Martingale Neural Operator (MNO)

• MNO is a neural-operator architecture that learns the terminal conditional law from stochastic PDEs using a Doob–Meyer-inspired mean/residual factorization.
• It maps initial conditions to both mean and covariance estimates in one forward pass, ensuring positive semidefiniteness via a low-rank Gaussian residual formulation.
• MNO provides fast, one-shot uncertainty quantification and resolution transfer, outperforming traditional methods while noting limitations in capturing heavy tails.

Searching arXiv for the specified topic and closely related papers. arxiv_search(query="Martingale Neural Operator Doob-Meyer Factorization", max_results=10) arxiv_search(query="Martingale Neural Operator Doob-Meyer Factorization", max_results=10) Martingale Neural Operator (MNO) is a neural-operator architecture for learning the terminal conditional law of stochastic dynamics, written as $u_0 \mapsto \mathcal{L}(u_T \mid u_0)$, rather than only the conditional mean. It is introduced by framing the Doob–Meyer decomposition as an architectural prior: a predictable, drift-like component captures the mean structure, while an unpredictable zero-mean component captures the residual stochasticity (Hidajat, 15 May 2026). In the formulation studied experimentally, MNO maps an initial condition directly to the conditional mean and covariance of the terminal law in one forward pass, preserving the one-shot and resolution-transfer properties associated with neural operators while extending them to uncertainty quantification for stochastic PDEs (Hidajat, 15 May 2026).

1. Definition and problem setting

Martingale Neural Operator is motivated by a basic limitation of standard neural operators on stochastic data. When trained with squared loss, a deterministic operator surrogate recovers only the conditional mean,

$$f^\star(u_0) = \arg\min_f \mathbb{E}\|u_T-f(u_0)\|_H^2 = \mathbb{E}[u_T\mid u_0].$$

For stochastic PDEs, this collapses the terminal law to a point estimate and discards the variance and correlation structure needed for uncertainty quantification (Hidajat, 15 May 2026).

The MNO construction therefore targets a different object: the terminal conditional law. In the paper’s formulation, the model predicts both a mean operator $m_\theta(u_0,t)$ and a covariance operator $\Gamma_\phi(u_0,t)$ in one forward pass (Hidajat, 15 May 2026). The emphasis is explicitly on terminal moments and marginal uncertainty, rather than on reconstructing the full pathwise stochastic process.

This makes MNO distinct from deterministic operator learners such as FNO and DeepONet. Those architectures are designed to learn deterministic maps between function spaces; under squared-error training on stochastic data, they estimate only $\mathbb{E}[u_T\mid u_0]$ (Hidajat, 15 May 2026). MNO retains the operator-learning paradigm but augments it with a mean-plus-covariance representation of the terminal law.

A central interpretive point in the paper is that MNO is a terminal marginal learner, not a universal generative sampler and not a pathwise stochastic process model (Hidajat, 15 May 2026). This narrower target is deliberate. It suggests a role for MNO as a fast moment surrogate for stochastic PDEs, especially in settings where conditional means, variance fields, and an approximation to the terminal marginal law are more operationally relevant than full sample-path generation.

2. Doob–Meyer factorization as an architectural prior

The paper motivates MNO from the canonical decomposition of a square-integrable special semimartingale,

$u_t = u_0 + A_t + M_t,$

where $A_t$ is predictable finite variation and $M_t$ is a local martingale (Hidajat, 15 May 2026). For an Itô SPDE,

$du_t = \mu(u_t,t)\,dt + \sigma(u_t,t)\,dW_t,$

the decomposition becomes

$$A_t = \int_0^t \mu(u_s,s)\,ds, \qquad M_t = \int_0^t \sigma(u_s,s)\,dW_s.$$

The architectural intuition is then immediate: the drift component encodes the predictable mean structure, while the martingale residual carries the unpredictable fluctuations (Hidajat, 15 May 2026). MNO does not attempt to recover the full semimartingale filtration or a pathwise stochastic evolution. Instead, it uses the drift/residual split as a structural prior for the terminal law.

