Основні поняття
This paper introduces Lie Algebra Canonicalization (LieLAC), a novel method for achieving equivariance in pre-trained neural networks by transforming inputs to a canonical form, leveraging Lie group theory to enhance performance in image classification and PDE solving tasks.
Анотація
Bibliographic Information: Shumaylov, Zakhar, et al. "LIE ALGEBRA CANONICALIZATION: EQUIVARIANT NEURAL OPERATORS UNDER ARBITRARY LIE GROUPS." arXiv preprint arXiv:2410.02698 (2024).
Research Objective: This paper aims to address the limitations of existing equivariant neural network architectures that struggle to encode complex symmetries found in scientific applications. The authors propose a novel method called Lie Algebra Canonicalization (LieLAC) to induce equivariance in pre-trained neural networks, particularly for image classification and PDE solving tasks.
Methodology: LieLAC leverages the concept of energy-based canonicalization, where an energy function is minimized to find a canonical representation of the input data. The authors extend this approach to handle non-compact Lie groups, which are common in scientific applications, by introducing weighted canonicalizations and closed canonicalizations. They provide theoretical grounding for their method and connect it to existing concepts like frames and orbit canonicalizations. The authors demonstrate the effectiveness of LieLAC on invariant image classification tasks using affine and homography transformations and on PDE solving tasks using the heat equation, Burgers' equation, and the Allen-Cahn equation. They compare LieLAC with existing equivariant architectures and demonstrate its superior performance.
Key Findings: The authors show that LieLAC can effectively induce equivariance in pre-trained neural networks, leading to improved performance on invariant image classification tasks. They demonstrate that LieLAC achieves higher test accuracy on affine-perturbed and homography-perturbed MNIST datasets compared to existing equivariant architectures. Furthermore, the authors show LieLAC's efficacy in solving PDEs with non-trivial symmetry groups, such as the heat equation, Burgers' equation, and the Allen-Cahn equation. They demonstrate that LieLAC can improve the performance of pre-trained physics-informed neural operators, such as POSEIDON, on these tasks.
Main Conclusions: LieLAC offers a practical and effective way to induce equivariance in pre-trained neural networks, leveraging the power of Lie group theory. This approach overcomes limitations of existing equivariant architectures and can be applied to various domains, including image classification and PDE solving. The authors suggest that LieLAC has the potential to significantly impact scientific machine learning by enabling the development of more robust and generalizable models.
Significance: This research significantly contributes to the field of equivariant neural networks by providing a practical method for inducing equivariance in pre-trained models. This approach has the potential to be widely adopted in various scientific domains, leading to more efficient and robust machine learning models.
Limitations and Future Research: While LieLAC shows promising results, the authors acknowledge the challenges posed by non-convex optimization in finding the optimal canonical representation. Future research could explore more sophisticated optimization techniques tailored for Lie groups to further enhance LieLAC's performance. Additionally, investigating the applicability of LieLAC to other scientific domains beyond image classification and PDE solving would be valuable.
Статистика
CNN achieves 0.985 test accuracy on MNIST and 0.629 on affNIST.
LieLAC [CNN] achieves 0.979 test accuracy on MNIST and 0.972 on affNIST.
affConv achieves 0.982 test accuracy on MNIST and 0.943 on affNIST.
CNN achieves 0.985 test accuracy on MNIST and 0.644 on homNIST.
LieLAC [CNN] achieves 0.982 test accuracy on MNIST and 0.960 on homNIST.
homConv achieves 0.980 test accuracy on MNIST and 0.927 on homNIST.
POSEIDON achieves 6.448 × 10−4 ID test error and 7.619 × 10−3 OOD test error on the ACE task.
LieLAC [POSEIDON] achieves 1.592 × 10−3 ID test error and 2.916 × 10−3 OOD test error on the ACE task.
LieLAC [POSEIDON+ ft.] achieves 9.667 × 10−4 ID test error and 1.143 × 10−3 OOD test error on the ACE task.
Цитати
"Incorporating these symmetries into the design of neural networks can enhance their performance and generalization capabilities."
"Prior work on equivariant neural networks focuses on “simple” groups, resulting in frameworks that are often not rich enough to encode the complex geometric structure found in scientific applications."
"This flexibility allows for integration with existing models, notably pre-trained models, with the potential of leveraging the benefits of geometric inductive biases beyond classical equivariant architectures."
"However, it still remains an open question, whether it is possible to make existing physics-informed approaches equivariant. In this paper we directly tackle this question."