AI and the World's Hard Problems · Theoretical computer science · arXiv · 2026-10-22
Tractable Inference with Kolmogorov-Arnold Networks
Machine-learning surrogates in computational science need not only accurate predictions but also expectations, variances, conserved integrals, and normalization constants, which conventional networks and KANs evaluate through costly Monte Carlo sampling or multidimensional quadrature, especially in high dimensions. TOUCKANs use finite-degree polynomial outer functions yielding a decomposable sum-product representation, so under factorizing input measures expectations and higher moments reduce to deterministic recursions over one-dimensional integrals, while conserved integrals and normalization can be imposed throughout training. Across uniform and Gaussian benchmarks through 50 dimensions, deterministic mean and variance errors stayed below median fixed-budget Monte Carlo errors in every case, Cahn-Hilliard mass was conserved to machine precision, and the lowest four Schrödinger eigenpairs were jointly computed in two, three, and five dimensions.