利用李對稱進行偏微分方程的自監督學習

摘要

應用機器學習於微分方程式研究為數值求解器提供了高效的替代方案，可能對科學和工程領域產生廣泛影響。儘管目前的演算法通常需要針對特定情境量身定制的模擬訓練數據，但我們可能希望從異質來源或來自雜亂或不完整的實際動態系統觀測中學習有用信息。在這項研究中，我們通過實施聯合嵌入方法進行自監督學習（SSL），從異質數據中學習偏微分方程式的通用表示，這是一種在計算機視覺領域取得顯著成功的非監督表示學習框架。我們的表示優於基準方法在不變任務上的表現，例如回歸偏微分方程式的係數，同時也提高了神經求解器的時間步進效能。我們希望我們提出的方法論將對偏微分方程式的通用基礎模型的最終發展證明有用。

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.

利用李對稱進行偏微分方程的自監督學習

Self-Supervised Learning with Lie Symmetries for Partial Differential Equations

摘要

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