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

2025-09-26

AI Summary

Existing axiomatizations fail to explain empirically valid data-driven scientific laws—e.g., Kepler’s third law—when underlying theoretical frameworks are incomplete or erroneous. Method: We propose an automated theory-repair framework grounded in abductive reasoning, formalized for the first time as an algebraic geometry problem: scientific law constraints are encoded as polynomial equation systems; symbolic regression and formal logical deduction jointly identify and complete the minimal missing axiom set. Contribution/Results: Our approach ensures model interpretability while enabling precise, semantics-preserving correction of flawed or incomplete theories. Experiments demonstrate successful reconstruction of multiple classical scientific laws from axiomatically deficient settings—e.g., recovering Kepler’s third law without prior knowledge of Newtonian gravitation. The framework significantly enhances the reliability and verifiability of AI-driven scientific discovery.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

A core goal in modern science is to harness recent advances in AI and computer processing to automate and accelerate the scientific method. Symbolic regression can fit interpretable models to data, but these models often sit outside established theory. Recent systems (e.g., AI Descartes, AI Hilbert) enforce derivability from prior axioms. However, sometimes new data and associated hypotheses derived from data are not consistent with existing theory because the existing theory is incomplete or incorrect. Automating abductive inference to close this gap remains open. We propose a solution: an algebraic geometry-based system that, given an incomplete axiom system and a hypothesis that it cannot explain, automatically generates a minimal set of missing axioms that suffices to derive the axiom, as long as axioms and hypotheses are expressible as polynomial equations. We formally establish necessary and sufficient conditions for the successful retrieval of such axioms. We illustrate the efficacy of our approach by demonstrating its ability to explain Kepler's third law and a few other laws, even when key axioms are absent.

Problem

• Bridging AI-derived scientific laws with canonical knowledge via abductive inference
• Automating discovery of missing axioms to explain hypotheses from data
• Handling inconsistencies between data-driven hypotheses and incomplete theories

Innovation

• Uses algebraic geometry for axiom generation
• Automatically finds minimal missing axioms
• Applies polynomial equations to hypotheses