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

TL;DR

The paper introduces Martingale Neural Operators (MNO), a novel approach leveraging Doob-Meyer decomposition to efficiently learn stochastic PDEs' distributions, capturing variance and tail structures in a single shot.

Contribution

MNO is the first neural operator architecture that directly models the conditional mean and covariance of stochastic PDE solutions using a martingale-based prior.

Findings

MNO reduces Wasserstein distance by up to 120× on φ^4 field theory.

MNO evaluates approximately 3× faster than conditional diffusion baselines.

MNO performs comparably to FNO on zero-shot resolution transfer and turbulent flow tasks.

Abstract

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 $B_\phi$ with $B_\phi^\top B_\phi$ 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 $120\times$ on $\phi^4$ field theory and $68\times$ on stochastic Burgers, evaluating $\sim 3\times$ 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.