Hybrid DNN-stochastic fixed point algorithm for a partially unknown function

This paper investigates the problem of approximating the fixed point of a mapping that depends on an unknown function available only through finitely many sampled values. To address this setting, we propose a hybrid framework combining machine learning techniques with a stochastic fixed-point algorithm. Specifically, a deep neural network is employed to construct a data-driven approximation of the unknown function. The resulting surrogate model is then incorporated into a stochastic iterative scheme based on the Krasnosel’skii algorithm to compute the fixed point.After determining the final error floor of the algorithm, an exponential inequality is established to analyze the convergence of the iterative scheme and to characterize the relationship between the learning error and the type of convergence. The theoretical results are then validated on three numerical examples in one, two and infinite dimensions.