用物理约束的稀疏算法，从噪声数据中精准发现非线性系统的微分方程。
Physics-informed AI and ML-based sparse system identification algorithm for discovery of PDE's representing nonlinear dynamic systems
- 结合B样条与逐次正则化导数去噪，提升导数精度并保留系统信息。
- 通过无相关成分分析剔除冗余函数，准确识别真实方程结构。
- 物理信息样条拟合逐步收敛，对高阶、刚性方程和噪声数据均有效。
非线性动态系统的稀疏系统辨识仍具挑战性，尤其针对含噪声测量数据的刚性及高阶微分方程。高度相关的基函数使真伪函数难以区分，限制了候选函数选择。本文提出一种方程发现方法，核心包括：a）采用B样条进行数据拟合并获取优于数值导数的解析导数；b）逐次正则化导数去噪（SRDD）算法，可有效去除信号噪声而不丢失系统信息；c）无相关成分分析（UCA）算法，能识别并剔除高度相关的函数，同时保留真实函数；d）物理信息样条拟合（PISF），在候选函数字典中逐步更新样条拟合，同时满足控制方程，实现方程的序列收敛。整个框架基于统一深度学习架构，简化优化过程。所提方法在多种噪声水平下成功发现三维、四阶及刚性微分方程，参数估计收敛至真实值，变异系数小，表现出强鲁棒性。
原文摘要 · Abstract (English)
Sparse system identification of nonlinear dynamic systems is still challenging, especially for stiff and high-order differential equations for noisy measurement data. The use of highly correlated functions makes distinguishing between true and false functions difficult, which limits the choice of functions. In this study, an equation discovery method has been proposed to tackle these problems. The key elements include a) use of B-splines for data fitting to get analytical derivatives superior to numerical derivatives, b) sequentially regularized derivatives for denoising (SRDD) algorithm, highly effective in removing noise from signal without system information loss, c) uncorrelated component analysis (UCA) algorithm that identifies and eliminates highly correlated functions while retaining the true functions, and d) physics-informed spline fitting (PISF) where the spline fitting is updated gradually while satisfying the governing equation with a dictionary of candidate functions to converge to the correct equation sequentially. The complete framework is built on a unified deep-learning architecture that eases the optimization process. The proposed method is demonstrated to discover various differential equations at various noise levels, including three-dimensional, fourth-order, and stiff equations. The parameter estimation converges accurately to the true values with a small coefficient of variation, suggesting robustness to the noise.
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