In the model’s parameterization, the mean is written as

$$f^\star(u_0) = \arg\min_f \mathbb{E}\|u_T-f(u_0)\|_H^2 = \mathbb{E}[u_T\mid u_0].$$0

where $$f^\star(u_0) = \arg\min_f \mathbb{E}\|u_T-f(u_0)\|_H^2 = \mathbb{E}[u_T\mid u_0].$$1 is an FNO-based drift head (Hidajat, 15 May 2026). The covariance is represented via a low-rank factor $$f^\star(u_0) = \arg\min_f \mathbb{E}\|u_T-f(u_0)\|_H^2 = \mathbb{E}[u_T\mid u_0].$$2, which plays the role of a martingale-like residual factor. This is not a literal enforcement of full martingale structure; the paper explicitly notes that the training includes a residual-centering “martingale” penalty, but that the method does not enforce full filtration-level martingale properties (Hidajat, 15 May 2026).

The semimartingale interpretation is also not universally literal across all tested domains. In rough volatility experiments, for example, the paper states that rough volatility is not semimartingale, so the Doob–Meyer interpretation there is not literal; MNO is instead used as a practical terminal mean/covariance factorization (Hidajat, 15 May 2026). This caveat is important because it clarifies that the method’s empirical utility is not restricted to settings where the underlying process exactly satisfies the motivating decomposition.

3. Mean–covariance parameterization and Gaussian residual instantiation

MNO maps $$f^\star(u_0) = \arg\min_f \mathbb{E}\|u_T-f(u_0)\|_H^2 = \mathbb{E}[u_T\mid u_0].$$3 to a Gaussian terminal law,

$$f^\star(u_0) = \arg\min_f \mathbb{E}\|u_T-f(u_0)\|_H^2 = \mathbb{E}[u_T\mid u_0].$$4

with covariance parameterized as

$$f^\star(u_0) = \arg\min_f \mathbb{E}\|u_T-f(u_0)\|_H^2 = \mathbb{E}[u_T\mid u_0].$$5

This factorization guarantees positive semidefiniteness by construction: $$f^\star(u_0) = \arg\min_f \mathbb{E}\|u_T-f(u_0)\|_H^2 = \mathbb{E}[u_T\mid u_0].$$6 The PSD guarantee is one of the method’s structural advantages (Hidajat, 15 May 2026).

Sampling is written through the residual

$$f^\star(u_0) = \arg\min_f \mathbb{E}\|u_T-f(u_0)\|_H^2 = \mathbb{E}[u_T\mid u_0].$$7

so that a sample takes the form

$$f^\star(u_0) = \arg\min_f \mathbb{E}\|u_T-f(u_0)\|_H^2 = \mathbb{E}[u_T\mid u_0].$$8

Under this construction,

$$f^\star(u_0) = \arg\min_f \mathbb{E}\|u_T-f(u_0)\|_H^2 = \mathbb{E}[u_T\mid u_0].$$9

The low-rank factor therefore serves simultaneously as a covariance parameterization and as a sampling mechanism for terminal-law realizations (Hidajat, 15 May 2026).

For a $m_\theta(u_0,t)$0-channel field, the diagonal variance used in training and reporting is

$m_\theta(u_0,t)$1

The paper emphasizes that this yields a field-valued uncertainty estimate directly, without requiring a dense covariance matrix (Hidajat, 15 May 2026). This is one reason MNO can remain computationally aligned with operator-learning practice despite representing stochastic marginals.

The experiments instantiate the residual as Gaussian throughout,

$m_\theta(u_0,t)$2

The stated consequences are closed-form Gaussian NLL, simple sampling, and compatibility with $m_\theta(u_0,t)$3 evaluation (Hidajat, 15 May 2026). The training loss includes Gaussian negative log-likelihood, variance consistency, residual centering, and factor regularization. A representative NLL term is given as

$m_\theta(u_0,t)$4

with

$m_\theta(u_0,t)$5

This formulation implies a specific modeling bias. Because the residual is Gaussian and low-rank, the learned terminal law is efficient and tractable, but the paper correspondingly notes that Gaussian residuals cannot capture heavy tails, jumps, or multimodality (Hidajat, 15 May 2026).

4. Relation to neural operators and generative uncertainty models

The paper positions MNO between two established paradigms. On one side are standard neural operators such as FNO and DeepONet; on the other are diffusion, score, and flow-based generative models (Hidajat, 15 May 2026).

