Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups (2410.02698v2)

Abstract: The quest for robust and generalizable machine learning models has driven recent interest in exploiting symmetries through equivariant neural networks. In the context of PDE solvers, recent works have shown that Lie point symmetries can be a useful inductive bias for Physics-Informed Neural Networks (PINNs) through data and loss augmentation. Despite this, directly enforcing equivariance within the model architecture for these problems remains elusive. This is because many PDEs admit non-compact symmetry groups, oftentimes not studied beyond their infinitesimal generators, making them incompatible with most existing equivariant architectures. In this work, we propose Lie aLgebrA Canonicalization (LieLAC), a novel approach that exploits only the action of infinitesimal generators of the symmetry group, circumventing the need for knowledge of the full group structure. To achieve this, we address existing theoretical issues in the canonicalization literature, establishing connections with frame averaging in the case of continuous non-compact groups. Operating within the framework of canonicalization, LieLAC can easily be integrated with unconstrained pre-trained models, transforming inputs to a canonical form before feeding them into the existing model, effectively aligning the input for model inference according to allowed symmetries. LieLAC utilizes standard Lie group descent schemes, achieving equivariance in pre-trained models. Finally, we showcase LieLAC's efficacy on tasks of invariant image classification and Lie point symmetry equivariant neural PDE solvers using pre-trained models.

• The paper introduces LieLAC, a novel method that leverages Lie algebra generators to transform inputs into a canonical form, achieving equivariance in neural network PDE solvers.
• It employs retraction mapping for compact groups and coordinate descent for non-compact cases, addressing the limitations of conventional equivariant architectures.
• Experimental results show significant performance gains in image classification and PDE tasks by improving model generalization on symmetry-transformed inputs.

Lie Algebra Canonicalization: Equivariant Neural Operators under Arbitrary Lie Groups

The paper "Lie Algebra Canonicalization: Equivariant Neural Operators under Arbitrary Lie Groups" tackles the challenge of incorporating symmetries in neural network architectures, specifically in the context of PDE solvers. The authors introduce Lie Algebra Canonicalization (LieLAC), a method that exploits the action of infinitesimal generators of symmetry groups to achieve equivariance. This method is positioned as a solution to the limitations of existing equivariant architectures, which often do not capture the complex symmetry groups of many PDEs.

Key Contributions

The paper's primary contribution is the introduction of LieLAC, a novel approach for achieving equivariance in pre-trained models without requiring complete knowledge of the full group structure. This approach focuses on transforming inputs to a canonical form, leveraging the action of infinitesimal generators. This enables alignment of inputs for model inference, making symmetries explicit and enhancing model performance.

LieLAC is particularly relevant in the setting of non-compact groups, which often arise in PDE contexts. The authors address theoretical issues in canonicalization by making connections with frame averaging for continuous groups. This theoretical foundation allows LieLAC to integrate seamlessly with existing models and improve their generalization capabilities.

Methodology

The paper articulates LieLAC's theoretical setup through a framework that differentiates between frames and canonicalizations for non-compact and finite groups. It introduces weighted canonicalization and closed orbit canonicalization, discussing their theoretical properties and implications.

LieLAC leverages a retraction mapping approach for compact or solvable groups and employs coordinate descent for non-compact cases where only certain vector actions are known. This flexibility makes LieLAC a practical tool for real-world applications.

Experimental Results

Empirical validation is provided through several experiments:

1. Image Classification: The method significantly improves the performance of pre-trained convolutional classifiers on affine and homography-perturbed datasets, achieving substantial boosts in accuracy compared to state-of-the-art methods.
2. Neural PDE Solvers: In PDE settings, LieLAC is applied to physics-informed neural operators for Heat and Burgers' equations, as well as the Allen-Cahn equation with the Poseidon foundation model. The results demonstrate LieLAC's efficacy in aligning pre-trained models with symmetry-transformed inputs, reducing errors, and showing improved generalization on out-of-distribution data.

Implications and Future Directions

The implications of this research are both practical and theoretical. Practically, LieLAC allows for efficient fine-tuning of foundation models with enhanced symmetry handling, which is especially beneficial in scientific computing where data can be limited. Theoretically, the paper offers a robust framework for equivalence via canonicalization, potentially applicable to a broad range of AI models and tasks.

Future research could focus on further optimizing the computational aspects of LieLAC and extending its applicability to other types of PDE symmetries and real-world scientific problems where explicit symmetry incorporation is key.

Conclusion

The introduction of LieLAC presents an effective strategy for achieving equivariance in neural networks using Lie group structures. It offers a solution to implement symmetry properties in existing models without restructuring them entirely. Theoretical insights combined with practical validations demonstrate the potential of LieLAC in advancing the design of equivariant neural operators, essential for bridging gaps in existing architectures unable to capture the rich symmetries of complex scientific datasets.