-source method to construct synthetic chordae tendineae geometries in the SlicerHeart Extension of 3D Slicer. The generated geometries are then used in FEBio finite element simulations of atrioventricular valve function to evaluate how variations in chordae tendineae geometry influence valve behavior. Effects are evaluated using functional and mechanical metrics. Our findings demonstrated that altering the chordae tendineae geometry of a stereotypical mitral valve led to changes in clinically relevant valve metrics and valve mechanics. Specifically, cross sectional area had the most influence over valve closure metrics, followed by chordae tendineae density, length, radius and branches. We then used this information to showcase the flexibility of our new workflow by altering the chordae tendineae geometry of two additional geometries (mitral valve with annular dilation and tricuspid valve) to improve finite element predictions. This study presents a flexible, open-source method for generating synthetic chordae tendineae with realistic branching structures. Further, we establish relationships between the chordae tendineae geometry and valve functional/mechanical metrics. This research contribution helps enrich our open-source workflow and brings the finite element simulations closer to use in a patient-specific clinical setting., Comment: 35 pages, 10 figures

The common practice of preprocessing text before feeding it into NLP models introduces many decision points which have unintended consequences on model performance. In this opinion piece, we focus on the handling of diacritics in texts originating in many languages and scripts. We demonstrate, through several case studies, the adverse effects of inconsistent encoding of diacritized characters and of removing diacritics altogether. We call on the community to adopt simple but necessary steps across all models and toolkits in order to improve handling of diacritized text and, by extension, increase equity in multilingual NLP., Comment: 6 pages

Manual parallelization of code remains a significant challenge due to the complexities of modern software systems and the widespread adoption of multi-core architectures. This paper introduces OMPar, an AI-driven tool designed to automate the parallelization of C/C++ code using OpenMP pragmas. OMPar integrates Large Language Models (LLMs) through two key components: OMPify, which assesses loop parallelization potential, and MonoCoder-OMP, a new fine-tuned model which generates precise OpenMP pragmas. The evaluation of OMPar follows the same rigorous process applied to traditional tools like source-to-source AutoPar and ICPC compilers: (1) ensuring the generated code compiles and runs correctly in serial form, (2) assessing performance with the gradual addition of threads and corresponding physical cores, and (3) verifying and validating the correctness of the code's output. Benchmarks from HeCBench and ParEval are used to evaluate accuracy and performance. Experimental results demonstrate that OMPar significantly outperforms traditional methods, achieving higher accuracy in identifying parallelizable loops and generating efficient pragmas. Beyond accuracy, OMPar offers advantages such as the ability to work on partial or incomplete codebases and the capacity to continuously learn from new code patterns, enhancing its parallelization capabilities over time. These results underscore the potential of LLMs in revolutionizing automatic parallelization techniques, paving the way for more efficient and scalable parallel computing systems.

Researchers face the trade-off between publishing mobility data along with their papers while simultaneously protecting the privacy of the individuals. In addition to the fundamental anonymization process, other techniques, such as spatial discretization and, in certain cases, location concealing or complete removal, are applied to achieve these dual objectives. The primary research question is whether concealing the observation area is an adequate form of protection or whether human mobility patterns in urban areas are inherently revealing of location. The characteristics of the mobility data, such as the number of activity records or the number of unique users in a given spatial unit, reveal the silhouette of the urban landscape, which can be used to infer the identity of the city in question. It was demonstrated that even without disclosing the exact location, the patterns of human mobility can still reveal the urban area from which the data was collected. The presented locating method was tested on other cities using different open data sets and against coarser spatial discretization units. While publishing mobility data is essential for research, it was demonstrated that concealing the observation area is insufficient to prevent the identification of the urban area. Furthermore, using larger discretization units alone is an ineffective solution to the problem of the observation area re-identification. Instead of obscuring the observation area, noise should be added to the trajectories to prevent user identification.

