arXiv:2610.29909v1 Announce Type: cross Abstract: Machine-learning surrogates in computational science often require not only accurate predictions but also expectations, variances, conserved integrals, and normalization constants. Conventional neural networks, including Kolmogorov--Arnold networks (KANs), generally evaluate these quantities through Monte Carlo sampling or multidimensional quadrature, making repeated inference costly, particularly in high dimensions. We introduce KANs with Tractable Operators for Uncertainty and Constraints (TOUCKANs), whose finite-degree polynomial outer functions yield a decomposable sum--product representation. Under factorizing input measures, expectations and higher moments then reduce algebraically to deterministic recursions over one-dimensional leaf integrals. Independent and shared inner-block variants form a parameter-efficiency hierarchy, and a universal-approximation proof for its most restricted member extends to the complete class. The same deterministic operators allow conserved integrals and normalization to be imposed throughout training. Across uniform and Gaussian benchmarks through 50 dimensions, deterministic mean and variance errors remained below the median fixed-budget Monte Carlo errors in every case. TOUCKANs also conserved Cahn--Hilliard mass to machine precision and imposed normalization and orthogonality through deterministic moments, enabling joint computation of the lowest four Schr\"odinger eigenpairs in two, three, and five dimensions. TOUCKANs therefore connect KAN approximation with tractable probabilistic circuits, making statistical operators and integral constraints directly available within an expressive neural surrogate.