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

Abstract:

While neural operators serve as highly effective deterministic surrogates, their application to stochastic partial differential equations (PDEs) often results in a collapse toward the conditional mean. This limitation discards the variance and tail structures essential for robust uncertainty quantification. Traditionally, recovering these statistical features necessitates computationally expensive Monte Carlo simulations or the integration of auxiliary generative models, both of which compromise the one-shot inference speed and resolution invariance that characterize the operator framework.

To address these challenges, we leverage the Doob-Meyer theorem, which posits that any semimartingale can be uniquely decomposed into a predictable drift component and an unpredictable, zero-mean martingale. By embedding this theoretical insight as an architectural prior, we propose the Martingale Neural Operator (MNO). The MNO architecture directly maps initial conditions to the conditional mean and covariance of the terminal distribution. This is achieved by parameterizing the output with a drift-like mean and a low-rank factor, $B_\phi$, where the product $B_\phi^\top B_\phi$ is constructed to be positive semi-definite. For our experimental validation, we employ a Gaussian residual instantiation of this framework.

Empirical results across 1D SPDEs, rough volatility models, and 2D operator tasks demonstrate significant performance gains. Specifically, MNO achieves up to a $120\times$ reduction in Wasserstein distance on $\phi^4$ field theory and a $68\times$ reduction on stochastic Burgers equations. Furthermore, it evaluates approximately $3\times$ faster than a conditional diffusion baseline when compared under equivalent wall-clock training constraints. In 2D applications, MNO performs on par with Fourier Neural Operators (FNO) regarding zero-shot resolution transfer and turbulent flow dynamics, although it continues to struggle with quasi-deterministic systems like the Gray-Scott model.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC