# Score Operator Newton transport

arXiv: https://arxiv.org/abs/2305.09792 · HTML: https://mlsanity.com/paper/2305.09792/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 76 references that are in the map.
- First posted on arXiv 16 May 2023.

## Builds on (most cited first, 24 of 24)

- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [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)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization](https://mlsanity.com/paper/2012.05942/) (2020)
- [On the geometry of Stein variational gradient descent](https://mlsanity.com/paper/1912.00894/) (2019)
- [SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence](https://mlsanity.com/paper/2006.02509/) (2020)
- [A Stein variational Newton method](https://mlsanity.com/paper/1806.03085/) (2018)
- [Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality](https://mlsanity.com/paper/2110.06897/) (2021)
- [Projected Stein Variational Gradient Descent](https://mlsanity.com/paper/2002.03469/) (2020)
- [Learning elliptic partial differential equations with randomized linear algebra](https://mlsanity.com/paper/2102.00491/) (2021)
- [Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problems](https://mlsanity.com/paper/2112.03749/) (2021)
- [q-Paths: Generalizing the Geometric Annealing Path using Power Means](https://mlsanity.com/paper/2107.00745/) (2021)
- [A stochastic version of Stein Variational Gradient Descent for efficient sampling](https://mlsanity.com/paper/1902.03394/) (2019)
- [Stein Variational Adaptive Importance Sampling](https://mlsanity.com/paper/1704.05201/) (2017)
- [Adaptive regularisation for ensemble Kalman inversion](https://mlsanity.com/paper/2006.14980/) (2020)

## Built on by (most cited first, 1 of 1)

- [Uncertainty quantification of neural network models of evolving processes via Langevin sampling](https://mlsanity.com/paper/2504.14854/) (2025)

## Abstract (arXiv)

We propose a new approach for sampling and Bayesian computation that uses the score of the target distribution to construct a transport from a given reference distribution to the target. Our approach is an infinite-dimensional Newton method, involving a linear PDE, for finding a zero of a ``score-residual'' operator. We prove sufficient conditions for convergence to a valid transport map. Our Newton iterates can be computed by exploiting fast solvers for elliptic PDEs, resulting in new algorithms for Bayesian inference and other sampling tasks. We identify elementary settings where score-operator Newton transport achieves fast convergence while avoiding mode collapse.