We report the discovery and characterization of a small planet, TOI-1408 c, on a 2.2-day orbit located interior to a previously known hot Jupiter, TOI-1408 b ($P=4.42$ d, $M=1.86\pm0.02\,M_\mathrm{Jup}$, $R=2.4\pm0.5\,R_\mathrm{Jup}$) that exhibits grazing transits. The two planets are near 2:1 period commensurability, resulting in significant transit timing variations (TTVs) for both planets and transit duration variations (TDVs) for the inner planet. The TTV amplitude for TOI-1408 c is 15% of the planet's orbital period, marking the largest TTV amplitude relative to the orbital period measured to date. Photodynamical modeling of ground-based radial velocity (RV) observations and transit light curves obtained with the Transiting Exoplanet Survey Satellite (TESS) and ground-based facilities leads to an inner planet radius of $2.22\pm0.06\,R_\oplus$ and mass of $7.6\pm0.2\,M_\oplus$ that locates the planet into the Sub-Neptune regime. The proximity to the 2:1 period commensurability leads to the libration of the resonant argument of the inner planet. The RV measurements support the existence of a third body with an orbital period of several thousand days. This discovery places the system among the rare systems featuring a hot Jupiter accompanied by an inner low-mass planet., Comment: Accepted to ApJL, 17 pages, 6 figures, 4 tables

TU-games with utility functions are considered. Generalizations of the prenucleolus, essential coalitions and the core: the u-prenucleolus, u-essential coalitions and the u-core respectively are introduced. We show that u-essential coalitions form a characterisation set for the u-prenucleolus in case of games with nonempty u-core.

In quantum mechanics, the Schrieffer--Wolff (SW) transformation (also called quasi-degenerate perturbation theory) is known as an approximative method to reduce the dimension of the Hamiltonian. We present a geometric interpretation of the SW transformation: We prove that it induces a local coordinate chart in the space of Hermitian matrices near a $k$-fold degeneracy submanifold. Inspired by this result, we establish a `distance theorem': we show that the standard deviation of $k$ neighboring eigenvalues of a Hamiltonian equals the distance of this Hamiltonian from the corresponding $k$-fold degeneracy submanifold, divided by $\sqrt{k}$. Furthermore, we investigate one-parameter perturbations of a degenerate Hamiltonian, and prove that the standard deviation and the pairwise differences of the eigenvalues lead to the same order of splitting of the energy eigenvalues, which in turn is the same as the order of distancing from the degeneracy submanifold. As applications, we prove the `protection' of Weyl points using the transversality theorem, and infer geometrical properties of certain degeneracy submanifolds based on results from quantum error correction and topological order., Comment: 48 pages, 9 figures

Using language models as a remote service entails sending private information to an untrusted provider. In addition, potential eavesdroppers can intercept the messages, thereby exposing the information. In this work, we explore the prospects of avoiding such data exposure at the level of text manipulation. We focus on text classification models, examining various token mapping and contextualized manipulation functions in order to see whether classifier accuracy may be maintained while keeping the original text unrecoverable. We find that although some token mapping functions are easy and straightforward to implement, they heavily influence performance on the downstream task, and via a sophisticated attacker can be reconstructed. In comparison, the contextualized manipulation provides an improvement in performance., Comment: PrivateNLP@ACL 2024

Recently, there has been increasing activity in using deep learning for software engineering, including tasks like code generation and summarization. In particular, the most recent coding Large Language Models seem to perform well on these problems. In this technical report, we aim to review how these models perform in code explanation/summarization, while also investigating their code generation capabilities (based on natural language descriptions)., Comment: technical report with 11 pages, 1 figure, 10 tables

Understanding and controlling the electrical properties of solution-processed 2D materials is key to further printed electronics progress. Here we demonstrate that the thermolysis of the aromatic intercalants utilized in nanosheet exfoliation for graphene laminates opens the route to achieving high intrinsic mobility and simultaneously controlling doping type ($n$- and $p$-) and concentration over a wide range. We establish that the intra-flake mobility is high by observing a linear magnetoresistance of such solution-processed graphene laminates and using it to devolve the inter-flake tunneling and intra-layer magnetotransport. Consequently, we determine the temperature dependences of the inter- and intra-layer characteristics, which both appear to be dominated by phonon-assisted processes at temperature $T>$20 Kelvin. In particular, we identify the efficiency of phonon-assisted tunneling as the main limiting factor for electrical conductivity in graphene laminates at room temperature. We also demonstrate a thermoelectric sensitivity of around 50 ${\mu}$V K$^{-1}$ in a solution-processed metal-free graphene-based thermocouple.

