AI Noether - Bridging the Gap Between Scientific Laws Derived by AI Systems and Canonical Knowledge via Abductive Inference
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.
Machine-assisted scientific discovery has been a popular topic in machine learning. Recent breakthroughs have shown that building a framework for exploiting known background theory (encoded as physical axioms) in addition to data-driven symbolic regression models can greatly improve machine-assisted discovery in the scientific context (1, 2). These methods not only generate hypotheses that fit data, but are also derivable from theory as well. This certificate of derivability, however, is only guaranteed when the background theory itself is complete. If there are essential axioms missing, then while these systems are able to recover the correct hypothesis from data, they are not able to generate a certificate of derivability. Therefore, there is a gap of explainability between machine-generated hypotheses and known theory if the known theory is incomplete. In this paper, we attempt to bridge this gap with AI Noether, a computational framework for abductively inferring missing axioms that are required for an incomplete background theory (encoded as polynomials) to explain a hypothesis.
You can find relevant code here.
Karan Srivastava
PhD Student, Mathematics
My research interests include machine learning, reinforcement learning, combinatorics, and algebraic geometry