Self-supervised learning (SSL) allows training data representations without a supervised signal and has become an important paradigm in machine learning. Most SSL methods employ the cosine similarity between embedding vectors and hence effectively embed data on a hypersphere. While this seemingly implies that embedding norms cannot play any role in SSL, a few recent works have suggested that embedding norms have properties related to network convergence and confidence. In this paper, we resolve this apparent contradiction and systematically establish the embedding norm's role in SSL training. Using theoretical analysis, simulations, and experiments, we show that embedding norms (i) govern SSL convergence rates and (ii) encode network confidence, with smaller norms corresponding to unexpected samples. Additionally, we show that manipulating embedding norms can have large effects on convergence speed. Our findings demonstrate that SSL embedding norms are integral to understanding and optimizing network behavior.

This paper explores the key factors that influence the performance of models working with point clouds, across different tasks of varying geometric complexity. In this work, we explore the trade-offs between flexibility and weight-sharing introduced by equivariant layers, assessing when equivariance boosts or detracts from performance. It is often argued that providing more information as input improves a model's performance. However, if this additional information breaks certain properties, such as SE(3) equivariance, does it remain beneficial? We identify the key aspects of equivariant and non-equivariant architectures that drive success in different tasks by benchmarking them on segmentation, regression, and generation tasks across multiple datasets with increasing complexity. We observe a positive impact of equivariance, which becomes more pronounced with increasing task complexity, even when strict equivariance is not required. The inductive bias of weight sharing in convolutions, as introduced in LeCun et al. (2010) traditionally refers to applying the same convolution kernel (a linear transformation) across all neighborhoods of an image. To extend this to transformations beyond translations, Cohen & Welling (2016) introduced Group Equivariant CNN (G-CNNs), adding group equivariance properties to encompass group actions and have weight-sharing across group convolution kernels. G-CNN layers are explicitly designed to maintain equivariance under group transformations, allowing the model to handle transformations naturally without needing to learn invariance to changes that preserve object identity.

We propose Pullback Flow Matching (PFM), a novel framework for generative modeling on data manifolds. Unlike existing methods that assume or learn restrictive closed-form manifold mappings for training Riemannian Flow Matching (RFM) models, PFM leverages pullback geometry and isometric learning to preserve the underlying manifold's geometry while enabling efficient generation and precise interpolation in latent space. This approach not only facilitates closed-form mappings on the data manifold but also allows for designable latent spaces, using assumed metrics on both data and latent manifolds. By enhancing isometric learning through Neural ODEs and proposing a scalable training objective, we achieve a latent space more suitable for interpolation, leading to improved manifold learning and generative performance. We demonstrate PFM's effectiveness through applications in synthetic data, protein dynamics and protein sequence data, generating novel proteins with specific properties. This method shows strong potential for drug discovery and materials science, where generating novel samples with specific properties is of great interest.

Advances in artificial intelligence (AI) are fueling a new paradigm of discoveries in natural sciences. Today, AI has started to advance natural sciences by improving, accelerating, and enabling our understanding of natural phenomena at a wide range of spatial and temporal scales, giving rise to a new area of research known as AI for science (AI4Science). Being an emerging research paradigm, AI4Science is unique in that it is an enormous and highly interdisciplinary area. Thus, a unified and technical treatment of this field is needed yet challenging. This work aims to provide a technically thorough account of a subarea of AI4Science; namely, AI for quantum, atomistic, and continuum systems. These areas aim at understanding the physical world from the subatomic (wavefunctions and electron density), atomic (molecules, proteins, materials, and interactions), to macro (fluids, climate, and subsurface) scales and form an important subarea of AI4Science. A unique advantage of focusing on these areas is that they largely share a common set of challenges, thereby allowing a unified and foundational treatment. A key common challenge is how to capture physics first principles, especially symmetries, in natural systems by deep learning methods. We provide an in-depth yet intuitive account of techniques to achieve equivariance to symmetry transformations. We also discuss other common technical challenges, including explainability, out-of-distribution generalization, knowledge transfer with foundation and large language models, and uncertainty quantification. To facilitate learning and education, we provide categorized lists of resources that we found to be useful. We strive to be thorough and unified and hope this initial effort may trigger more community interests and efforts to further advance AI4Science.

The most prevalent class of neural networks operating on graphs are message passing neural networks (MPNNs), in which the representation of a node is updated iteratively by aggregating information in the 1-hop neighborhood. Since this paradigm for computing node embeddings may prevent the model from learning coarse topological structures, the initial features are often augmented with structural information of the graph, typically in the form of Laplacian eigenvectors or Random Walk transition probabilities. In this work, we explore the contribution of message passing when strong structural encodings are provided. We introduce a novel way of modeling the interaction between feature and structural information based on their tensor product rather than the standard concatenation. The choice of interaction is compared in common scenarios and in settings where the capacity of the message-passing layer is severely reduced and ultimately the message-passing phase is removed altogether. Our results indicate that using tensor-based encodings is always at least on par with the concatenation-based encoding and that it makes the model much more robust when the message passing layers are removed, on some tasks incurring almost no drop in performance. This suggests that the importance of message passing is limited when the model can construct strong structural encodings.

Electrocardiography (ECG) is an effective and non-invasive diagnostic tool that measures the electrical activity of the heart. Interpretation of ECG signals to detect various abnormalities is a challenging task that requires expertise. Recently, the use of deep neural networks for ECG classification to aid medical practitioners has become popular, but their black box nature hampers clinical implementation. Several saliency-based interpretability techniques have been proposed, but they only indicate the location of important features and not the actual features. We present a novel interpretability technique called qLST, a query-based latent space traversal technique that is able to provide explanations for any ECG classification model. With qLST, we train a neural network that learns to traverse in the latent space of a variational autoencoder trained on a large university hospital dataset with over 800,000 ECGs annotated for 28 diseases. We demonstrate through experiments that we can explain different black box classifiers by generating ECGs through these traversals.

Including covariant information, such as position, force, velocity or spin is important in many tasks in computational physics and chemistry. We introduce Steerable E(3) Equivariant Graph Neural Networks (SEGNNs) that generalise equivariant graph networks, such that node and edge attributes are not restricted to invariant scalars, but can contain covariant information, such as vectors or tensors. This model, composed of steerable MLPs, is able to incorporate geometric and physical information in both the message and update functions. Through the definition of steerable node attributes, the MLPs provide a new class of activation functions for general use with steerable feature fields. We discuss ours and related work through the lens of equivariant non-linear convolutions, which further allows us to pin-point the successful components of SEGNNs: non-linear message aggregation improves upon classic linear (steerable) point convolutions; steerable messages improve upon recent equivariant graph networks that send invariant messages. We demonstrate the effectiveness of our method on several tasks in computational physics and chemistry and provide extensive ablation studies.