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

Learning mappings between probability distributions arises naturally in settings where both inputs and outputs are represented by populations of samples rather than individual observations. This paper develops an approximation-theoretic framework for such distribution-to-distribution learning problems and extends it to mappings between stochastic processes. For continuous operators acting on $W_2$ -compact families of finite-dimensional input laws, we establish uniform neural approximation in the 2-Wasserstein metric. Our construction relies on finite law statistics as input representations, a simplex-valued neural map, and a shared atomic output support that guarantees the output is always a valid probability measure. We then extend this approximation principle to probability laws on separable Hilbert spaces via fixed finite-rank orthogonal projections. Together, these results establish the representational feasibility of learning transformations whose inputs and outputs are probability laws rather than deterministic vectors or functions. To assess practical relevance, we construct numerical problems whose target transformations are naturally defined at the level of distributions or probability laws: predicting the first-passage-time distribution of an Ornstein–Uhlenbeck process and the nonlinear response-path laws of a Duffing oscillator. Since the theoretical framework is deliberately model-agnostic and covers a substantially broader class of operators than can be efficiently represented by any single practical architecture, our experiments instantiate the framework using task-adapted neural models rather than reproducing the theoretical construction verbatim. Both models outperform a fixed-feature MLP baseline and distribution-space kernel regression. These proof-of-concept demonstrations complement the approximation theory by supporting the learnability and practical relevance of distribution-to-distribution transformations for random systems.