NestyNet. III. 基于解析神经代理的符号回归
NestyNet. III. Symbolic Regression from Analytic Neural Surrogates
AI总结：
本文提出NestyNet-SR方法，通过带解析导数的神经代理结合分层符号搜索，在SRBench AI Feynman基准实现无噪声方程精确恢复，还能从星系质量数据发现重子加速度坐标等物理规律。
AI中文摘要：
许多物理定律只有在找到合适的表示、分解或内部坐标后才会变得简单，但从数据中发现这种结构具有组合复杂性，这项任务就是符号回归（SR），即搜索无需假设固定模型类别的、能拟合数据的闭式表达式。本文提出NestyNet-SR：一种带有解析导数的神经代理，用于检测可分性，递归地将多元问题简化为更简单的神经原子；这些原子通过分层符号搜索栈被提炼为闭式形式，该栈的最后一层是一种新颖的因子化符号搜索，可将结构与校准分离。该方法自由组合候选内部坐标，通过其校准函数（如多项式、幂律、正弦函数等）拟合数据的程度对每个坐标打分，因此这些校准映射的常数无论在最终表达式中嵌套多深，都能被拟合而非搜索。该方法支持多数据集回归、自动特征发现和量纲分析剪枝。在SRBench AI Feynman基准测试中，NestyNet-SR实现了全部120个无噪声方程的精确符号恢复，这是首个达成该结果的方法；在有噪声情况下，统计审计可验证哪些结构能保留。作为真实数据示例，仅给定SPARC巡天星系的独立质量模型分量，该算法可发现重子加速度坐标，复现已确立的质光比、加速度标度以及径向加速度关系的非唯一形式，还能实现保留星系的泛化、校准对称性弃权（不执行）以及定律局部斜率的后验分布。因此，解析导数提供了从神经代理到可解释闭式经验定律的实用途径。
英文摘要：
Many physical laws are simple only after the right representation, decomposition or internal coordinate has been found, but discovering that structure from data is combinatorially hard. This task is symbolic regression (SR), the search for closed-form expressions that fit data without assuming a fixed model class. Here we present NestyNet-SR. A neural surrogate with analytic derivatives is used to detect separability, recursively reducing multivariate problems to simpler neural atoms. These atoms are distilled into closed form by a tiered symbolic-search stack, whose final tier is a novel factorized symbolic search that separates structure from calibration. Composing candidate internal coordinates freely, it scores each coordinate by how well calibrated functions of it (e.g., polynomials, power laws, sinusoids) fit the data, so the constants of those calibrated maps, however deeply nested in the final expression, are fitted rather than searched. The method supports multi-dataset regression, automated feature discovery, and dimensional-analysis pruning. On the SRBench AI~Feynman benchmark, NestyNet-SR achieves exact symbolic recovery of all 120 noiseless equations, the first such result, and under noise a statistical audit certifies which structures survive. As a real-data vignette, given only the separate mass-model components of SPARC-survey galaxies, the algorithm discovers the baryonic acceleration coordinate, reproduces the established mass-to-light and acceleration scales and the non-unique form of the radial acceleration relation, and adds held-out-galaxy generalization, a calibrated symmetry abstention, and a posterior for the local slope of the law. Analytic derivatives thus provide a practical route from neural surrogates to interpretable closed-form empirical laws.
