Low-Rank Kolmogorov-Arnold Generator Flows for Interpretable Operator Learning

Key points

• Neural operators that learn physical laws from data should generalize beyond their training distribution and expose the learned law in closed form.1
• 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.1
• 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.1
• 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.1

Sources (1)

• [1]Low-Rank Kolmogorov-Arnold Generator Flows for Interpretable Operator Learning
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 9, 05:37 AM

  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.

  Neural operators that learn physical laws from data should generalize beyond their training distribution and expose the learned law in closed form.

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