Equivariant Symmetry Breaking Sets (2402.02681v3)

Abstract: Equivariant neural networks (ENNs) have been shown to be extremely effective in applications involving underlying symmetries. By construction ENNs cannot produce lower symmetry outputs given a higher symmetry input. However, symmetry breaking occurs in many physical systems and we may obtain a less symmetric stable state from an initial highly symmetric one. Hence, it is imperative that we understand how to systematically break symmetry in ENNs. In this work, we propose a novel symmetry breaking framework that is fully equivariant and is the first which fully addresses spontaneous symmetry breaking. We emphasize that our approach is general and applicable to equivariance under any group. To achieve this, we introduce the idea of symmetry breaking sets (SBS). Rather than redesign existing networks, we design sets of symmetry breaking objects which we feed into our network based on the symmetry of our inputs and outputs. We show there is a natural way to define equivariance on these sets, which gives an additional constraint. Minimizing the size of these sets equates to data efficiency. We prove that minimizing these sets translates to a well studied group theory problem, and tabulate solutions to this problem for the point groups. Finally, we provide some examples of symmetry breaking to demonstrate how our approach works in practice. The code for these examples is available at \url{https://github.com/atomicarchitects/equivariant-SBS}.

• The paper introduces a novel symmetry breaking framework that integrates symmetry breaking sets into equivariant neural networks to address spontaneous symmetry breaking.
• It systematically aligns the optimization of symmetry breaking sets with group theory principles to enhance data efficiency.
• The approach demonstrates improved performance in applications like molecular dynamics and protein folding through practical implementations.

Equivariant Symmetry Breaking Sets

Introduction to Equivariant Neural Networks

Equivariant neural networks (ENNs) stand out as essential models for processing data that exhibit intrinsic symmetries. Through their innate design, these networks respect the symmetries embedded within the data, unlike traditional neural networks that must discern these patterns from augmented datasets or custom training strategies. ENNs have therefore established superior performance benchmarks in various applications, ranging from molecular dynamics and generation to protein folding. A consistent characteristic of these networks is that the output retains the symmetry of the input or exhibits a higher level of symmetry.

Challenges with Spontaneous Symmetry Breaking

However, spontaneous symmetry breaking—a phenomenon observed in numerous physical systems—poses a complication. The Hamiltonian in such systems can exhibit symmetrical properties, while individual ground states may not. Therefore, tackling tasks where the output has less symmetry than the input becomes challenging for standard ENN frameworks, as they are generally not well-equipped to produce or sample from a set of lower symmetry outputs.

Novel Symmetry Breaking Framework

This paper introduces an innovative framework for symmetry breaking within ENNs that maintains full equivariance. Our approach systematically breaks the symmetry by incorporating symmetry breaking sets (SBS) into the model based on the symmetry of inputs and desired outputs. We introduce the concept of the equivariant SBS and detail constraints that they must fulfill. A significant finding is the equivalence of optimizing equivariant SBSs with a fundamental problem in group theory, bringing a precise characterization of the data efficiency of the symmetry breaking process.

Implications and Use Cases

The proposed method has several advantages, including straightforward implementation and the ability to generalize across various groups. It proves particularly beneficial in scenarios where existing methods fall short, such as when equivariance is broken or when the data distribution lacks systematic symmetry breaking. Furthermore, we provide actionable insights by tabulating solutions for point groups and demonstrating our approach through practical examples, showcasing the robustness and versatility of our method. It should be noted that the formulation assumes prior knowledge of input and output symmetries, which could be feasible through pre-processing steps in various applications.

Overall, this paper offers a powerful framework for a model architecture that confronts the challenge of spontaneous symmetry breaking within a fully equivariant system, laying the groundwork for future explorations in complex symmetry-related tasks.