Relative to standard neural operators, the distinction is structural. FNO and DeepONet are deterministic surrogates; if trained with squared error on stochastic targets, they estimate the conditional mean and discard residual uncertainty. MNO preserves the familiar neural-operator properties of being one-shot, resolution transferable, and able to output fields on arbitrary grids, but augments the operator output with a mean field plus PSD low-rank covariance factor (Hidajat, 15 May 2026). In the paper’s summary formulation: a standard operator yields one field output, whereas MNO yields a terminal marginal law approximation.

Relative to diffusion and other generative uncertainty models, the difference is one of target object and computational profile. Diffusion, score, and flow-based models can represent rich distributions, but they typically require iterative sampling, are often much slower at inference, and are aimed at learning the full density or sample-generation process (Hidajat, 15 May 2026). MNO instead targets terminal moments and marginal uncertainty. The paper describes it as a moment surrogate that is faster and directly provides mean and variance fields.

A matched-budget comparison is reported against a conditional diffusion model with FNO score backbone (Hidajat, 15 May 2026). Under the stated setup, MNO predicted moments in $m_\theta(u_0,t)$6 versus $m_\theta(u_0,t)$7 for conditional diffusion at NFE $m_\theta(u_0,t)$8, corresponding to about a $m_\theta(u_0,t)$9 speedup, while also achieving better $\Gamma_\phi(u_0,t)$0: $\Gamma_\phi(u_0,t)$1 for MNO versus $\Gamma_\phi(u_0,t)$2 for diffusion (Hidajat, 15 May 2026). This comparison is specific to the paper’s experimental instantiation, but it sharpens the conceptual placement of MNO as an efficient terminal-marginal learner rather than a full generative sampler.

A plausible implication is that MNO is best understood not as a competitor to all generative models in full-distribution modeling, but as an operator-theoretic uncertainty model optimized for one-shot terminal-law estimation.

5. Experimental domains and reported empirical results

The experimental program spans 1D stochastic surrogate tasks, rough volatility, 2D operator tasks, and a generative-efficiency comparison (Hidajat, 15 May 2026). The 1D stochastic benchmarks include stochastic Burgers, $\Gamma_\phi(u_0,t)$3 field theory / SPDEBench, and 1D zero-shot superresolution; baselines include Neural SPDE, Wiener-chaos-style surrogates, SDENO-style surrogates, and FNO for mean prediction. Rough volatility is evaluated for Hurst parameters $\Gamma_\phi(u_0,t)$4 against Neural SDE and Neural CDE. The 2D tasks include 2D turbulent flow, 2D resolution transfer, and Gray-Scott reaction-diffusion, with FNO, ResNet, and U-Net as baselines (Hidajat, 15 May 2026).

The main quantitative findings reported in the paper are summarized below.

Benchmark	Reported result	Comparison
1D stochastic Burgers	$\Gamma_\phi(u_0,t)$5	vs Neural SPDE $\Gamma_\phi(u_0,t)$6, $\Gamma_\phi(u_0,t)$7 improvement
$\Gamma_\phi(u_0,t)$8 / SPDEBench	$\Gamma_\phi(u_0,t)$9	vs Neural SPDE $\mathbb{E}[u_T\mid u_0]$0, $\mathbb{E}[u_T\mid u_0]$1 improvement
Rough volatility at $\mathbb{E}[u_T\mid u_0]$2	$\mathbb{E}[u_T\mid u_0]$3	vs Neural SDE $\mathbb{E}[u_T\mid u_0]$4, Neural CDE $\mathbb{E}[u_T\mid u_0]$5, $\mathbb{E}[u_T\mid u_0]$6 improvement
Generative efficiency	$\mathbb{E}[u_T\mid u_0]$7	vs diffusion $\mathbb{E}[u_T\mid u_0]$8, about $\mathbb{E}[u_T\mid u_0]$9 faster
2D turbulent flow	mean RMSE $u_t = u_0 + A_t + M_t,$0	roughly on par with FNO $u_t = u_0 + A_t + M_t,$1
2D Gray-Scott	$u_t = u_0 + A_t + M_t,$2	underperforms FNO $u_t = u_0 + A_t + M_t,$3

On stochastic Burgers, MNO achieved $u_t = u_0 + A_t + M_t,$4 versus $u_t = u_0 + A_t + M_t,$5 for Neural SPDE, reported as a $u_t = u_0 + A_t + M_t,$6 improvement (Hidajat, 15 May 2026). Mean RMSE remained close to FNO, with MNO at $u_t = u_0 + A_t + M_t,$7 and FNO at $u_t = u_0 + A_t + M_t,$8. The significance attributed in the paper is that uncertainty modeling improves substantially without materially degrading mean prediction.

