Data-Driven Telescoping Approximations via Learned Kernels A Rigorous Framework for Operator Learning with Provable Convergence

We extend the telescoping approximation framework to data-driven settings, where the underlyingevolution kernel is not known analytically but is inferred from observations. Ratherthan learning full solution operators, we propose learning low-complexity kernel representationsconstrained to admit telescoping structure.We show that if a learned kernel operator approximates a true semigroup with uniform errorε(M) depending on model rank M, and if the telescoping approximants preserve incrementaldecay, then the resulting approximation converges with explicit rates O(n−k) for order-k telescoping.This yields provable stability for data-driven approximations of evolution equations,bridging deterministic analysis, operator learning, and modern scientific machine learning.The framework establishes rigorous error bounds under both deterministic and stochasticdata regimes, provides algorithmic implementations for kernel learning with telescoping constraints,and demonstrates applications to operator identification from time-series data, reducedordermodeling, and uncertainty-aware surrogate solvers.