用张量方法自动发现高维系统的不变性控制方程，精度效率双提升。
CTSR: Cartesian tensor-based sparse regression for data-driven discovery of high-dimensional invariant governing equations
- 基于笛卡尔张量的稀疏回归，实现旋转反射不变性
- 2D和3D测试中精度优于传统方法，计算高效
- 适合高维复杂系统动力学建模，无需额外先验知识
精确且简洁的控制方程对理解系统动态至关重要。近年来，稀疏回归等数据驱动方法被用于从数据中自动发现控制方程，标志着从传统第一性原理建模的重大转变。然而，现有方法多聚焦于标量方程，难以应用于复杂高维场景，且在不引入显著计算成本或额外先验知识的前提下，难以保证旋转与反射不变性。本文提出一种基于笛卡尔张量的稀疏回归（CTSR）方法，可高效准确地发现复杂高维系统的控制方程，并确保其不变性。在两个二维（2D）和两个三维（3D）测试案例上的评估表明，该方法在精度和效率上均优于传统技术。
原文摘要 · Abstract (English)
Accurate and concise governing equations are crucial for understanding system dynamics. Recently, data-driven methods such as sparse regression have been employed to automatically uncover governing equations from data, representing a significant shift from traditional first-principles modeling. However, most existing methods focus on scalar equations, limiting their applicability to simple, low-dimensional scenarios, and failing to ensure rotation and reflection invariance without incurring significant computational cost or requiring additional prior knowledge. This paper proposes a Cartesian tensor-based sparse regression (CTSR) technique to accurately and efficiently uncover complex, high-dimensional governing equations while ensuring invariance. Evaluations on two two-dimensional (2D) and two three-dimensional (3D) test cases demonstrate that the proposed method achieves superior accuracy and efficiency compared to the conventional technique.
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