论文标题
从噪声数据中对部分微分方程的机器学习
Machine Learning of Partial Differential Equations from Noise Data
论文作者
论文摘要
从数据中对部分微分方程的机器学习是解决复杂动态系统缺乏物理方程的潜在突破，但是由于数值差异不适合噪声数据，因此噪声已成为应用部分微分方程识别方法应用的最大障碍。为了克服这个问题，我们提出了基于傅立叶变换的频域识别方法，通过使用频域数据的低频组件来确定频域中的部分微分方程，从而有效地消除了噪声的影响。我们还提出了一个新的稀疏识别标准，可以从低信噪比数据中准确识别等式中的术语。通过识别跨越许多科学领域的各种规范方程，事实证明，该方法具有高度准确性和鲁棒性，用于方程结构和参数识别，以识别低信噪比数据。该方法提供了一种有希望的技术，可以从嘈杂的实验数据中发现潜在的偏微分方程。
Machine learning of partial differential equations from data is a potential breakthrough to solve the lack of physical equations in complex dynamic systems, but because numerical differentiation is ill-posed to noise data, noise has become the biggest obstacle in the application of partial differential equation identification method. To overcome this problem, we propose Frequency Domain Identification method based on Fourier transforms, which effectively eliminates the influence of noise by using the low frequency component of frequency domain data to identify partial differential equations in frequency domain. We also propose a new sparse identification criterion, which can accurately identify the terms in the equation from low signal-to-noise ratio data. Through identifying a variety of canonical equations spanning a number of scientific domains, the proposed method is proved to have high accuracy and robustness for equation structure and parameters identification for low signal-to-noise ratio data. The method provides a promising technique to discover potential partial differential equations from noisy experimental data.
