用符号回归+逻辑编程,从流体数据中自动发现物理公式。
ASP-Assisted Symbolic Regression: Uncovering Hidden Physics in Fluid Mechanics
- 结合符号回归与答案集编程,生成既准确又符合物理规律的方程。
- 成功复现矩形通道内抛物线速度分布和线性压降,与解析解高度一致。
- 适合关注可解释性建模的流体力学研究者或物理信息机器学习方向。
符号回归(SR)为传统机器学习提供了可解释的替代方案,后者常被批评为“黑箱”。与需预设函数形式的标准回归模型不同,SR通过用户定义的数学基本项构造表达式,实现对数据拟合且揭示潜在物理关系的自动化。在流体力学中,理解底层物理与预测精度同等重要。本研究将SR应用于三维层流在矩形通道中的轴向速度与压力场建模,从数值模拟数据中导出紧凑的符号方程,准确再现预期的抛物线速度分布与线性压力梯度,并与文献中的解析解高度吻合。针对纯数据驱动的SR可能忽略领域约束的问题,提出一种创新的混合框架:将符号回归与答案集编程(ASP)相结合。该整合利用SR的生成能力与ASP的声明式推理能力,确保推导出的方程兼具统计准确性与物理合理性。所提出的SR/ASP方法展示了数据驱动与知识表示融合在提升流体动力学等领域的可解释性、可靠性及物理一致性方面的潜力。
原文摘要 · Abstract (English)
Symbolic Regression (SR) offers an interpretable alternative to conventional Machine-Learning (ML) approaches, which are often criticized as ``black boxes''. In contrast to standard regression models that require a prescribed functional form, SR constructs expressions from a user-defined set of mathematical primitives, enabling the automated discovery of compact formulas that fit the data and reveal underlying physical relationships. In fluid mechanics, where understanding the underlying physics is as crucial as predictive accuracy, this study applies SR to model three-dimensional (3D) laminar flow in a rectangular channel, focusing on the axial velocity and pressure fields. Compact symbolic equations were derived from numerical simulation data, accurately reproducing the expected parabolic velocity profile and linear pressure drop, and showing excellent agreement with analytical solutions from the literature. To address the limitation that purely data-driven SR models may overlook domain-specific constraints, an innovative hybrid framework that integrates SR with Answer Set Programming (ASP) is also introduced. This integration combines the generative power of SR with the declarative reasoning capabilities of ASP, ensuring that derived equations remain both statistically accurate and physically plausible. The proposed SR/ASP methodology demonstrates the potential of combining data-driven and knowledge-representation approaches to enhance interpretability, reliability, and alignment with physical principles in fluid dynamics and related domains.
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