The popular subword tokenizers of current language models, such as Byte-Pair Encoding (BPE), are known not to respect morpheme boundaries, which affects the downstream performance of the models. While many improved tokenization algorithms have been proposed, their evaluation and cross-comparison is still an open problem. As a solution, we propose a combined intrinsic-extrinsic evaluation framework for subword tokenization. Intrinsic evaluation is based on our new UniMorph Labeller tool that classifies subword tokenization as either morphological or alien. Extrinsic evaluation, in turn, is performed via the Out-of-Vocabulary Generalization Challenge 1.0 benchmark, which consists of three newly specified downstream text classification tasks. Our empirical findings show that the accuracy of UniMorph Labeller is 98%, and that, in all language models studied (including ALBERT, BERT, RoBERTa, and DeBERTa), alien tokenization leads to poorer generalizations compared to morphological tokenization for semantic compositionality of word meanings.

We explore threshold vocabulary trimming in Byte-Pair Encoding subword tokenization, a postprocessing step that replaces rare subwords with their component subwords. The technique is available in popular tokenization libraries but has not been subjected to rigorous scientific scrutiny. While the removal of rare subwords is suggested as best practice in machine translation implementations, both as a means to reduce model size and for improving model performance through robustness, our experiments indicate that, across a large space of hyperparameter settings, vocabulary trimming fails to improve performance, and is even prone to incurring heavy degradation., Comment: 15 pages

Changes in the number of Weyl nodes in Weyl semimetals occur through merging processes, usually involving a pair of oppositely charged nodes. More complicated processes involving multiple Weyl nodes are also possible, but they typically require fine tuning and are thus less stable. In this work, we study how symmetries affect the allowed merging processes and their stability, focusing on the combination of a two-fold rotation and time-reversal ($C_2\mathcal{T}$) symmetry. We find that, counter-intuitively, processes involving a merging of three nodes are more generic than processes involving only two nodes. Our work suggests that multi-Weyl-merging may be observed in a large variety of quantum materials, and we discuss SrSi$_2$ and bilayer graphene as potential candidates

Neural machine translation (NMT) has progressed rapidly in the past few years, promising improvements and quality translations for different languages. Evaluation of this task is crucial to determine the quality of the translation. Overall, insufficient emphasis is placed on the actual sense of the translation in traditional methods. We propose a bidirectional semantic-based evaluation method designed to assess the sense distance of the translation from the source text. This approach employs the comprehensive multilingual encyclopedic dictionary BabelNet. Through the calculation of the semantic distance between the source and its back translation of the output, our method introduces a quantifiable approach that empowers sentence comparison on the same linguistic level. Factual analysis shows a strong correlation between the average evaluation scores generated by our method and the human assessments across various machine translation systems for English-German language pair. Finally, our method proposes a new multilingual approach to rank MT systems without the need for parallel corpora., Comment: LREC-COLING 2024

While subword tokenizers such as BPE and WordPiece are typically used to build vocabularies for NLP models, the method of decoding text into a sequence of tokens from these vocabularies is often left unspecified, or ill-suited to the method in which they were constructed. We provide a controlled analysis of seven tokenizer inference methods across four different algorithms and three vocabulary sizes, performed on a novel intrinsic evaluation suite we curated for English, combining measures rooted in morphology, cognition, and information theory. We show that for the most commonly used tokenizers, greedy inference performs surprisingly well; and that SaGe, a recently-introduced contextually-informed tokenizer, outperforms all others on morphological alignment., Comment: ACL 2024 (main)

Tokenization is a foundational step in natural language processing (NLP) tasks, bridging raw text and language models. Existing tokenization approaches like Byte-Pair Encoding (BPE) originate from the field of data compression, and it has been suggested that the effectiveness of BPE stems from its ability to condense text into a relatively small number of tokens. We test the hypothesis that fewer tokens lead to better downstream performance by introducing PathPiece, a new tokenizer that segments a document's text into the minimum number of tokens for a given vocabulary. Through extensive experimentation we find this hypothesis not to be the case, casting doubt on the understanding of the reasons for effective tokenization. To examine which other factors play a role, we evaluate design decisions across all three phases of tokenization: pre-tokenization, vocabulary construction, and segmentation, offering new insights into the design of effective tokenizers. Specifically, we illustrate the importance of pre-tokenization and the benefits of using BPE to initialize vocabulary construction. We train 64 language models with varying tokenization, ranging in size from 350M to 2.4B parameters, all of which are made publicly available., Comment: EMNLP 2024