On $u_t = u_0 + A_t + M_t,$9 field theory / SPDEBench, MNO achieved $A_t$0 versus $A_t$1 for Neural SPDE, reported as a $A_t$2 improvement, which the paper identifies as the strongest distributional gain in the study (Hidajat, 15 May 2026).

On rough volatility at $A_t$3, MNO achieved $A_t$4, compared with $A_t$5 for Neural SDE and $A_t$6 for Neural CDE, corresponding to a $A_t$7 improvement (Hidajat, 15 May 2026). As noted earlier, the paper explicitly cautions that rough volatility is not semimartingale, so the Doob–Meyer interpretation is heuristic rather than literal in this setting.

On 2D turbulent flow, MNO was roughly on par with FNO on mean RMSE, $A_t$8 versus $A_t$9 (Hidajat, 15 May 2026). The reported interpretation is that the method scales to 2D fields and preserves operator-style transfer behavior, though it is not best-in-class on mean prediction alone.

The negative case is 2D Gray-Scott reaction-diffusion, where MNO underperformed FNO, $M_t$0 versus $M_t$1 (Hidajat, 15 May 2026). The paper treats this as a failure mode, likely because Gray-Scott is relatively quasi-deterministic and pattern-dominated; under that regime, the residual-factor machinery can misallocate deterministic fine-scale structure into uncertainty. This is explicitly framed as a limitation rather than as an isolated anomaly.

6. Strengths, limitations, and nomenclatural ambiguity

The paper lists several strengths of the Martingale Neural Operator: one-shot uncertainty quantification, direct mean and variance fields, PSD covariance by construction through $M_t$2, low-rank computational efficiency, resolution invariance and zero-shot transfer in the neural-operator sense, and strong performance on stochastic PDE terminal laws (Hidajat, 15 May 2026). These properties jointly place MNO within the neural-operator tradition while extending that tradition from deterministic surrogacy to terminal-marginal uncertainty representation.

The limitations are equally explicit. MNO is a terminal marginal learner, not a full pathwise stochastic process model (Hidajat, 15 May 2026). Its Gaussian residual instantiation cannot capture heavy tails, jumps, or multimodality. It does not enforce full filtration-level martingale properties. It can struggle when the main challenge is deterministic fine-scale pattern prediction, as in Gray-Scott. The paper also states that off-diagonal covariance recovery is limited because training emphasizes diagonal variance consistency rather than full covariance fidelity (Hidajat, 15 May 2026).

These caveats address a common misconception. The “martingale” in Martingale Neural Operator does not mean that the architecture learns a complete martingale process in the stochastic-analysis sense. Rather, the name refers to a Doob–Meyer-inspired mean/residual factorization used as a structural prior for terminal-law modeling (Hidajat, 15 May 2026).

A separate source of confusion is acronym overload. Contemporary arXiv literature uses MNO for several unrelated architectures. Multiple Neural Operator denotes a separable multi-task or multiple-operator learning architecture for operator families indexed by a descriptor $M_t$3 (Weihs et al., 2 Apr 2026, Weihs et al., 21 May 2026, Weihs et al., 29 Oct 2025). Mamba Neural Operator denotes an SSM-based neural-operator framework that replaces or augments Transformer attention with Mamba/S6 blocks for PDEs (Cheng et al., 2024). Multi-modal Neural Operator denotes an FMM-inspired architecture for parametric nonlinear boundary value problems with multiple coupled PDE inputs (Madala et al., 16 Jul 2025). Martingale Neural Operator is therefore a distinct construction associated specifically with stochastic terminal-law learning via Doob–Meyer factorization (Hidajat, 15 May 2026).

Taken together, the available evidence suggests a precise niche for Martingale Neural Operator: it extends neural operators from deterministic mean estimation to fast, one-shot approximation of terminal stochastic marginals, with explicit mean and variance fields, while accepting the modeling constraints implied by low-rank Gaussian residual structure (Hidajat, 15 May 2026).