Background
Suppose you want to use a neural network to simulate a stochastic partial differential equation (SPDE)—say, a physical field evolving under random noise. Traditional neural operators (e.g., the Fourier Neural Operator, FNO) train quickly and produce a prediction in a single forward pass.
But there's a problem: if your PDE is stochastic—tiny fluctuations in initial conditions cause huge differences in outcomes—the neural operator only outputs the conditional mean. It tells you "on average, the temperature at this location is 23.5 degrees." It doesn't tell you "there's a 10% chance the temperature exceeds 40 degrees." No variance, no tail risk. For uncertainty quantification, that's nearly as useless as saying nothing.
The usual fix is to bolt on a generative model—a diffusion model or a flow model—to learn the conditional distribution. But this sacrifices the neural operator's most attractive property: single-pass efficiency and resolution invariance.
Hidajat—the same author who posted a grokking paper on arXiv recently—proposed a new architecture (2605.15806) that solves this in a surprising way: encode a probability theorem directly into the neural network's architecture.
The Doob-Meyer Decomposition
The Doob-Meyer theorem is a core result of martingale theory developed in the 1960s. It states: any semimartingale (a sufficiently general stochastic process) can be uniquely decomposed into a predictable drift term and an unpredictable zero-expectation martingale.
> "Semimartingale" sounds intimidating, but it just means "a stochastic process with a regular drift part and a completely random noise part." Stock prices are an example—price movements decompose into a "fundamentals-driven trend" (drift) and "unpredictable market noise" (the martingale part).
What Hidajat did is simple: use this decomposition as an architecture design principle. The neural operator's main body predicts the drift term—the conditional mean. A lightweight low-rank branch predicts the martingale term—the conditional covariance.
The key word is "lightweight." The standard approach trains a separate generative model to predict the full conditional distribution. MNO instead makes the network output mean and covariance simultaneously, with the covariance parameterized by a low-rank factor B_φ, so that B_φ^T B_φ is positive semi-definite by construction—no extra constraints needed.
Striking Results
They tested on three classes of tasks: 1D stochastic PDEs, rough volatility (a stochastic process from finance), and 2D neural operator tasks.
Results: on the φ⁴ field theory standard benchmark, MNO's Wasserstein distance beat the conditional diffusion baseline by 120x. On the stochastic Burgers equation, by 68x. It is also roughly 3x faster than the conditional diffusion baseline under the same training budget.
On 2D tasks, MNO matches standard FNO on zero-shot resolution transfer and turbulence simulation.
One interesting failure mode: "quasi-deterministic systems" such as the Gray-Scott reaction-diffusion equation. In these systems the process has very little stochasticity—it's mostly deterministic dynamics. MNO's martingale decomposition offers little advantage here, because there is little randomness to learn.
Open Questions
1. Does the decomposition transfer? The Doob-Meyer decomposition is defined for stochastic processes in time. But in many SPDE tasks, "randomness" may live in initial conditions, boundary conditions, or the driving noise. MNO applies the Doob-Meyer architecture to the map "initial condition → terminal distribution"—treating the initial condition as drift and packing all uncertainty into the martingale. Whether this map always preserves the Doob-Meyer structure mathematically is unclear, given possibly complex nonlinear paths between initial conditions and terminal distributions.
2. The low-rank covariance assumption. B_φ^T B_φ is a low-rank matrix—meaning it assumes stochastic uncertainty is concentrated in a low-dimensional subspace. For many physical systems this holds: the dominant uncertainty modes are few. But which systems violate it? Systems with uncertainty energy spread uniformly across all modes—fully developed turbulence, for instance—may not be well captured by low-rank covariance.
3. Gaussian residual instantiation. The paper assumes conditional Gaussianity of the terminal distribution. Many physical systems have non-Gaussian conditional distributions—bimodal, skewed, heavy-tailed. The Gaussian assumption simplifies covariance parameterization but limits the ability to capture non-Gaussian structure.
Overall, though, this is an elegant architectural idea: rather than treating uncertainty quantification as a post-training add-on, encode it into the network's architecture. A 1960s financial mathematics theorem, used in 2026 to accelerate SPDE simulation.
References
1. Hidajat, K. (2026). *Martingale Neural Operators: Learning Stochastic Marginals via Doob-Meyer Factorization*. arXiv:2605.15806 [cs.LG]. https://arxiv.org/abs/2605.15806 2. Doob, J. L. (1953). *Stochastic Processes*. Wiley. 3. Li, Z., et al. (2021). *Fourier Neural Operator for Parametric Partial Differential Equations*. ICLR 2021. 4. Kovachki, N., et al. (2023). *Neural Operator: Learning Maps Between Function Spaces*. JMLR, 24(1), 1-97. 5. Meyer, P. A. (1966). *Probability and Potentials*. Blaisdell.