The imperative need to scale computation across numerous nodes highlights the significance of efficient parallel computing, particularly in the realm of Message Passing Interface (MPI) integration. The challenging parallel programming task of generating MPI-based parallel programs has remained unexplored. This study first investigates the performance of state-of-the-art language models in generating MPI-based parallel programs. Findings reveal that widely used models such as GPT-3.5 and PolyCoder (specialized multi-lingual code models) exhibit notable performance degradation, when generating MPI-based programs compared to general-purpose programs. In contrast, domain-specific models such as MonoCoder, which are pretrained on MPI-related programming languages of C and C++, outperform larger models. Subsequently, we introduce a dedicated downstream task of MPI-based program generation by fine-tuning MonoCoder on HPCorpusMPI. We call the resulting model as MPIrigen. We propose an innovative preprocessing for completion only after observing the whole code, thus enabling better completion with a wider context. Comparative analysis against GPT-3.5 zero-shot performance, using a novel HPC-oriented evaluation method, demonstrates that MPIrigen excels in generating accurate MPI functions up to 0.8 accuracy in location and function predictions, and with more than 0.9 accuracy for argument predictions. The success of this tailored solution underscores the importance of domain-specific fine-tuning in optimizing language models for parallel computing code generation, paving the way for a new generation of automatic parallelization tools. The sources of this work are available at our GitHub MPIrigen repository: https://github.com/Scientific-Computing-Lab-NRCN/MPI-rigen

We compute the signature of the Milnor fiber of certain type of non-isolated complex surface singularities, namely, images of finitely determined holomorphic germs. An explicit formula is given in algebraic terms. As a corollary we show that the signature of the Milnor fiber is a topological invariant for these singularities. The proof combines complex analytic and smooth topological techniques. The main tools are Thom-Mather theory of map germs and the Ekholm-Sz\H{u}cs-Takase-Saeki formula for immersions. We give a table with many examples for which the signature is computed using our formula., Comment: 36 pages, 7 figures, 1 table

We consider finite information structures, and quest for the answer of the question: What is the proper definition of prior? In the single player setting we conclude that a probability distribution is a prior if it is disintegrable, because this definition excludes money pump. In the multiplayer setting our analysis does not boil down to one proper notion of common prior (the multiplayer version of prior). The appropriate notion is a choice of the modeller in this setting. We consider three variants of money pump, each "defines" a notion of common prior. Furthermore, we also consider three variants of trade, each correspond to one of the money pump variants, hence to one of the common prior variants.

With easier access to powerful compute resources, there is a growing trend in AI for software development to develop large language models (LLMs) to address a variety of programming tasks. Even LLMs applied to tasks from the high-performance computing (HPC) domain are huge in size and demand expensive compute resources for training. This is partly because LLMs for HPC tasks are obtained by finetuning existing LLMs that support several natural and/or programming languages. We found this design choice confusing - why do we need LLMs trained on natural languages and programming languages unrelated to HPC for HPC-specific tasks? In this line of work, we aim to question choices made by existing LLMs by developing smaller language models (LMs) for specific domains - we call them domain-specific LMs. Specifically, we start with HPC as a domain and build an HPC-specific LM, named MonoCoder, which is orders of magnitude smaller than existing LMs but delivers better performance on non-HPC and HPC codes. Specifically, we pre-trained MonoCoder on an HPC-specific dataset (named HPCorpus) of C and C++ programs mined from GitHub. We evaluated the performance of MonoCoder against state-of-the-art multi-lingual LLMs. Results demonstrate that MonoCoder, although much smaller than existing LMs, outperforms other LLMs on normalized-perplexity tests (in relation to model size) while also delivering competing CodeBLEU scores for high-performance and parallel code generations. In other words, results suggest that MonoCoder understands HPC code better than state-of-the-art LLMs.

Barriers in cities, such as administrative boundaries, natural obstacles, railways or major roads are thought to induce segregation. However, the empirical knowledge about this phenomenon is limited. Here, we present a network science framework to assess barriers to urban mobility along their hierarchy, across residential areas and visited amenities. Using GPS mobility data, we construct a network of blocks from the sequence of individual stays in a major European city. A community detection algorithm allows us to partition this network into non-overlapping areas of dense mobility clusters, in which the effect of transportation hubs can be tuned with a parameter. We apply the Symmetric Area Difference index to quantify the overlap between these mobility clusters and the polygons of urban area separated by barriers. Reducing the effect of transportation hubs results in smaller scale mobility clusters that fit better to lower rank administrative or road barriers compared to their higher rank pairs. We find that characteristic urban barriers can replace each other in dividing mobility clusters of different scales. Next, we define the Barrier Crossing Ratio, the fraction of barrier crossings that bridge mobility clusters. The decomposition of this indicator by origins and destinations suggests a significantly higher impact of barriers on those who live closer to the city center and smaller impact on visits to complex amenities. These results contribute to the ongoing discourse on urban segregation, emphasizing the importance of barriers to urban mobility in shaping interactions and mixing.

