# From Distributions to Stochastic Processes: Neural Approximation of Measure-Valued Maps

arXiv: https://arxiv.org/abs/2609.33649 · HTML: https://mlsanity.com/paper/2609.33649/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 24 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Builds on (most cited first, 6 of 6)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Learning Theory for Distribution Regression](https://mlsanity.com/paper/1411.2066/) (2014)
- [Measure-to-measure interpolation using Transformers](https://mlsanity.com/paper/2411.04551/) (2024)
- [Mean-field neural networks: learning mappings on Wasserstein space](https://mlsanity.com/paper/2210.15179/) (2022)
- [An Approximation Theory for Metric Space-Valued Functions With A View Towards Deep Learning](https://mlsanity.com/paper/2304.12231/) (2023)

## Abstract (arXiv)

Learning mappings between probability distributions arises naturally when inputs and outputs are represented by populations of samples rather than individual observations. We develop an approximation-theoretic framework for distribution-to-distribution learning and extend it to mappings between stochastic processes. For continuous operators on W2-compact families of finite-dimensional probability laws, we establish uniform neural approximation in the 2-Wasserstein metric using finite law statistics, a simplex-valued neural map, and a shared atomic output support that guarantees valid probability measures. We further extend this principle to probability laws on separable Hilbert spaces through finite-rank orthogonal projections. These results establish the representational feasibility of learning transformations between probability laws rather than deterministic vectors or functions. To demonstrate practical relevance, we study two problems naturally defined at the distribution level: prediction of first-passage-time distributions for an Ornstein--Uhlenbeck process and nonlinear response-path laws of a Duffing oscillator. Because the theory is model-agnostic and broader than any single practical architecture, the experiments use task-adapted neural models rather than reproducing the theoretical construction exactly. In both problems, the proposed models outperform a fixed-feature MLP baseline and distribution-space kernel regression. These experiments complement the theory by demonstrating the practical learnability of distribution-to-distribution transformations in random systems.