PDEs (Some PDE solutions)

从非常有限和高噪声的观察中发现时空数据集背后的偏微分方程是许多科学领域最感兴趣的。然而，了解基于稀疏回归的模型发现算法何时可以真正恢复底层物理过程仍然是一个悬而未决的问题。在这项工作中，我们展示了用于通过稀疏回归推断方程的设计矩阵可能违反 Lasso 的不可表示性条件 (IRC)，即使是从解析 PDE 解（即没有额外噪声）导出的也是如此。因此，需要能够在违反 IRC 条件下恢复真实基础模型的稀疏回归技术，从而引入了随机自适应 Lasso。我们表明，一旦将后者集成到深度学习模型发现框架 DeepMod 中，就可以恢复各种非线性和混沌规范 PDE：(1) 噪声与样本比比 state-of- 高 O(2)最先进的算法，(2) 具有一组超参数，这为真正自动化的模型发现铺平了道路。Discovering partial differential equations (PDEs) underlying spatiotemporal datasets from extremely limited and highly noisy observations is a topic of great interest across numerous scientific fields. However, it remains an open question to determine when sparse regression-based model discovery algorithms can truly recover the underlying physical processes. In this work, we demonstrate that the design matrices used for inferring equations via sparse regression may violate the Irrepresentable Condition (IRC) of Lasso, even when derived from analytical PDE solutions (i.e., without additional noise). Accordingly, there is a need for sparse regression techniques that can recover the true underlying model even when the IRC is violated, leading us to introduce the randomized adaptive Lasso. We show that once this method is integrated into the deep learning-based model discovery framework DeepMod, a wide range of nonlinear and chaotic canonical PDEs can be recovered: (1) at noise-to-sample ratios higher than those of state-of-the-art algorithms, and (2) using a fixed set of hyperparameters, which paves the way for truly automated model discovery.

搜集汇总

背景与挑战

背景概述

该数据集专注于从有限且高噪声的时空观测中，通过稀疏回归技术（如随机自适应Lasso）发现背后的偏微分方程（PDE），并将其集成到深度学习框架DeepMod中，以恢复非线性和混沌PDE。它由巴黎大学于2021年发布，旨在提高模型发现的自动化水平。