Gender-inclusive NLP research has documented the harmful limitations of gender binary-centric large language models (LLM), such as the inability to correctly use gender-diverse English neopronouns (e.g., xe, zir, fae). While data scarcity is a known culprit, the precise mechanisms through which scarcity affects this behavior remain underexplored. We discover LLM misgendering is significantly influenced by Byte-Pair Encoding (BPE) tokenization, the tokenizer powering many popular LLMs. Unlike binary pronouns, BPE overfragments neopronouns, a direct consequence of data scarcity during tokenizer training. This disparate tokenization mirrors tokenizer limitations observed in multilingual and low-resource NLP, unlocking new misgendering mitigation strategies. We propose two techniques: (1) pronoun tokenization parity, a method to enforce consistent tokenization across gendered pronouns, and (2) utilizing pre-existing LLM pronoun knowledge to improve neopronoun proficiency. Our proposed methods outperform finetuning with standard BPE, improving neopronoun accuracy from 14.1% to 58.4%. Our paper is the first to link LLM misgendering to tokenization and deficient neopronoun grammar, indicating that LLMs unable to correctly treat neopronouns as pronouns are more prone to misgender., Comment: Accepted to NAACL 2024 findings

We explore large-scale cosmic structure using the spatial distribution of 542 gamma-ray bursts (GRBs) having accurately measured positions and spectroscopic redshifts. Prominent cosmological clusters are identified in both the northern and southern galactic hemispheres (avoiding extinction effects in the plane of the Milky Way) using the Bootstrap Point-Radius method. The Northern Galactic hemisphere contains a significant group of four GRBs in the redshift range 0.59 < z < 0.62 (with a Bootstrap probability of p = 0.012) along with the previously-identified Hercules-Corona Borealis GreatWall (in the revised redshift range 0.9 < z < 2.1, p = 0.017). The Southern Galactic hemisphere contains the previously-identified Giant GRB Ring (p = 0.022) along with another possible cluster of 7 - 9 GRBs at 1.179 < z < 1.444 (p = 0.031). Additionally, both the Hercules-Corona Borealis Great Wall and the Giant GRB Ring have become more prominent as the GRB sample size has grown. The approach used here underscores the potential value of GRB clustering as a probe of large-scale cosmic structure, complementary to galaxy and quasar clustering. Because of the vast scale on which GRB clustering provides valuable insights, it is important that optical GRB monitoring continue so that additional spectroscopic redshift measurements should be obtained., Comment: 11 pages, 15 figures, submitted to MNRAS

We made statistical analysis of the Fermi GBM and Swift BAT observational material, accumulated over 15 years. We studied how GRB parameters (T$_{90}$ duration, fluence, peak flux) that were observed by only one satellite differ from those observed by both. In the latter case, it was possible to directly compare the values of the parameters that both satellites measured. The GRBs measured by both satellites were identified using the knn() k-nearest neighbour algorithm in the FNN library of the R statistical package. In the parameter space we determined the direction in which the jointly detected GRBs differ most from those detected by only one of the instruments using the lda() in MASS library of R. To get the strength of the relationship between the parameters obtained from the GBM and BAT, a canonical correlation was performed using the cc() procedure in the CCA library in R. The GBM and BAT T$_{90}$ distributions were fitted with a linear combination of lognormal functions. The optimal number of such functions required for fit is two for GBM and three for BAT. Contrary to the widely accepted view, we found that the number of lognormal functions required for fitting the observed distribution of GRB durations does not allow us to deduce the number of central engine types responsible for GRBs., Comment: Submitted to MNRAS

We introduce Universal NER (UNER), an open, community-driven project to develop gold-standard NER benchmarks in many languages. The overarching goal of UNER is to provide high-quality, cross-lingually consistent annotations to facilitate and standardize multilingual NER research. UNER v1 contains 18 datasets annotated with named entities in a cross-lingual consistent schema across 12 diverse languages. In this paper, we detail the dataset creation and composition of UNER; we also provide initial modeling baselines on both in-language and cross-lingual learning settings. We release the data, code, and fitted models to the public., Comment: NAACL 2024 Camera-ready

Motivated by the work of David Singmaster, we study the number of times an integer can appear among the Stirling numbers of both kinds. We provide an upper bound for the occurrences of all the positive integers, and present certain questions for further study. Some numerical results and conjectures concerning the related diohantine equations are collected.

