math.PRJul 15, 2026

NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processes

Authors: Anastasis KratsiosGiulia LivieriPhilipp Schmocker

Abstract

We address fundamental challenges in representing and computing Rd\mathbb{R}^{d}-valued predictable square-integrable processes over [0,T][0,T], collected in the space HT2(Rd)\mathcal{H}^2_T(\mathbb{R}^{d}). 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 HT2(Rd)\mathcal{H}^2_T(\mathbb{R}^{d}) using only finitely many evaluations of the driving Brownian motion, while preserving predictability and square-integrability. We prove that NeuralChaos is dense in HT2(Rd)\mathcal{H}^2_T(\mathbb{R}^{d}) and achieves the best NN-term chaoslet approximation rates for compressible and Malliavin--Sobolev regular processes. Moreover, compressibility is shown to be typical for processes from HT2(Rd)\mathcal{H}^2_T(\mathbb{R}^{d}) 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 HT2(Rd)\mathcal{H}^2_T(\mathbb{R}^{d}), 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.

Explore similar work

Jul 25, 2026stat.ML

Operator Neural Jump ODEs: L^2-optimal prediction in function spaces

In this paper, we study the extension of Neural Jump ODEs to infinite-dimensional function spaces. In particular, the underlying process XX now takes values in L2(Ξ,RdX)L^2(Ξ, \mathbb{R}^{d_X}) instead of RdX\mathbb{R}^{d_X} and the Operator NJ-ODE approximates the optimal predictor of this process by producing a representative of the conditional expectation. The NJ-ODE model is a framework for online learning the optimal prediction of continuous-time stochastic processes, given discrete, possibly irregular and incomplete past observations. In a series of works, this model has been extended to deal with generic path-dependent processes, with observation noise and dependent observations, with long-term predictions, and with input-output systems. However, throughout all of these works, the underlying processes were restricted to be finite-dimensional. In particular, function-valued problems, like yield curve or volatility surface predictions, could only be handled through discretization, which inherently leads to a loss of information. In this work, we build on ideas from Neural Operator methods that allow us to extend the NJ-ODE framework to an infinite-dimensional output process. To prove convergence of the NJ-ODE to the optimal prediction process, we develop a new approximation strategy that also generalizes previous works in the finite-dimensional setting by considerably weakening the assumptions.
Florian Krach, Oliver Löthgren, Josef Teichmann
May 15, 2026cs.LG

Martingale Neural Operators: Learning Stochastic Marginals via Doob-Meyer Factorization

Neural operators excel as deterministic surrogates, but inevitably collapse to the conditional mean when applied to stochastic PDEs, discarding the variance and tail structure upon which uncertainty quantification depends. Recovering this structure typically requires Monte Carlo rollouts or grafted generative models, both of which surrender the one-shot efficiency and resolution invariance that define the operator paradigm. To resolve this, we draw on the Doob-Meyer theorem, which establishes that any semimartingale fundamentally decomposes into a predictable drift and an unpredictable, zero-mean martingale. Translating this theorem into an architectural prior, we introduce the Martingale Neural Operator (MNO). MNO maps an initial condition directly to the conditional mean and covariance of the terminal law, parameterized by a drift-like mean and a low-rank factor BφB_φ with BφBφB_φ^\top B_φ positive semi-definite by construction. For our experiments, we use a Gaussian residual instantiation. Across 1D SPDEs, rough volatility, and 2D operator tasks, MNO reduces Wasserstein distance by up to 120×120\times on φ4φ^4 field theory and 68×68\times on stochastic Burgers, evaluating 3×\sim 3\times faster than a conditional diffusion baseline at matched wall-clock training budgets. On 2D tasks, MNO is comparable to FNO on zero-shot resolution transfer and turbulent flow, while quasi-deterministic systems such as Gray-Scott remain a failure mode.
Kai Hidajat
Jun 22, 2026cs.LG

The Fractal Neural Operator: Overcoming Spectral Bias in Chaotic Attractors via Prime-Harmonic Weierstrass Encodings

Deep learning models, particularly Transformers and Neural Operators, exhibit a well-documented "spectral bias," effectively acting as low-pass filters that smooth out high-frequency information. While benign in fluid dynamics, this bias is catastrophic for Chaotic Dynamical Systems, where the underlying strange attractor is characterized by fractal geometry and infinite spectral density. We introduce the Fractal Neural Operator (FNO), a novel architecture that utilizes a non-resonant prime number basis to approximate continuous dynamical systems. Unlike geometric encodings (2k2^k), which suffer from spectral gaps and resonance, our Harmonic Weierstrass Encoder injects infinite spectral resolution into the latent space. We demonstrate that FNO extends the valid prediction horizon of the Lorenz-63 system to 347 Lyapunov times, exceeding state-of-the-art Reservoir Computing baselines by a factor of 2.3x. These results suggest that "chaos" is not inherently unpredictable to neural networks, but rather requires non-differentiable, fractal embedding manifolds.
Kanishk Awadhiya