符号神经常微分方程
Symbolic Neural Ordinary Differential Equations
AI总结：
本文提出符号神经常微分方程（SNODEs）框架，通过梯度流匹配预训练、神经常微分方程微调和残差网络三阶段学习复杂系统动力学，兼具通用逼近性、可解释性和外推能力，可应用于分岔控制、重构预测及方程发现等科学问题。
摘要：
微分方程被广泛用于描述自然界和工程中参数不断演化的复杂动力系统。有效学习从参数函数到系统动力学的一族映射具有重要意义。在本研究中，我们提出了一种新颖的符号连续深度神经网络学习框架，称为符号神经常微分方程（SNODEs），以有效且准确地学习复杂系统的底层动力学。具体而言，我们的学习框架包括三个阶段：首先，通过梯度流匹配策略预训练一个预定义的符号神经网络；随后，使用神经常微分方程对该网络进行微调；最后，构建一个通用神经网络来捕捉残差。在这一过程中，我们通过傅里叶分析将SNODEs框架应用于偏微分方程系统，实现了分辨率不变的建模。此外，该框架融合了符号主义与联结主义的优势，具备通用逼近定理，同时相对于最先进的基线方法显著增强了可解释性和外推能力。我们通过在若干代表性复杂系统上的实验证明了这一点。因此，我们的框架可以进一步应用于广泛的科学问题，例如系统分岔与控制、重构与预测，以及新方程的发现。
英文摘要：
Differential equations are widely used to describe complex dynamical systems with evolving parameters in nature and engineering. Effectively learning a family of maps from the parameter function to the system dynamics is of great significance. In this study, we propose a novel learning framework of symbolic continuous-depth neural networks, termed Symbolic Neural Ordinary Differential Equations (SNODEs), to effectively and accurately learn the underlying dynamics of complex systems. Specifically, our learning framework comprises three stages: initially, pre-training a predefined symbolic neural network via a gradient flow matching strategy; subsequently, fine-tuning this network using Neural ODEs; and finally, constructing a general neural network to capture residuals. In this process, we apply the SNODEs framework to partial differential equation systems through Fourier analysis, achieving resolution-invariant modeling. Moreover, this framework integrates the strengths of symbolism and connectionism, boasting a universal approximation theorem while significantly enhancing interpretability and extrapolation capabilities relative to state-of-the-art baseline methods. We demonstrate this through experiments on several representative complex systems. Therefore, our framework can be further applied to a wide range of scientific problems, such as system bifurcation and control, reconstruction and forecasting, as well as the discovery of new equations.
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