Universal Approximation

Deep Neural Operator Learning for Probabilistic Models ArXiv ID: 2511.07235 “View on arXiv” Authors: Erhan Bayraktar, Qi Feng, Zecheng Zhang, Zhaoyu Zhang Abstract We propose a deep neural-operator framework for a general class of probability models. Under global Lipschitz conditions on the operator over the entire Euclidean space-and for a broad class of probabilistic models-we establish a universal approximation theorem with explicit network-size bounds for the proposed architecture. The underlying stochastic processes are required only to satisfy integrability and general tail-probability conditions. We verify these assumptions for both European and American option-pricing problems within the forward-backward SDE (FBSDE) framework, which in turn covers a broad class of operators arising from parabolic PDEs, with or without free boundaries. Finally, we present a numerical example for a basket of American options, demonstrating that the learned model produces optimal stopping boundaries for new strike prices without retraining. ...

Branched Signature Model ArXiv ID: 2511.00018 “View on arXiv” Authors: Munawar Ali, Qi Feng Abstract In this paper, we introduce the branched signature model, motivated by the branched rough path framework of [“Gubinelli, Journal of Differential Equations, 248(4), 2010”], which generalizes the classical geometric rough path. We establish a universal approximation theorem for the branched signature model and demonstrate that iterative compositions of lower-level signature maps can approximate higher-level signatures. Furthermore, building on the existence of the extension map proposed in [“Hairer-Kelly. Annales de l’Institue Henri Poincaré, Probabilités et Statistiques 51, no. 1 (2015)”], we show how to explicitly construct the extension of the original paths into higher-dimensional spaces via a map $Ψ$, so that the branched signature can be realized as the classical geometric signature of the extended path. This framework not only provides an efficient computational scheme for branched signatures but also opens new avenues for data-driven modeling and applications. ...