mlsanityData to 7 Oct 2026

Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit

arXiv 1905.09883 ↗First posted 23 May 2019cs.LG stat.MLcited by 168

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Cited by 21 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

In deep latent Gaussian models, the latent variable is generated by a time-inhomogeneous Markov chain, where at each time step we pass the current state through a parametric nonlinear map, such as a feedforward neural net, and add a small independent Gaussian perturbation. This work considers the diffusion limit of such models, where the number of layers tends to infinity, while the step size and the noise variance tend to zero. The limiting latent object is an Ito diffusion process that solves a stochastic differential equation (SDE) whose drift and diffusion coefficient are implemented by neural nets. We develop a variational inference framework for these textitneural SDEs via stochastic automatic differentiation in Wiener space, where the variational approximations to the posterior are obtained by Girsanov (mean-shift) transformation of the standard Wiener process and the computation of gradients is based on the theory of stochastic flows. This permits the use of black-box SDE solvers and automatic differentiation for end-to-end inference. Experimental results with synthetic data are provided.

Key facts. Cited by 168 papers in the map (data through 7 Oct 2026). First posted on arXiv 23 May 2019.

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Citing papers per year: 2021: 24; 2022: 27; 2023: 30; 2024: 33; 2025: 33; 2026: 21.

168 papers.

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  1. 2022Photorealistic Text-to-Image Diffusion Models with Deep Language Understandingcited by 6,322
  2. 2022Video Diffusion Modelscited by 2,280
  3. 2022Imagen Video: High Definition Video Generation with Diffusion Modelscited by 1,765
  4. 2021Maximum Likelihood Training of Score-Based Diffusion Modelscited by 670
  5. 2022MAGVIT: Masked Generative Video Transformercited by 341
  6. 2021GRAND: Graph Neural Diffusioncited by 258
  7. 2021Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Modelscited by 251
  8. 2021A Variational Perspective on Diffusion-Based Generative Models and Score Matchingcited by 182
  9. 2021Neural SDEs as Infinite-Dimensional GANscited by 159
  10. 2024Iterated Denoising Energy Matching for Sampling from Boltzmann Densitiescited by 113
  11. 2025Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matchingcited by 78
  12. 2021Efficient and Accurate Gradients for Neural SDEscited by 68
38 more
  1. 2022Progressive Distillation for Fast Sampling of Diffusion Modelscited by 61
  2. 2021Neural ODE Processescited by 57
  3. 2021Bayesian Learning via Neural Schr"odinger-F"ollmer Flowscited by 56
  4. 2024Beyond ELBOs: A Large-Scale Evaluation of Variational Methods for Samplingcited by 55
  5. 2022Generative Adversarial Neural Operatorscited by 51
  6. 2024Improved off-policy training of diffusion samplerscited by 47
  7. 2021Noisy Recurrent Neural Networkscited by 42
  8. 2021Heavy Ball Neural Ordinary Differential Equationscited by 42
  9. 2022Normalizing flow neural networks by JKO schemecited by 39
  10. 2021Infinitely Deep Bayesian Neural Networks with Stochastic Differential Equationscited by 38
  11. 2023Non-adversarial training of Neural SDEs with signature kernel scorescited by 36
  12. 2024Stable Neural Stochastic Differential Equations in Analyzing Irregular Time Series Datacited by 35
  13. 2022Signature Methods in Machine Learningcited by 33
  14. 2022On Analyzing Generative and Denoising Capabilities of Diffusion-based Deep Generative Modelscited by 31
  15. 2021Attentive Neural Controlled Differential Equations for Time-series Classification and Forecastingcited by 28
  16. 2023Bayesian Neural Controlled Differential Equations for Treatment Effect Estimationcited by 26
  17. 2022Unifying Generative Models with GFlowNets and Beyondcited by 24
  18. 2022Predicting the impact of treatments over time with uncertainty aware neural differential equationscited by 23
  19. 2024Adaptive teachers for amortized samplerscited by 23
  20. 2022EXIT: Extrapolation and Interpolation-based Neural Controlled Differential Equations for Time-series Classification and Forecastingcited by 22
  21. 2023Learning Data Representations with Joint Diffusion Modelscited by 22
  22. 2023Generalization bounds for neural ordinary differential equations and deep residual networkscited by 21
  23. 2023Transport meets Variational Inference: Controlled Monte Carlo Diffusionscited by 21
  24. 2025Comprehensive Review of Neural Differential Equations for Time Series Analysiscited by 20
  25. 2025SDE Matching: Scalable and Simulation-Free Training of Latent Stochastic Differential Equationscited by 19
  26. 2021Policy Analysis using Synthetic Controls in Continuous-Timecited by 18
  27. 2021Learning the temporal evolution of multivariate densities via normalizing flowscited by 18
  28. 2023Deep Generative Models for Detector Signature Simulation: A Taxonomic Reviewcited by 18
  29. 2025From discrete-time policies to continuous-time diffusion samplers: Asymptotic equivalences and faster trainingcited by 17
  30. 2023Latent SDEs on Homogeneous Spacescited by 16
  31. 2025Outsourced diffusion sampling: Efficient posterior inference in latent spaces of generative modelscited by 16
  32. 2021Variational multiple shooting for Bayesian ODEs with Gaussian processescited by 15
  33. 2023ERNIE-Music: Text-to-Waveform Music Generation with Diffusion Modelscited by 15
  34. 2021Scalable Inference in SDEs by Direct Matching of the Fokker-Planck-Kolmogorov Equationcited by 14
  35. 2023Neural Structure Learning with Stochastic Differential Equationscited by 14
  36. 2023Neural Ideal Large Eddy Simulation: Modeling Turbulence with Neural Stochastic Differential Equationscited by 13
  37. 2021Differentiable Physics: A Position Piececited by 12
  38. 2023STDiff: Spatio-temporal Diffusion for Continuous Stochastic Video Predictioncited by 12

50 most cited of 168 All 168 (JSON)