# Score Neural Operator: A Generative Model for Learning and Generalizing Across Multiple Probability Distributions

arXiv: https://arxiv.org/abs/2410.08549 · HTML: https://mlsanity.com/paper/2410.08549/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 28 references that are in the map.
- First posted on arXiv 11 Oct 2024.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [A note on the evaluation of generative models](https://mlsanity.com/paper/1511.01844/) (2015)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [One-Shot Generalization in Deep Generative Models](https://mlsanity.com/paper/1603.05106/) (2016)

## Built on by (most cited first, 4 of 4)

- [DiScoFormer: Plug-In Density and Score Estimation with Transformers](https://mlsanity.com/paper/2511.05924/) (2025)
- [SetONet: A Set-Based Operator Network for Solving PDEs with Variable-Input Sampling](https://mlsanity.com/paper/2505.04738/) (2025)
- [From Data to Program: Fast & Direct Generative Program Inference from Empirical Data](https://mlsanity.com/paper/2609.35348/) (2026)
- [Transformer-based In-Context Score Operator Learning for Diffusion Models](https://mlsanity.com/paper/2610.13690/) (2026)

## Abstract (arXiv)

Most existing generative models are limited to learning a single probability distribution from the training data and cannot generalize to novel distributions for unseen data. An architecture that can generate samples from both trained datasets and unseen probability distributions would mark a significant breakthrough. Recently, score-based generative models have gained considerable attention for their comprehensive mode coverage and high-quality image synthesis, as they effectively learn an operator that maps a probability distribution to its corresponding score function. In this work, we introduce the Score Neural Operator, which learns the mapping from multiple probability distributions to their score functions within a unified framework. We employ latent space techniques to facilitate the training of score matching, which tends to over-fit in the original image pixel space, thereby enhancing sample generation quality. Our trained Score Neural Operator demonstrates the ability to predict score functions of probability measures beyond the training space and exhibits strong generalization performance in both 2-dimensional Gaussian Mixture Models and 1024-dimensional MNIST double-digit datasets. Importantly, our approach offers significant potential for few-shot learning applications, where a single image from a new distribution can be leveraged to generate multiple distinct images from that distribution.