用复数乘积单元直接从数据中发现复杂动力系统方程
Model discovery for dynamical systems with complex-valued product units
- 通过复数单项式网络自动学习动态系统方程，无需预设函数库
- 在4个混沌系统上90%试验中准确恢复方程，分数阶情形70-90%成功
- 适用于高维真实数据，如人体步态信号，预测误差仅占信号幅值12-14%
从观测轨迹中发现动力系统的控制方程，比单纯预测未来状态更能揭示其内在结构。本文提出一种基于复数乘积单元网络的数据驱动模型发现方法，每个单元代表一个复数单项式，网络输出为这些单项式的稀疏线性组合。与传统的基于候选函数库的方法（如SINDy）不同，该方法无需预先设定函数集，可直接从数据中学习包含分数或负指数的有用单项式。在四个混沌基准系统（Lorenz63、Lorenz84、Four-Wing吸引子及分数阶Lorenz63）上，前三个系统在至少3000个训练点下90%试验中精确恢复方程，分数阶情形成功率70%-90%。应用于真实人体步态加速度信号时，模型生成稳定轨迹，预测误差在测试区间长达训练长度三倍的情况下仍保持在信号幅值范围的12%-14%之间，展现出对无解析方程的高维系统建模潜力。
原文摘要 · Abstract (English)
Discovering the governing equations of a dynamical system from observed trajectories provides deeper insight into its structure than mere prediction of future states. We present a data-driven approach to model discovery based on complex-valued product-unit networks, in which each unit represents a complex monomial and the network output is a sparse linear combination of such monomials. In contrast to established library-based methods such as SINDy, our approach does not require a predefined set of candidate functions: the relevant monomials, including those with fractional or negative exponents, are learned directly from data. Across four chaotic benchmark systems (Lorenz63, Lorenz84, the Four-Wing attractor, and a fractional variant of Lorenz63), we recover the exact governing equations in 90% of trials for the first three systems, and in 70-90% of trials for the fractional case, using at least 3000 training points. Applied to real-world human-gait accelerometer signals, the model produced stable trajectories with bounded prediction errors, corresponding to an RMSE of approximately 12-14% of the signal amplitude range over a test horizon three times longer than the training interval, demonstrating its potential for high-dimensional systems in which analytic equations are unavailable.
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