Abstract: Scientific progress is defined by a fundamental asymmetry: a theory feels complete until the moment it is proven wrong. In mathematical modeling, we often operate within axiomatic systems, (conservation laws, symmetries, and invariants), that are inevitably incomplete. The central challenge of automated discovery is abduction: can we reason backward from observed phenomena to identify the missing principles that our axioms fail to capture?

In this talk, I will outline a program for the systematic discovery of physical laws through the lens of algebraic geometry. Building on the AI-Noether framework, which utilized Hilbert's Nullstellensatz and Noether's primary decomposition (I=⋂qi) for abductive discovery in exact polynomial settings, we now extend this machinery into two new frontiers.

First, we address the "brittleness" of symbolic computation by embracing Numerical Algebraic Geometry. Using homotopy continuation and witness sets, we move beyond exact arithmetic to achieve scalable, noise-robust inference in empirical data. Second, we transition from static constraints to dynamical systems via Differential Algebra. By shifting from Hilbert’s Basis Theorem to Ritt’s, and from Gröbner bases to Rosenfeld-Gröbner characteristic sets, we enable the native discovery of differential equations directly from observations.

The power of this synthesis is already emerging. Preliminary applications to a series of long-standing open problems in mathematical physics suggest that this framework can identify candidate laws and hidden symmetries where classical derivation has stalled. We are finally positioned to venture into genuinely unknown territory: the discovery of fundamental principles that have remained, until now, just beyond human reach.

Contact: Shravan Veerapaneni