标题： DUE：用于建模未知方程的深度学习框架与库 显示英文标题

标题： DUE: A Deep Learning Framework and Library for Modeling Unknown Equations

摘要： 方程，特别是微分方程，对于理解自然现象和预测各科学与工程学科中的复杂动力学至关重要。 然而，由于复杂的底层机制，许多复杂系统的控制方程仍然是未知的。 机器学习和数据科学领域的最新进展为从测量或模拟数据中建模未知方程提供了新的范式。 这一范式转变，即数据驱动的发现或建模，处于科学人工智能的前沿，并在过去几年取得了显著进展。 本文介绍了一个用于使用深度学习进行未知方程数据驱动建模的系统框架。 该多功能框架能够学习未知的常微分方程（ODEs）、偏微分方程（PDEs）、微分代数方程（DAEs）、积分微分方程（IDEs）、随机微分方程（SDEs）、降阶或部分观测系统以及非自治微分方程。 基于此框架，我们开发了Deep Unknown Equations (DUE)，这是一个开源软件包，旨在利用现代深度学习技术促进未知方程的数据驱动建模。 DUE 是一种教学工具，可用于课堂讲授，使学生和初学者能够获得关于微分方程、数据驱动建模以及当代深度学习方法（如FNN、ResNet、广义ResNet、算子半群网络（OSG-Net）和Transformer）的实际经验。 此外，DUE 是一个多功能且易于访问的研究工具，适用于各个科学和工程领域的研究人员。 它不仅适用于从数据中学习未知方程，还适用于代理建模那些已知但复杂且使用传统数值方法求解成本高昂的方程。 我们详细描述了DUE，并通过多样化的示例展示了其能力，这些示例可作为模板轻松适应其他应用。 显示英文摘要

摘要： Equations, particularly differential equations, are fundamental for understanding natural phenomena and predicting complex dynamics across various scientific and engineering disciplines. However, the governing equations for many complex systems remain unknown due to intricate underlying mechanisms. Recent advancements in machine learning and data science offer a new paradigm for modeling unknown equations from measurement or simulation data. This paradigm shift, known as data-driven discovery or modeling, stands at the forefront of AI for science, with significant progress made in recent years. In this paper, we introduce a systematic framework for data-driven modeling of unknown equations using deep learning. This versatile framework is capable of learning unknown ODEs, PDEs, DAEs, IDEs, SDEs, reduced or partially observed systems, and non-autonomous differential equations. Based on this framework, we have developed Deep Unknown Equations (DUE), an open-source software package designed to facilitate the data-driven modeling of unknown equations using modern deep learning techniques. DUE serves as an educational tool for classroom instruction, enabling students and newcomers to gain hands-on experience with differential equations, data-driven modeling, and contemporary deep learning approaches such as FNN, ResNet, generalized ResNet, operator semigroup networks (OSG-Net), and Transformers. Additionally, DUE is a versatile and accessible toolkit for researchers across various scientific and engineering fields. It is applicable not only for learning unknown equations from data but also for surrogate modeling of known, yet complex, equations that are costly to solve using traditional numerical methods. We provide detailed descriptions of DUE and demonstrate its capabilities through diverse examples, which serve as templates that can be easily adapted for other applications.