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

TL;DR

This work addresses how to repair incomplete scientific theories when AI-generated hypotheses do not derive from existing axioms. It introduces AI-Noether, an algebraic-geometry–driven abductive inference system that encodes axioms and hypotheses as polynomial relations, decomposes the resulting solution variety to expose minimal missing axioms, and verifies candidates with exact projections or existential reasoning. The method demonstrates high axiom-recovery rates across diverse physical systems (including Kepler’s law, relativistic effects, and carrier-resolved Hall effect), and shows robustness to noise via numerical irreducible decomposition and regression. Compared with solver-based abduction and large-language-model baselines, AI-Noether provides formal guarantees, works in data-free settings, and scales to complex theories, marking a principled path for minimal theory revision in AI-assisted scientific discovery. Open-source availability and detailed benchmarks further enable reproducibility and future extension to broader scientific domains.

Abstract

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.

Figures (14)

• Figure 1: AI-Noether--augmented scientific discovery loop. Given background theory axioms and observational data, a new candidate hypothesis is generated that may or may not be compatible with the existing theory (under different notions of derivability and consistency) and may fit the data with varying approximation error. AI-Noether performs abductive inference to identify missing axioms that reconcile these hypotheses with canonical knowledge, while symmetrically, on the data-driven end, experimental design prescribes the generation of new data. Both procedures are instrumental to further refine both hypotheses and axioms, closing the discovery-verification loop.
• Figure 2: AI-Noether system overview. Given background theory axioms and one or more candidate hypotheses that are not supported by the existing theory, AI-Noether identifies additional axioms that reconcile the hypotheses with the background knowledge. The system first encodes axioms and hypotheses into a geometric representation, decomposes this representation into fundamental components, and then reasons over these components to determine which candidate axioms, when added to the theory, are sufficient to explain the hypotheses. These axioms are returned as output after verification, which proves the hypotheses, using both algebraic and logic-based methods.
• Figure 3: Geometric visualization of decomposition. On the left, we have an example of a variety which is the solution to a system of polynomial equations. The middle and right panels show that this variety can be decomposed into irreducible components, which, geometrically, is the process of finding smaller varieties whose union is the original variety.
• Figure 4: Geometric visualization of algebraic reasoning. On the left, we have some surface defined by an axiom (here, $F_c=F_g$ as an example). In the middle, we have an additional equation that defines a new surface. In order to apply the axiom to the new surface, we project the surface onto the space defined by the known axiom(s).
• Figure 5: Illustration of the experimental setup for the Carrier-Resolved Photo-Hall Effect. AI-Noether successfully recovers the necessary Hall coefficient relation required to prove the relationship between the various parameters of a semiconducting surface.

Theorems & Definitions (1)

• Example 1