Interpreter-friendly method improves learning of physical laws from data
Low-Rank Kolmogorov-Arnold Generator Flows for Interpretable Operator Learning
Machine Learning
Summary
Learning physical laws from data can be hard, especially if you want your model to work well on new situations and explain itself clearly. The paper introduces a new method called KAFlow that makes these models simpler and more accurate by focusing on a special type of mathematical pattern called low-rank symbols. This approach not only improves prediction accuracy by a large margin but also automatically provides clear formulas describing the learned laws. It works well on both synthetic tests and complex quantum systems.
What this means in practice
- •For computational physicists: Model position-dependent physical systems more accurately and get interpretable operator formulas automatically from data.
- •For quantum simulation engineers: Simulate long-term quantum dynamics with significantly reduced errors using learned operators from limited measurements.
Authors
Minjong Cheon
Abstract
Neural operators that learn physical laws from data should generalize beyond their training distribution and expose the learned law in closed form. The Kolmogorov-Arnold neural operator achieves both for position-dependent dynamics by learning a dense pseudo-differential symbol over all positions and frequencies, at a cost that grows with the square of the resolution and with closed forms that must be read out by hand. This dense symbol is unnecessary: a finite-order variable-coefficient differential operator has a symbol whose separation rank is bounded by the number of its coefficient functions, independently of the resolution, and the symbols that this network learns on its own benchmarks have rank two. We introduce the Kolmogorov-Arnold flow (KAFlow), which learns the generator of a physical operator as a low-rank symbol built from univariate Kolmogorov-Arnold functions, fixes the gauge of these functions so that every term has a unique place, integrates the dynamics with exact exponential integrators, and distills its closed form automatically. On the synthetic benchmarks of the Kolmogorov-Arnold neural operator, KAFlow lowers the out-of-distribution loss by 3.4 to 11 times and recovers every coefficient to within 10^-4 without symbolic templates, reaching the precision floor of the data. On long-horizon quantum dynamics, it lowers the state infidelity by up to 6 orders of magnitude, and by 30 times when trained only on position and momentum measurements. KAFlow also learns a variable-order fractional operator 15 times more accurately than existing operator networks and discovers nonlocal, fractional and non-polynomial dispersion relations on which sparse regression fails.