Symmetry Breaking and Equivariant Neural Networks

Using symmetry as an inductive bias in deep learning has been proven to be a principled approach for sample-efficient model design. However, the relationship between symmetry and the imperative for equivariance in neural networks is not always obvious. Here, we analyze a key limitation that arises in equivariant functions: their incapacity to break symmetry at the level of individual data samples. In response, we introduce a novel notion of 'relaxed equivariance' that circumvents this limitation. We further demonstrate how to incorporate this relaxation into equivariant multilayer perceptrons (E-MLPs), offering an alternative to the noise-injection method. The relevance of symmetry breaking is then discussed in various application domains: physics, graph representation learning, combinatorial optimization and equivariant decoding.

Equivariant Adaptation of Large Pre-Trained Models

Equivariant networks are specifically designed to ensure consistent behavior with respect to a set of input transformations, leading to higher sample efficiency and more accurate and robust predictions. However, redesigning each component of prevalent deep neural network architectures to achieve chosen equivariance is a difficult problem and can result in a computationally expensive network during both training and inference. A recently proposed alternative towards equivariance that removes the architectural constraints is to use a simple canonicalization network that transforms the input to a canonical form before feeding it to an unconstrained prediction network. We show here that this approach can effectively be used to make a large pre-trained network equivariant. However, we observe that the produced canonical orientations can be misaligned with those of the training distribution, hindering performance. Using dataset-dependent priors to inform the canonicalization function, we are able to make large pre-trained models equivariant while maintaining their performance. This significantly improves the robustness of these models to deterministic transformations of the data, such as rotations. We believe this equivariant adaptation of large pre-trained models can help their domain-specific applications with known symmetry priors.

Using Multiple Vector Channels Improves E(n)-Equivariant Graph Neural Networks

We present a natural extension to E(n)-equivariant graph neural networks that uses multiple equivariant vectors per node. We formulate the extension and show that it improves performance across different physical systems benchmark tasks, with minimal differences in runtime or number of parameters. The proposed multichannel EGNN outperforms the standard singlechannel EGNN on N-body charged particle dynamics, molecular property predictions, and predicting the trajectories of solar system bodies. Given the additional benefits and minimal additional cost of multi-channel EGNN, we suggest that this extension may be of practical use to researchers working in machine learning for the physical sciences

Equivariant Networks for Crystal Structures

Supervised learning with deep models has tremendous potential for applications in materials science. Recently, graph neural networks have been used in this context, drawing direct inspiration from models for molecules. However, materials are typically much more structured than molecules, which is a feature that these models do not leverage. In this work, we introduce a class of models that are equivariant with respect to crystalline symmetry groups. We do this by defining a generalization of the message passing operations that can be used with more general permutation groups, or that can alternatively be seen as defining an expressive convolution operation on the crystal graph. Empirically, these models achieve competitive results with state-of-the-art on property prediction tasks.

Equivariance with Learned Canonicalization Functions

Symmetry-based neural networks often constrain the architecture in order to achieve invariance or equivariance to a group of transformations. In this paper, we propose an alternative that avoids this architectural constraint by learning to produce a canonical representation of the data. These canonicalization functions can readily be plugged into non-equivariant backbone architectures. We offer explicit ways to implement them for many groups of interest. We show that this approach enjoys universality while providing interpretable insights. Our main hypothesis is that learning a neural network to perform canonicalization is better than using predefined heuristics. Our results show that learning the canonicalization function indeed leads to better results and that the approach achieves excellent performance in practice.

Prediction of Large Magnetic Moment Materials With Graph Neural Networks and Random Forests

Magnetic materials are crucial components of many technologies that could drive the ecological transition, including electric motors, wind turbine generators and magnetic refrigeration systems. Discovering materials with large magnetic moments is therefore an increasing priority. Here, using state-of-the-art machine learning methods, we scan the Inorganic Crystal Structure Database (ICSD) of hundreds of thousands of existing materials to find those that are ferromagnetic and have large magnetic moments. Crystal graph convolutional neural networks (CGCNN), materials graph network (MEGNet) and random forests are trained on the Materials Project database that contains the results of high-throughput DFT predictions. For random forests, we use a stochastic method to select nearly one hundred relevant descriptors based on chemical composition and crystal structure. This turns out to give results for the test sets that are comparable to those of neural networks. The comparison between these different machine learning approaches gives an estimate of the errors for our predictions on the ICSD database.

Gradient Starvation: A Learning Proclivity in Neural Networks

We identify and formalize a fundamental gradient descent phenomenon resulting in a learning proclivity in over-parameterized neural networks. Gradient Starvation arises when cross-entropy loss is minimized by capturing only a subset of features relevant for the task, despite the presence of other predictive features that fail to be discovered. This work provides a theoretical explanation for the emergence of such feature imbalance in neural networks. Using tools from Dynamical Systems theory, we identify simple properties of learning dynamics during gradient descent that lead to this imbalance, and prove that such a situation can be expected given certain statistical structure in training data. Based on our proposed formalism, we develop guarantees for a novel regularization method aimed at decoupling feature learning dynamics, improving accuracy and robustness in cases hindered by gradient starvation. We illustrate our findings with simple and real-world out-of-distribution (OOD) generalization experiments.