# Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Models arXiv: https://arxiv.org/abs/2303.17109 · HTML: https://mlsanity.com/paper/2303.17109/ ## Key facts - Cited by 1 papers in the map (data through 14 Oct 2026). - Builds on 45 of its 100 references that are in the map. - First posted on arXiv 30 Mar 2023. ## Builds on (most cited first, 30 of 45) - [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017) - [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017) - [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021) - [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017) - [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017) - [A Tail-Index Analysis of Stochastic Gradient Noise in Deep Neural Networks](https://mlsanity.com/paper/1901.06053/) (2019) - [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014) - [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015) - [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015) - [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016) - [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021) - [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018) - [Towards Theoretically Understanding Why SGD Generalizes Better Than ADAM in Deep Learning](https://mlsanity.com/paper/2010.05627/) (2020) - [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017) - [Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization](https://mlsanity.com/paper/1707.06618/) (2017) - [A Theoretical Analysis of Deep Neural Networks and Parametric PDEs](https://mlsanity.com/paper/1904.00377/) (2019) - [An overview on deep learning-based approximation methods for partial differential equations](https://mlsanity.com/paper/2012.12348/) (2020) - [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022) - [Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlo](https://mlsanity.com/paper/2202.05214/) (2022) - [Hausdorff Dimension, Heavy Tails, and Generalization in Neural Networks](https://mlsanity.com/paper/2006.09313/) (2020) - [Mirrored Langevin Dynamics](https://mlsanity.com/paper/1802.10174/) (2018) - [Stochastic gradient Markov chain Monte Carlo](https://mlsanity.com/paper/1907.06986/) (2019) - [On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothness](https://mlsanity.com/paper/2005.13097/) (2020) - [First Exit Time Analysis of Stochastic Gradient Descent Under Heavy-Tailed Gradient Noise](https://mlsanity.com/paper/1906.09069/) (2019) - [Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo](https://mlsanity.com/paper/1705.08964/) (2017) - [Non-parametric Models for Non-negative Functions](https://mlsanity.com/paper/2007.03926/) (2020) - [Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality](https://mlsanity.com/paper/2110.06897/) (2021) - [Primal Dual Interpretation of the Proximal Stochastic Gradient Langevin Algorithm](https://mlsanity.com/paper/2006.09270/) (2020) - [Projected Stochastic Gradient Langevin Algorithms for Constrained Sampling and Non-Convex Learning](https://mlsanity.com/paper/2012.12137/) (2020) - [Langevin Monte Carlo without smoothness](https://mlsanity.com/paper/1905.13285/) (2019) ## Built on by (most cited first, 1 of 1) - [BridgeNet: A Hybrid, Physics-Informed Machine Learning Framework for Solving High-Dimensional Fokker-Planck Equations](https://mlsanity.com/paper/2506.04354/) (2025) ## Abstract (arXiv) This paper deals with the problem of efficient sampling from a stochastic differential equation, given the drift function and the diffusion matrix. The proposed approach leverages a recent model for probabilities citerudi2021psd (the positive semi-definite -- PSD model) from which it is possible to obtain independent and identically distributed (i.i.d.) samples at precision varepsilon with a cost that is m2 d log(1/varepsilon) where m is the dimension of the model, d the dimension of the space. The proposed approach consists in: first, computing the PSD model that satisfies the Fokker-Planck equation (or its fractional variant) associated with the SDE, up to error varepsilon, and then sampling from the resulting PSD model. Assuming some regularity of the Fokker-Planck solution (i.e. β-times differentiability plus some geometric condition on its zeros) We obtain an algorithm that: (a) in the preparatory phase obtains a PSD model with L2 distance varepsilon from the solution of the equation, with a model of dimension m = varepsilon-(d+1)/(β-2s) (log(1/varepsilon))d+1 where 1/2≤ s≤1 is the fractional power to the Laplacian, and total computational complexity of O(m3.5 log(1/varepsilon)) and then (b) for Fokker-Planck equation, it is able to produce i.i.d. samples with error varepsilon in Wasserstein-1 distance, with a cost that is O(d varepsilon-2(d+1)/β-2 log(1/varepsilon)2d+3) per sample. This means that, if the probability associated with the SDE is somewhat regular, i.e. β ≥ 4d+2, then the algorithm requires O(varepsilon-0.88 log(1/varepsilon)4.5d) in the preparatory phase, and O(varepsilon-1/2log(1/varepsilon)2d+2) for each sample. Our results suggest that as the true solution gets smoother, we can circumvent the curse of dimensionality without requiring any sort of convexity.