Notes on Self-Supervised Learning with Lie Symmetries for Partial Differential Equations
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Link to paper: https://arxiv.org/abs/2307.05432
Paper published on: 2023-07-11
Paper's authors: Grégoire Mialon, Quentin Garrido, Hannah Lawrence, Danyal Rehman, Yann LeCun, Bobak T. Kiani
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Summary and Introduction
The paper in focus provides a novel approach to learning representations for Partial Differential Equations (PDEs) using self-supervised learning (SSL) with joint embedding methods. The authors propose a framework based on the joint-embedding SSL paradigm, leveraging symmetry groups of PDEs to create invariant embeddings and improve the performance of neural solvers.
Consider the scenario where we have a large dataset of images of cars taken from various angles. A traditional machine learning model might struggle to recognize that the same car from different angles is still the same car. However, by using symmetry groups (in this case, the different angles the car can be viewed from), we can train a model to understand that these different perspectives are all representations of the same object.
In this case, the authors apply a similar concept to PDEs, using symmetry groups to create invariant embeddings, which leads to a more efficient and robust model.
Understanding PDEs and Symmetry Groups
A PDE is a system of equations in independent variables x and dependent variables u with derivatives of u with respect to x. A symmetry group of a PDE acts on the total space and takes solutions of the PDE to other solutions of the PDE.
In simpler terms, think of a PDE like a complex mathematical puzzle, where different pieces (variables and their derivatives) interact to form a complete picture (the solution). A symmetry group is like a rule that allows us to rearrange or transform these pieces without changing the overall picture.
The authors focus on a specific type of symmetry group called Lie point symmetries, which act smoothly at any given point in the total space. These symmetries are used as SSL augmentations in the proposed framework.
Lie Point Symmetries and Augmentations
The authors present a general framework for SSL for PDEs using symmetry groups and demonstrate its utility on various downstream tasks. The methodology includes self-supervised pretraining and evaluation frameworks, as well as adaptations of SSL loss functions and augmentations for PDE data.
Lie point transformations can be written as one-parameter transforms, with the transformation at ϵ = 0 recovering the identity map. PDE symmetry groups can be used as SSL augmentations, but there are challenges in selecting the right symmetries and tuning the augmentations. The authors found that Lie point augmentations applied at relatively small strengths perform the best.
Implementing the Framework
The authors consider four equations for their methodology: viscous Burgers' equation, Korteweg-de Vries (KdV) equation, Kuramoto-Sivashinsky (KS) equation, and incompressible Navier-Stokes equation. They use SSL to pretrain a ResNet18 model on unlabeled data, and then use this pretrained model for downstream tasks such as equation parameter regression and time-stepping.
For equation parameter regression, the research shows that the SSL approach outperforms the supervised baseline in regressing coefficients for Burgers' equation and Navier-Stokes equation, as well as initial conditions for KdV and KS equations. For time-stepping, the research shows that conditioning the neural operator with the pretrained representation improves performance for KdV and KS equations, while the results are mixed for Burgers' equation and Navier-Stokes equation.
The VICReg Loss Function
The authors introduce a new loss function called VICReg for SSL. The VICReg loss function consists of three terms: a similarity term, a variance term, and a covariance term. The similarity term ensures that the embeddings are similar for pairs of inputs related by an augmentation. The variance term ensures that all dimensions in the representations are used and serves as a normalization. The covariance term decorrelates the different dimensions to spread out information across the embeddings.
Future Directions and Limitations
The authors discuss potential future directions, including expanding SSL to other types of PDEs and learning explicitly equivariant features. They also acknowledge that theoretical insights can help improve the results, such as preserving boundary conditions during augmentation and considering other types of symmetries. The work has limitations and can be expanded to real-world data and other applications.
Conclusion
This research provides a novel approach to learning representations for PDEs using SSL with joint embedding methods. By leveraging symmetry groups of PDEs to create invariant embeddings, the authors propose a framework that outperforms baseline approaches and improves the performance of neural solvers. The introduced VICReg loss function and the use of Lie point symmetries as augmentations contribute to the effectiveness of the proposed methodology.
