# Bridging the Gap Between Scientific Laws Derived by AI Systems and Canonical Knowledge via Abductive Inference with AI-Noether

arXiv: https://arxiv.org/abs/2509.23004 · HTML: https://mlsanity.com/paper/2509.23004/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 96 references that are in the map.
- First posted on arXiv 26 Sep 2025.

## Builds on (most cited first, 25 of 25)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [The AI Scientist: Towards Fully Automated Open-Ended Scientific Discovery](https://mlsanity.com/paper/2408.06292/) (2024)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [LeanDojo: Theorem Proving with Retrieval-Augmented Language Models](https://mlsanity.com/paper/2306.15626/) (2023)
- [Generative Language Modeling for Automated Theorem Proving](https://mlsanity.com/paper/2009.03393/) (2020)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Abduction-Based Explanations for Machine Learning Models](https://mlsanity.com/paper/1811.10656/) (2018)
- [LLM-SR: Scientific Equation Discovery via Programming with Large Language Models](https://mlsanity.com/paper/2404.18400/) (2024)
- [Mapping the Increasing Use of LLMs in Scientific Papers](https://mlsanity.com/paper/2404.01268/) (2024)
- [The Consciousness Prior](https://mlsanity.com/paper/1709.08568/) (2017)
- [SymbolicGPT: A Generative Transformer Model for Symbolic Regression](https://mlsanity.com/paper/2106.14131/) (2021)
- [Interpretable Scientific Discovery with Symbolic Regression: A Review](https://mlsanity.com/paper/2211.10873/) (2022)
- [On scientific understanding with artificial intelligence](https://mlsanity.com/paper/2204.01467/) (2022)
- [A Bayesian machine scientist to aid in the solution of challenging scientific problems](https://mlsanity.com/paper/2004.12157/) (2020)
- [Towards Scientific Discovery with Generative AI: Progress, Opportunities, and Challenges](https://mlsanity.com/paper/2412.11427/) (2024)
- [Are LLMs Ready for Real-World Materials Discovery?](https://mlsanity.com/paper/2402.05200/) (2024)
- [Learning Sparse Nonlinear Dynamics via Mixed-Integer Optimization](https://mlsanity.com/paper/2206.00176/) (2022)
- [Towards Cognitive AI Systems: a Survey and Prospective on Neuro-Symbolic AI](https://mlsanity.com/paper/2401.01040/) (2024)
- [DrSR: LLM based Scientific Equation Discovery with Dual Reasoning from Data and Experience](https://mlsanity.com/paper/2506.04282/) (2025)
- [Evolving Scientific Discovery by Unifying Data and Background Knowledge with AI Hilbert](https://mlsanity.com/paper/2308.09474/) (2023)
- [Symbolic Regression Driven by Training Data and Prior Knowledge](https://mlsanity.com/paper/2004.11947/) (2020)
- [The Need for Verification in AI-Driven Scientific Discovery](https://mlsanity.com/paper/2509.01398/) (2025)
- [A Generic Framework for Implicate Generation Modulo Theories](https://mlsanity.com/paper/1807.04557/) (2018)
- [Nonground Abductive Logic Programming with Probabilistic Integrity Constraints](https://mlsanity.com/paper/2108.03033/) (2021)

## Abstract (arXiv)

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.