Architectures for mathematical data
TL;DR
This work develops the principle that choosing a neural architecture is not a matter of choosing the largest model, but of choosing an inductive bias that matches the structure the data already has through three self-contained case studies.
Abstract
Mathematical objects rarely arrive as unstructured feature vectors. They arrive as sequences indexed by primes, as directed graphs closed under a combinatorial move, as functions constrained by a differential equation. Choosing a neural architecture is therefore not a matter of choosing the largest model, but of choosing an inductive bias that matches the structure the data already has. We develop this principle through three self-contained case studies, each implemented from scratch in JAX and each runnable on a laptop CPU. (i) Sequential data: a 1D CNN and a single-block transformer predict the rank of an elliptic curve [Formula: see text] from its normalized Frobenius traces [Formula: see text]; a saliency analysis tracked across training shows the network localizing the discriminative signal in the small primes, where the murmuration phenomenon lives. (ii) Graph data: a directed graph isomorphism network classifies quivers by Dynkin mutation type and is permutation equivariant to machine zero by construction; trained only on quivers with [Formula: see text] and [Formula: see text] vertices, it transfers to quivers on [Formula: see text] vertices. (iii) Continuous data: a physics-informed neural network solves a two-point boundary value problem, after which interval arithmetic bounds the residual rigorously over the entire domain rather than at the collocation points. Based on a tutorial delivered at DANGER: Data, Numbers, and Geometry (Banff, April 2026).
