维度一致的替代建模：通过量纲分析与调和展开
Dimensionally consistent surrogate modelling through dimensional analysis and harmonic expansions
- Universidad Europea de Valencia（瓦伦西亚欧洲大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出一种通过量纲分析和调和展开构建满足量纲齐次性约束的替代模型方法，在多个物理基准上验证了其改善条件数、鲁棒性和样本效率的优势。
AI中文摘要：
量纲齐次性是物理上有意义模型的基本约束，要求模型在单位变化下保持不变。我们提出了一种数据驱动方法，用于在假设类层面构建满足此约束的替代模型。该方法从测量变量的量纲矩阵出发，推导出白金汉Π群，构造可接受的量纲前因子，并在归一化不变域上使用截断调和展开来近似剩余的无量纲依赖关系。一旦前因子和字典固定，系数通过正则化线性回归问题获得。我们在单摆、普朗克黑体辐射定律、双摆李雅普诺夫场以及实验性COBE/FIRAS黑体辐射光谱数据集上测试了该方法。结果表明，与无约束基线相比，量纲约束改善了条件数、对噪声的鲁棒性和样本效率，而字典的选择在非周期或多不变设置中变得重要。学习到的表达式是显式的且评估成本低，这使得它们作为结构化物理问题的替代模型很有用。
英文摘要：
Dimensional homogeneity is a fundamental constraint on physically meaningful models, requiring invariance under changes of units. We present a data-driven method for constructing surrogate models that satisfy this constraint at the level of the hypothesis class. Starting from a dimension matrix of measured variables, the method derives Buckingham $Π$-groups, constructs admissible dimensional prefactors, and approximates the remaining dimensionless dependence using truncated harmonic expansions on normalized invariant domains. Once the prefactor and dictionary are fixed, the coefficients are obtained from a regularized linear regression problem. We test the approach on the simple pendulum, Planck's black-body law, the double-pendulum Lyapunov field, and an experimental COBE/FIRAS black-body spectrum dataset. The results show that dimensional constraints improve conditioning, robustness to noise, and sample efficiency relative to unconstrained baselines, while the choice of dictionary becomes important in non-periodic or multi-invariant settings. The learned expressions are explicit and inexpensive to evaluate, which makes them useful as surrogate models for structured physical problems.
