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 (like the Fourier Neural Operator, FNO) train quickly and give predictions 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—a neural operator only outputs the conditional mean. It tells you "on average, the temperature at this location is 23.5 degrees." It does not tell you "there's a 10% chance the temperature exceeds 40 degrees." No variance, no tail risk. For uncertainty quantification, that's barely better than saying nothing.
The usual fix is to attach a generative model—a diffusion model or flow model—to learn the conditional distribution. But that sacrifices the most attractive property of neural operators: single-forward-pass efficiency and resolution invariance.
Hidajat—the same author who posted a grokking paper in the same period—recently proposed a new architecture (arXiv:2605.15806) that solves this in a surprising way: encoding 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 scary, but it just means "a stochastic process with a regular drift part and a purely random noise part." Stock prices are an example—their movement decomposes into a "fundamentals-driven trend" (drift) and "unpredictable market noise" (the martingale part).
What Hidajat did is simple: treat this decomposition as an architectural design principle. The neural operator backbone 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. Instead, MNO has the network output mean and covariance simultaneously, with the covariance parameterized by a low-rank factor B_φ, where B_φ^T B_φ is automatically positive semidefinite—no extra constraint handling 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 benchmark, MNO's Wasserstein distance beats the conditional diffusion baseline by 120×. On the stochastic Burgers equation, 68×. And it's roughly 3× faster than the conditional diffusion baseline—at the same training budget.
On 2D tasks, MNO matches standard FNO on zero-shot resolution transfer and turbulence simulation.
One interesting failure mode: "accurate systems" like the Gray-Scott reaction-diffusion equation. In these systems the process has very little stochasticity—the dynamics are essentially deterministic. MNO's martingale decomposition offers little advantage there, because there's no randomness to learn.
Open Questions
1. Does Doob-Meyer actually apply? The decomposition is defined for stochastic processes indexed by time. But in many SPDE solving tasks, the "randomness" lives in initial conditions, boundary conditions, or driving noise. MNO applies the Doob-Meyer architecture to the map "from initial conditions to terminal distribution"—treating initial conditions as the "drift" and lumping all uncertainty into the "martingale." Whether this mapping always preserves the Doob-Meyer structure mathematically is unclear, since the path between initial conditions and terminal distributions can be highly nonlinear.
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 may hold: the dominant uncertainty modes really are few. But which systems violate it? Systems where uncertainty energy is spread uniformly across all modes—fully developed turbulence, for instance—may not be well captured by a low-rank covariance.
3. The 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.
Still, overall this is an elegant architectural idea: instead of treating uncertainty quantification as a post-training add-on, encode it into the architecture itself. A 1960s theorem from financial mathematics, 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.