# Deep operator learning for efficient sampling from invariant measures of stochastic differential equations

arXiv: https://arxiv.org/abs/2609.11376 · HTML: https://mlsanity.com/paper/2609.11376/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 38 references that are in the map.
- First posted on arXiv 10 Sep 2026.

## Builds on (most cited first, 20 of 20)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Perceiver: General Perception with Iterative Attention](https://mlsanity.com/paper/2103.03206/) (2021)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Are Transformers universal approximators of sequence-to-sequence functions?](https://mlsanity.com/paper/1912.10077/) (2019)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Generative Adversarial Neural Operators](https://mlsanity.com/paper/2205.03017/) (2022)
- [Continuum Attention for Neural Operators](https://mlsanity.com/paper/2406.06486/) (2024)
- [Variable-Input Deep Operator Networks](https://mlsanity.com/paper/2205.11404/) (2022)
- [Stochastic Process Learning via Operator Flow Matching](https://mlsanity.com/paper/2501.04126/) (2025)
- [Attention Mechanism, Max-Affine Partition, and Universal Approximation](https://mlsanity.com/paper/2504.19901/) (2025)
- [Weak Generative Sampler to Efficiently Sample Invariant Distribution of Stochastic Differential Equation](https://mlsanity.com/paper/2405.19256/) (2024)
- [FlowKac: An Efficient Neural Fokker-Planck solver using Temporal Normalizing Flows and the Feynman-Kac Formula](https://mlsanity.com/paper/2503.11427/) (2025)
- [Learn to Evolve: Self-supervised Neural JKO Operator for Wasserstein Gradient Flow](https://mlsanity.com/paper/2601.05583/) (2026)

## Abstract (arXiv)

We introduce an amortized neural sampler that combines operator learning with flow methods for sampling. It maps SDE coefficient functions to pushforwards from a reference measure to the invariant measures, enabling efficient sampling across families of stochastic differential equations. Our framework shifts traditional sampling cost to an initial training phase, after which new SDE instances require only one encoder pass and a few ODE solver steps, independent of mixing time. To handle problems in high dimensions, we use Lagrangian trajectory sensors for the coefficient functions and cross attention in the architecture. We also theoretically establish the expressivity and resolution invariance of our framework. Experiments on 1D and 2D SDE families show competitive accuracy with substantial speedups over MCMC in regimes with slow mixing, transfer across sensor counts, and demonstration results on a 64D interacting particle SDE where traditional grid approaches are infeasible.