# NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processes

arXiv: https://arxiv.org/abs/2607.14361 · HTML: https://mlsanity.com/paper/2607.14361/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 51 of its 113 references that are in the map.
- First posted on arXiv 15 Jul 2026.

## Builds on (most cited first, 30 of 51)

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- [Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysis](https://mlsanity.com/paper/2210.13300/) (2022)
- [Arbitrage-free neural-SDE market models](https://mlsanity.com/paper/2105.11053/) (2021)
- [Global universal approximation of functional input maps on weighted spaces](https://mlsanity.com/paper/2306.03303/) (2023)
- [Deep learning architectures for nonlinear operator functions and nonlinear inverse problems](https://mlsanity.com/paper/1912.11090/) (2019)
- [Globally injective and bijective neural operators](https://mlsanity.com/paper/2306.03982/) (2023)
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## Abstract (arXiv)

We address fundamental challenges in representing and computing ℝd-valued predictable square-integrable processes over [0,T], collected in the space H2T(ℝd). These processes are central to continuous-time stochastic control, reinforcement learning, and mathematical finance. Although Wiener-chaos expansions offer strong theoretical tools, traditional computational methods are hindered by the need for large chaos dictionaries and high-order iterated integrals. To overcome these obstacles, we introduce NeuralChaos -- a neural operator architecture that produces elements of H2T(ℝd) using only finitely many evaluations of the driving Brownian motion, while preserving predictability and square-integrability. We prove that NeuralChaos is dense in H2T(ℝd) and achieves the best N-term chaoslet approximation rates for compressible and Malliavin--Sobolev regular processes. Moreover, compressibility is shown to be typical for processes from H2T(ℝd) under non-degenerate sub-Gaussian sampling. In contrast, we show that finite-dimensional Markovian neural SDE models constitute a meagre and Gaussian-null subset in H2T(ℝd), regardless of discretization, whereas compressible processes are generic. Numerical experiments on a stochastic optimal control problem and dynamic hedging highlight the practical effectiveness of our approach. Our results enable more efficient and expressive modelling in stochastic analysis and mathematical finance.