用代数几何自动补全科学公理，让AI发现的规律可被理论推导。
Bridging the Gap Between Scientific Laws Derived by AI Systems and Canonical Knowledge via Abductive Inference with AI-Noether
- 基于多项式公理系统，用代数几何推导缺失公理
- 能复现光霍尔效应、相对论等关键物理定律的推导路径
- 适合需要理论自洽性的物理与数学建模研究者
人工智能在加速科学发现方面展现出巨大潜力。符号回归可拟合可解释模型，但这些模型未必能从已有理论推导出。近期系统（如AI-Descartes、AI-Hilbert）虽强制要求从先验知识推导，但当现有理论不完整或错误时，生成的假说可能超出理论范围。自动修正公理系统以弥合这一差距仍是科学发现的核心挑战。我们提出一种开源的代数几何基础系统：给定一个可表示为多项式的不完整公理体系及一个无法由该体系推导出的假说，系统能生成最小候选公理集，加入后可严格证明该假说（即使存在噪声）。我们通过实证表明，该方法可重构推导载流子分辨光电霍尔效应、爱因斯坦相对论定律及其他若干定律所需的关键公理。
原文摘要 · Abstract (English)
Advances in AI have shown great potential in contributing to the acceleration of scientific discovery. Symbolic regression can fit interpretable models to data, but these models are not necessarily derivable from established theory. Recent systems (e.g., AI-Descartes, AI-Hilbert) enforce derivability from prior knowledge. However, when existing theories are incomplete or incorrect, these machine-generated hypotheses may fall outside the theoretical scope. Automatically finding corrections to axiom systems to close this gap remains a central challenge in scientific discovery. We propose a solution: an open-source algebraic geometry-based system that, given an incomplete axiom system expressible as polynomials and a hypothesis that the axioms cannot derive, generates a minimal set of candidate axioms that, when added to the theory, provably derive the (possibly noisy) hypothesis. We illustrate the efficacy of our approach by showing that it can reconstruct key axioms required to derive the carrier-resolved photo-Hall effect, Einstein's relativistic laws, and several other laws.
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