# Deep Neural Operator Learning for Probabilistic Models

arXiv: https://arxiv.org/abs/2511.07235 · HTML: https://mlsanity.com/paper/2511.07235/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 46 references that are in the map.
- First posted on arXiv 10 Nov 2025.

## Builds on (most cited first, 17 of 17)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces](https://mlsanity.com/paper/2201.00217/) (2022)
- [Operator learning with PCA-Net: upper and lower complexity bounds](https://mlsanity.com/paper/2303.16317/) (2023)
- [Optimal Stopping via Randomized Neural Networks](https://mlsanity.com/paper/2104.13669/) (2021)
- [The Parametric Complexity of Operator Learning](https://mlsanity.com/paper/2306.15924/) (2023)
- [Solving high-dimensional optimal stopping problems using deep learning](https://mlsanity.com/paper/1908.01602/) (2019)
- [Neural Scaling Laws of Deep ReLU and Deep Operator Network: A Theoretical Study](https://mlsanity.com/paper/2410.00357/) (2024)
- [Convergence of the Backward Deep BSDE Method with Applications to Optimal Stopping Problems](https://mlsanity.com/paper/2210.04118/) (2022)
- [Deep neural network expressivity for optimal stopping problems](https://mlsanity.com/paper/2210.10443/) (2022)
- [Deep Signature Algorithm for Multi-dimensional Path-Dependent Options](https://mlsanity.com/paper/2211.11691/) (2022)
- [Neural Operators Can Play Dynamic Stackelberg Games](https://mlsanity.com/paper/2411.09644/) (2024)

## Built on by (most cited first, 3 of 3)

- [Deep Learning for the Multiple Optimal Stopping Problem](https://mlsanity.com/paper/2512.22961/) (2025)
- [Martingale Neural Operators: Learning Stochastic Marginals via Doob-Meyer Factorization](https://mlsanity.com/paper/2605.15806/) (2026)
- [Policy Gradient Learning for Distributionally Robust Markov Decision Processes under Wasserstein Ambiguity](https://mlsanity.com/paper/2606.27610/) (2026)

## Abstract (arXiv)

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.