Graph Neural Networks (GNNs) are often used for tasks involving the 3D geometry of a given graph, such as molecular dynamics simulation. While incorporating Euclidean distance into Message Passing Neural Networks (referred to as Vanilla DisGNN) is a straightforward way to learn the geometry, it has been demonstrated that Vanilla DisGNN is geometrically incomplete. In this work, we first construct families of novel and symmetric geometric graphs that Vanilla DisGNN cannot distinguish even when considering all-pair distances, which greatly expands the existing counterexample families. Our counterexamples show the inherent limitation of Vanilla DisGNN to capture symmetric geometric structures. We then propose $k$-DisGNNs, which can effectively exploit the rich geometry contained in the distance matrix. We demonstrate the high expressive power of $k$-DisGNNs from three perspectives: 1. They can learn high-order geometric information that cannot be captured by Vanilla DisGNN.

Geometric deep learning has broad applications in biology, a domain where relational structure in data is often intrinsic to modelling the underlying phenomena. Currently, efforts in both geometric deep learning and, more broadly, deep learning applied to biomolecular tasks have been hampered by a scarcity of appropriate datasets accessible to domain specialists and machine learning researchers alike. To address this, we introduce Graphein as a turn-key tool for transforming raw data from widely-used bioinformatics databases into machine learning-ready datasets in a high-throughput and flexible manner. Graphein is a Python library for constructing graph and surface-mesh representations of biomolecular structures, such as proteins, nucleic acids and small molecules, and biological interaction networks for computational analysis and machine learning. Graphein provides utilities for data retrieval from widely-used bioinformatics databases for structural data, including the Protein Data Bank, the AlphaFold Structure Database, chemical data from ZINC and ChEMBL, and for biomolecular interaction networks from STRINGdb, BioGrid, TRRUST and RegNetwork. The library interfaces with popular geometric deep learning libraries: DGL, Jraph, PyTorch Geometric and PyTorch3D though remains framework agnostic as it is built on top of the PyData ecosystem to enable inter-operability with scientific computing tools and libraries. Graphein is designed to be highly flexible, allowing the user to specify each step of the data preparation, scalable to facilitate working with large protein complexes and interaction graphs, and contains useful pre-processing tools for preparing experimental files.

This work studies operators mapping vector and scalar fields defined over a manifold $\mathcal{M}$, and which commute with its group of diffeomorphisms $\text{Diff}(\mathcal{M})$. We prove that in the case of scalar fields $L^p_\omega(\mathcal{M,\mathbb{R}})$, those operators correspond to point-wise non-linearities, recovering and extending known results on $\mathbb{R}^d$. In the context of Neural Networks defined over $\mathcal{M}$, it indicates that point-wise non-linear operators are the only universal family that commutes with any group of symmetries, and justifies their systematic use in combination with dedicated linear operators commuting with specific symmetries. In the case of vector fields $L^p_\omega(\mathcal{M},T\mathcal{M})$, we show that those operators are solely the scalar multiplication. It indicates that $\text{Diff}(\mathcal{M})$ is too rich and that there is no universal class of non-linear operators to motivate the design of Neural Networks over the symmetries of $\mathcal{M}$.

Existing EEG foundation models mainly treat neural signals as generic time series in Euclidean space, ignoring the intrinsic geometric structure of neural dynamics that constrains brain activity to low-dimensional manifolds. This fundamental mismatch between model assumptions and neural geometry limits representation quality and cross-subject generalization. ManifoldFormer addresses this limitation through a novel geometric deep learning framework that explicitly learns neural manifold representations. The architecture integrates three key innovations: a Riemannian VAE for manifold embedding that preserves geometric structure, a geometric Transformer with geodesic-aware attention mechanisms operating directly on neural manifolds, and a dynamics predictor leveraging neural ODEs for manifold-constrained temporal evolution. Extensive evaluation across four public datasets demonstrates substantial improvements over state-of-the-art methods, with 4.6-4.8% higher accuracy and 6.2-10.2% higher Cohen's Kappa, while maintaining robust cross-subject generalization. The geometric approach reveals meaningful neural patterns consistent with neurophysiological principles, establishing geometric constraints as essential for effective EEG foundation models.

Shape analysis, or geometric data science, is a field dedicated to building statistical and machine learning methods able to retrieve and analyze the morphological variability in geometric structures. This is a particularly central problem in applications such as computer vision or biomedical imaging where observations often come as segmented curves, surfaces, densities or other types of complex geometric data. Various approaches have been proposed, including Riemannian and elastic shape space models [23, 41, 31, 56], topology based methods [22, 15, 20], and metric/functional matching frameworks [11, 40, 42]. These different methods have proved quite successful in tackling problems such as pairwise comparison, regression, classification or clustering for datasets of shapes. However, with the constant advances in acquisition protocols and the explosion in the size and resolution of datasets that followed, many such methods do not always scale well to recent applications that may involve databases with up to tens of thousands of subjects, each made of potentially hundreds of thousands of vertices. In view of the rapid development of new machine learning paradigms, in particular neural network models, and their impressive achievements in image processing and analysis tasks, one can reasonably expect similar tools to be able to address those challenges on geometric data. Yet, the very particular and intricate nature of shape spaces poses unique challenges when it comes to designing robust neural network models for shape analysis tasks.

Since the dawn of civilization, humans have tried to understand the nature of intelligence. With the advent of computers, there have been attempts to emulate human intelligence using computer algorithms - a field that was dubbed'Artificial Intelligence' or'AI' by the computer scientist John McCarthy in 1956 and has recently enjoyed an explosion of popularity. Many efforts in AI research have focused on the study and replication of what is considered the hallmark of human cognition, such as playing intelligent games, the faculty of language, visual perception, and creativity. While at the time of writing we have multiple successful takes at the above - computers nowadays play chess and Go better than any human, can translate English into Chinese without a dictionary, automatically drive a car in a crowded city, and generate poetry and art that wins artistic competitions - it is fair to say that we still do not have a full understanding of what human-like or'general' intelligence entails and how to replicate it.

Rapid advances in 3D model scanning have enabled the mass digitization of dental clay models. However, most clinicians and researchers continue to use manual morphometric analysis methods on these models such as landmarking. This is a significant step in treatment planning for craniomaxillofacial conditions. We aimed to develop and test a geometric deep learning model that would accurately and reliably label landmarks on a complicated and specialized patient population -- infants, as accurately as a human specialist without a large amount of training data. Our developed pipeline demonstrated an accuracy of 94.44% with an absolute mean error of 1.676 +/- 0.959 mm on a set of 100 models acquired from newborn babies with cleft lip and palate. Our proposed pipeline has the potential to serve as a fast, accurate, and reliable quantifier of maxillary arch morphometric features, as well as an integral step towards a future fully automated dental treatment pipeline.

Graph Neural Networks (GNNs) are often used for tasks involving the 3D geometry of a given graph, such as molecular dynamics simulation. While incorporating Euclidean distance into Message Passing Neural Networks (referred to as Vanilla DisGNN) is a straightforward way to learn the geometry, it has been demonstrated that Vanilla DisGNN is geometrically incomplete. In this work, we first construct families of novel and symmetric geometric graphs that Vanilla DisGNN cannot distinguish even when considering all-pair distances, which greatly expands the existing counterexample families. Our counterexamples show the inherent limitation of Vanilla DisGNN to capture symmetric geometric structures. We then propose k -DisGNNs, which can effectively exploit the rich geometry contained in the distance matrix.