Subword tokenization has become the de-facto standard for tokenization, although comparative evaluations of subword vocabulary quality across languages are scarce. Existing evaluation studies focus on the effect of a tokenization algorithm on the performance in downstream tasks, or on engineering criteria such as the compression rate. We present a new evaluation paradigm that focuses on the cognitive plausibility of subword tokenization. We analyze the correlation of the tokenizer output with the response time and accuracy of human performance on a lexical decision task. We compare three tokenization algorithms across several languages and vocabulary sizes. Our results indicate that the UnigramLM algorithm yields less cognitively plausible tokenization behavior and a worse coverage of derivational morphemes, in contrast with prior work., Comment: EMNLP 2023 (main)

We call into question the recently popularized method of direct model editing as a means of correcting factual errors in LLM generations. We contrast model editing with three similar but distinct approaches that pursue better defined objectives: (1) retrieval-based architectures, which decouple factual memory from inference and linguistic capabilities embodied in LLMs; (2) concept erasure methods, which aim at preventing systemic bias in generated text; and (3) attribution methods, which aim at grounding generations into identified textual sources. We argue that direct model editing cannot be trusted as a systematic remedy for the disadvantages inherent to LLMs, and while it has proven potential in improving model explainability, it opens risks by reinforcing the notion that models can be trusted for factuality. We call for cautious promotion and application of model editing as part of the LLM deployment process, and for responsibly limiting the use cases of LLMs to those not relying on editing as a critical component., Comment: Findings of ACL: EMNLP 2023

We report the discovery, mass, and radius determination of TOI-1801 b, a temperate mini-Neptune around a young M dwarf. TOI-1801 b was observed in TESS sectors 22 and 49, and the alert that this was a TESS planet candidate with a period of 21.3 days went out in April 2020. However, ground-based follow-up observations, including seeing-limited photometry in and outside transit together with precise radial velocity (RV) measurements with CARMENES and HIRES revealed that the true period of the planet is 10.6 days. These observations also allowed us to retrieve a mass of 5.74 $\pm$ 1.46 $M_\oplus$, which together with a radius of 2.08 $\pm$ 0.12 $R_\oplus$, means that TOI-1801 b is most probably composed of water and rock, with an upper limit of 2\% by mass of H$_{2}$ in its atmosphere. The stellar rotation period of 16 days is readily detectable in our RV time series and in the ground-based photometry. We derived a likely age of 600--800 Myr for the parent star TOI-1801, which means that TOI-1801 b is the least massive young mini-Neptune with precise mass and radius determinations. Our results suggest that if TOI-1801 b had a larger atmosphere in the past, it must have been removed by some evolutionary mechanism on timescales shorter than 1 Gyr., Comment: Accepted in A&A. 29 pages, 21 figures

Weyl points (WP) are robust spectral degeneracies, which can not be split by small perturbations, as they are protected by their non-zero topological charge. For larger perturbations, WPs can disappear via pairwise annihilation, where two oppositely charged WPs merge, and the resulting neutral degeneracy disappears. The neutral degeneracy is unstable, meaning that it requires the fine-tuning of the perturbation. Fine-tuning of more than one parameter can lead to more exotic WP mergers. In this work, we reveal and analyze a fundamental connection of the WP mergers and singularity theory: phase boundary points of Weyl phase diagrams, i.e., control parameter values where Weyl point mergers happen, can be classified according to singularity classes of maps between manifolds of equal dimension. We demonstrate this connection on a Weyl--Josephson circuit where the merger of 4 WPs draw a swallowtail singularity, and in a random BdG Hamiltonian which reveal a rich pattern of fold lines and cusp points. Our results predict universal geometrical features of Weyl phase diagrams, and generalize naturally to creation and annihilation of Weyl points in electronic (phononic, magnonic, photonic, etc) band-structure models, where Weyl phase transitions can be triggered by control parameters such as mechanical strain., Comment: 11 pages, 3 figures, 2 tables