利用李对称性进行偏微分方程的自监督学习

摘要

针对微分方程的机器学习为计算效率高的数值求解器提供了替代方案，可能在科学和工程领域产生广泛影响。尽管当前的算法通常需要针对特定情境定制的模拟训练数据，但有人可能希望从异构来源或来自杂乱或不完整的实际动态系统观测中学习有用信息。在这项工作中，我们通过实施联合嵌入方法进行自监督学习（SSL），从异构数据中学习偏微分方程的通用表示，这是一种在计算机视觉领域取得显著成功的无监督表示学习框架。我们的表示优于基线方法在不变任务上的表现，例如回归偏微分方程的系数，同时也提高了神经求解器的时间步性能。我们希望我们提出的方法论能够在最终发展偏微分方程通用基础模型方面发挥作用。

English

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.

利用李对称性进行偏微分方程的自监督学习

Self-Supervised Learning with Lie Symmetries for Partial Differential Equations

摘要

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