aiqbench analysis
This paper introduces TOUCKANs, a variant of Kolmogorov-Arnold networks that replaces stochastic sampling with deterministic polynomial operators to efficiently compute expectations, variances, and conserved integrals in high-dimensional computational physics. By reducing multidimensional problems to one-dimensional recursions, the method achieves superior accuracy over Monte Carlo sampling while enforcing physical constraints like mass conservation directly during training. This advancement bridges the gap between expressive neural surrogates and tractable probabilistic circuits for scientific applications.
Tractable Inference with Kolmogorov-Arnold Networks
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.
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 Schrödinger 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.
Computational Physics (physics.comp-ph); Machine Learning (cs.LG); Numerical Analysis (math.NA)
arXiv:2610.29909 [physics.comp-ph]
(or arXiv:2610.29909v1 [physics.comp-ph] for this version)
https://doi.org/10.48550/arXiv.2610.29909 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 297 words