从多参数模拟数据中自动发现可解释的湍流闭包模型，无需预设形式。
Parameter-Aware Ensemble SINDy for Interpretable Symbolic SGS Closure
- 基于SINDy框架，引入参数化稀疏回归，统一处理多参数数据。
- 在过滤伯格斯方程上发现$τ_{\mathrm{SGS}} = 0.1604\cdotΔ^2(\partial \bar{u}/\partial x)^2$，$R^2=0.885$。
- 能自动识别物理结构并校准系数，适合数据驱动的湍流建模研究。
本文设计了一种可扩展、参数感知的稀疏回归框架，用于从多参数模拟数据中发现可解释的偏微分方程与亚网格尺度（SGS）闭包。基于SINDy（非线性动力学稀疏识别），通过四项改进克服关键局限：一、符号参数化使物理参数在统一回归中变化；二、量纲相似性过滤器在保持单位一致性的同时减少候选项；三、内存高效的格矩阵累积实现大规模数据批量处理；四、集成共识与系数稳定性分析确保模型鲁棒性。在经典一维基准测试中，跨参数范围稳定发现控制方程。应用于过滤伯格斯数据集时，框架自主发现$τ_{\mathrm{SGS}} = 0.1604\cdotΔ^2(\partial \bar{u}/\partial x)^2$，SINDy发现的Smagorinsky常数$C_s^{\text{SINDy}} \approx 0.4005$，无预设假设，直接从数据恢复出Smagorinsky型结构。该模型在滤波尺度上$R^2 = 0.885$，预测精度优于经典闭包方法。该框架识别物理意义闭包形式并校准系数的能力，为现有湍流建模方法提供了补充，推动数据驱动湍流闭包发现的发展。
原文摘要 · Abstract (English)
This work designs a scalable, parameter-aware sparse regression framework for discovering interpretable partial differential equations and subgrid-scale closures from multi-parameter simulation data. Building on SINDy (Sparse Identification of Nonlinear Dynamics), the approach addresses key limitations through four enhancements. First, symbolic parameterisation enables physical parameters to vary within unified regression. Second, the Dimensional Similarity Filter enforces unit consistency while reducing candidate libraries. Third, memory-efficient Gram-matrix accumulation enables batch processing of large datasets. Fourth, ensemble consensus with coefficient stability analysis ensures robust model identification. Validation on canonical one-dimensional benchmarks demonstrates consistent discovery of governing equations across parameter ranges. Applied to filtered Burgers datasets, the framework autonomously discovers the SGS closure $τ_{\mathrm{SGS}} = 0.1604\cdotΔ^2\left(\frac{\partial \bar{u}}{\partial x}\right)^2$ with the SINDy-discovered Smagorinsky constant $C_s^{\text{SINDy}} \approx 0.4005$ without predefined closure assumptions, recovering Smagorinsky-type structure directly from data. The discovered model achieves $R^2 = 0.885$ across filter scales and demonstrates improved prediction accuracy compared to classical SGS closures. The ability of the framework to identify physically meaningful SGS forms and calibrate coefficients offers a complementary approach to existing turbulence modelling methods, contributing to the broader field of data-driven turbulence closure discovery.
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