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

Machine Learning 2026-05-18 v1

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 BϕB_\phi with BϕBϕB_\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 120×120\times on ϕ4\phi^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.

Keywords

Cite

@article{arxiv.2605.15806,
  title  = {Martingale Neural Operators: Learning Stochastic Marginals via Doob-Meyer Factorization},
  author = {Kai Hidajat},
  journal= {arXiv preprint arXiv:2605.15806},
  year   = {2026}
}
R2 v1 2026-07-22T07:14:09.272Z