# Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generation

arXiv: https://arxiv.org/abs/2303.04772 · HTML: https://mlsanity.com/paper/2303.04772/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 35 of its 85 references that are in the map.
- First posted on arXiv 8 Mar 2023.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 35)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [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)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Simple diffusion: End-to-end diffusion for high resolution images](https://mlsanity.com/paper/2301.11093/) (2023)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [The Vendi Score: A Diversity Evaluation Metric for Machine Learning](https://mlsanity.com/paper/2210.02410/) (2022)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021)
- [Generative Modelling With Inverse Heat Dissipation](https://mlsanity.com/paper/2206.13397/) (2022)
- [Generative Models as Distributions of Functions](https://mlsanity.com/paper/2102.04776/) (2021)
- [Score-based Diffusion Models in Function Space](https://mlsanity.com/paper/2302.07400/) (2023)
- [SinDDM: A Single Image Denoising Diffusion Model](https://mlsanity.com/paper/2211.16582/) (2022)
- [Generative Adversarial Neural Operators](https://mlsanity.com/paper/2205.03017/) (2022)
- [Diffusion Generative Models in Infinite Dimensions](https://mlsanity.com/paper/2212.00886/) (2022)
- [Continuous-Time Functional Diffusion Processes](https://mlsanity.com/paper/2303.00800/) (2023)
- [Progressive Deblurring of Diffusion Models for Coarse-to-Fine Image Synthesis](https://mlsanity.com/paper/2207.11192/) (2022)
- [Stochastic Normalizing Flows for Inverse Problems: a Markov Chains Viewpoint](https://mlsanity.com/paper/2109.11375/) (2021)
- [Generalized Normalizing Flows via Markov Chains](https://mlsanity.com/paper/2111.12506/) (2021)

## Built on by (most cited first, 30 of 37)

- [Functional Flow Matching](https://mlsanity.com/paper/2305.17209/) (2023)
- [Wavelet Diffusion Neural Operator](https://mlsanity.com/paper/2412.04833/) (2024)
- [∞-Diff: Infinite Resolution Diffusion with Subsampled Mollified States](https://mlsanity.com/paper/2303.18242/) (2023)
- [Warped Diffusion: Solving Video Inverse Problems with Image Diffusion Models](https://mlsanity.com/paper/2410.16152/) (2024)
- [Stochastic Optimal Control for Diffusion Bridges in Function Spaces](https://mlsanity.com/paper/2405.20630/) (2024)
- [Conditioning non-linear and infinite-dimensional diffusion processes](https://mlsanity.com/paper/2402.01434/) (2024)
- [Functional Diffusion](https://mlsanity.com/paper/2311.15435/) (2023)
- [Geometric Neural Diffusion Processes](https://mlsanity.com/paper/2307.05431/) (2023)
- [Probabilistic operator learning: generative modeling and uncertainty quantification for foundation models of differential equations](https://mlsanity.com/paper/2509.05186/) (2025)
- [Universal Functional Regression with Neural Operator Flows](https://mlsanity.com/paper/2404.02986/) (2024)
- [Taming Score-Based Diffusion Priors for Infinite-Dimensional Nonlinear Inverse Problems](https://mlsanity.com/paper/2405.15676/) (2024)
- [Infinite-dimensional Diffusion Bridge Simulation via Operator Learning](https://mlsanity.com/paper/2405.18353/) (2024)
- [Flow Straight and Fast in Hilbert Space: Functional Rectified Flow](https://mlsanity.com/paper/2509.10384/) (2025)
- [Probability-Flow ODE in Infinite-Dimensional Function Spaces](https://mlsanity.com/paper/2503.10219/) (2025)
- [Preconditioned Langevin Dynamics with Score-Based Generative Models for Infinite-Dimensional Linear Bayesian Inverse Problems](https://mlsanity.com/paper/2505.18276/) (2025)
- [Scale-Adaptive Generative Flows for Multiscale Scientific Data](https://mlsanity.com/paper/2509.02971/) (2025)
- [Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport](https://mlsanity.com/paper/2509.22138/) (2025)
- [Functional Mean Flow in Hilbert Space](https://mlsanity.com/paper/2511.12898/) (2025)
- [Physics-informed diffusion models in spectral space](https://mlsanity.com/paper/2602.09708/) (2026)
- [Bayesian computation with generative diffusion models by Multilevel Monte Carlo](https://mlsanity.com/paper/2409.15511/) (2024)
- [Supervised Guidance Training for Infinite-Dimensional Diffusion Models](https://mlsanity.com/paper/2601.20756/) (2026)
- [Dimension-Free Multimodal Sampling via Preconditioned Annealed Langevin Dynamics](https://mlsanity.com/paper/2602.01449/) (2026)
- [A Malliavin-Gamma calculus approach to Score Based Diffusion Generative models for random fields](https://mlsanity.com/paper/2505.13189/) (2025)
- [A Jump-Diffusion Framework for Irregular Time Series Generation](https://mlsanity.com/paper/2505.23215/) (2025)
- [Provable Mixed-Noise Learning with Flow-Matching](https://mlsanity.com/paper/2508.18122/) (2025)
- [Multi-Lattice Sampling of Quantum Field Theories via Neural Operator-based Flows](https://mlsanity.com/paper/2401.00828/) (2024)
- [DFU: scale-robust diffusion model for zero-shot super-resolution image generation](https://mlsanity.com/paper/2401.06144/) (2023)
- [An Unconditional Representation of the Conditional Score in Infinite-Dimensional Linear Inverse Problems](https://mlsanity.com/paper/2405.15643/) (2024)
- [Resolution-Robust 3D MRI Reconstruction with 2D Diffusion Priors: Diverse-Resolution Training Outperforms Interpolation](https://mlsanity.com/paper/2412.18584/) (2024)
- [Relative Entropy Estimation in Function Space: Theory and Applications to Trajectory Inference](https://mlsanity.com/paper/2604.20775/) (2026)

## Abstract (arXiv)

Score-based diffusion models (SBDM) have recently emerged as state-of-the-art approaches for image generation. Existing SBDMs are typically formulated in a finite-dimensional setting, where images are considered as tensors of finite size. This paper develops SBDMs in the infinite-dimensional setting, that is, we model the training data as functions supported on a rectangular domain. In addition to the quest for generating images at ever-higher resolutions, our primary motivation is to create a well-posed infinite-dimensional learning problem that we can discretize consistently on multiple resolution levels. We thereby intend to obtain diffusion models that generalize across different resolution levels and improve the efficiency of the training process. We demonstrate how to overcome two shortcomings of current SBDM approaches in the infinite-dimensional setting. First, we modify the forward process using trace class operators to ensure that the latent distribution is well-defined in the infinite-dimensional setting and derive the reverse processes for finite-dimensional approximations. Second, we illustrate that approximating the score function with an operator network is beneficial for multilevel training. After deriving the convergence of the discretization and the approximation of multilevel training, we demonstrate some practical benefits of our infinite-dimensional SBDM approach on a synthetic Gaussian mixture example, the MNIST dataset, and a dataset generated from a nonlinear 2D reaction-diffusion equation.