Self-Supervised Learning with Lie Symmetries for Partial Differential Equations (2307.05432v2)

Abstract: Machine learning for differential equations paves the way for computationally efficient alternatives to numerical solvers, with potentially broad impacts in science and engineering. Though current algorithms typically require simulated training data tailored to a given setting, one may instead wish to learn useful information from heterogeneous sources, or from real dynamical systems observations that are messy or incomplete. In this work, we learn general-purpose representations of PDEs from heterogeneous data by implementing joint embedding methods for self-supervised learning (SSL), a framework for unsupervised representation learning that has had notable success in computer vision. Our representation outperforms baseline approaches to invariant tasks, such as regressing the coefficients of a PDE, while also improving the time-stepping performance of neural solvers. We hope that our proposed methodology will prove useful in the eventual development of general-purpose foundation models for PDEs. Code: https://github.com/facebookresearch/SSLForPDEs.

• The paper introduces a framework that integrates self-supervised learning with Lie symmetry-based augmentations to extract invariant representations from PDE data.
• It demonstrates significant improvements in regression accuracy across equations like Burgers’ and Navier-Stokes compared to traditional supervised methods.
• The approach paves the way for foundation models in scientific computing by harnessing intrinsic mathematical structures for robust, generalizable solutions.

Self-Supervised Learning with Lie Symmetries for Partial Differential Equations

The paper of partial differential equations (PDEs) is a foundational aspect of understanding dynamical systems in various scientific and engineering disciplines. The paper "Self-Supervised Learning with Lie Symmetries for Partial Differential Equations" introduces a novel approach to enhancing the utility and efficiency of machine learning models in this domain. The authors propose a framework for self-supervised learning (SSL) that leverages Lie symmetries inherent in PDEs to derive general-purpose representations from a diverse pool of data.

Methodological Advancements

The core contribution of this work lies in the adaptation of self-supervised learning, a paradigm widely hailed for its success in machine learning tasks for computer vision, particularly in unsupervised representation learning. By integrating joint embedding methods with PDE data, the authors develop a unique model capable of outperforming traditional baseline approaches in invariant tasks, such as regressing PDE coefficients.

In exploring the intrinsic symmetries of PDEs, the framework utilizes Lie point symmetries to define data augmentations, which are crucial in the SSL process. These symmetries allow for transformations that maintain the solution set of a PDE, thus mimicking natural variations without losing the essential properties of the data. The use of Lie point symmetries facilitates the learning of representations that are invariant under specific transformations, capturing the underlying dynamics of the PDEs effectively.

Numerical Results and Implications

Empirical evaluations demonstrate the efficacy of this approach. The paper reports a significant improvement in parameter regression tasks across different equations, namely the Korteweg-de Vries (KdV), Kuramoto-Sivashinsky (KS), viscous Burgers', and Navier-Stokes equations. Notably, for kinematic viscosity regression in Burgers' equation, the proposed method reduces the relative error below that achievable through supervised methods. The inclusion of SSL representations also enhances the time-stepping performance of neural solvers for these equations.

Moreover, the successful implementation of SSL in extracting meaningful representations indicates potential for broader application. The authors suggest that these methods may be extended to form the basis of foundation models for PDEs, analogous to foundational models in other machine learning domains.

Broader Implications and Future Directions

The integration of Lie symmetries into SSL frameworks paves the way for advancements in scientific computing, particularly in contexts requiring the analysis of noisy or incomplete real-world data. The results are promising for the development of computationally efficient and generalizable models capable of understanding complex systems governed by PDEs.

The paper speculates on several future directions, including extending the approach to accommodate more complex dynamical systems, applying the methodology to real-world scientific data where governing equations are not explicitly known, and exploring advanced types of symmetries beyond Lie point symmetries. A noteworthy potential direction is the development of models with equivariant representations, which could strongly enhance the robustness and transferability of learned features across different tasks and systems.

Overall, this research contributes a vital step towards creating robust machine learning models that can serve numerous scientific fields by providing computational tools that harness the rich mathematical structures present in PDEs.