# Single-seed generation of Brownian paths and integrals for adaptive and high order SDE solvers

arXiv: https://arxiv.org/abs/2405.06464 · HTML: https://mlsanity.com/paper/2405.06464/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 57 references that are in the map.
- First posted on arXiv 10 May 2024.

## Builds on (most cited first, 14 of 14)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1402.4102/) (2014)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Efficient and Accurate Gradients for Neural SDEs](https://mlsanity.com/paper/2105.13493/) (2021)
- [On sampling from a log-concave density using kinetic Langevin diffusions](https://mlsanity.com/paper/1807.09382/) (2018)
- [Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyond](https://mlsanity.com/paper/1906.07868/) (2019)
- [Sampling with Mollified Interaction Energy Descent](https://mlsanity.com/paper/2210.13400/) (2022)
- [Estimating Convergence of Markov chains with L-Lag Couplings](https://mlsanity.com/paper/1905.09971/) (2019)
- [Unbiased Markov chain Monte Carlo with couplings](https://mlsanity.com/paper/1708.03625/) (2017)
- [Unbiased Kinetic Langevin Monte Carlo with Inexact Gradients](https://mlsanity.com/paper/2311.05025/) (2023)
- [Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations](https://mlsanity.com/paper/2104.12384/) (2021)
- [Generative Modelling of L'evy Area for High Order SDE Simulation](https://mlsanity.com/paper/2308.02452/) (2023)

## Built on by (most cited first, 3 of 3)

- [Strong Stochastic Flow Maps](https://mlsanity.com/paper/2606.01086/) (2026)
- [A Langevin sampling algorithm inspired by the Adam optimizer](https://mlsanity.com/paper/2504.18911/) (2025)
- [Approximating Measures on Function Spaces: Transport and Truncation](https://mlsanity.com/paper/2609.17802/) (2026)

## Abstract (arXiv)

Despite the success of adaptive time-stepping in ODE simulation, it has so far seen few applications for Stochastic Differential Equations (SDEs). To simulate SDEs adaptively, methods such as the Virtual Brownian Tree (VBT) have been developed, which can generate Brownian motion (BM) non-chronologically. However, in most applications, knowing only the values of Brownian motion is not enough to achieve a high order of convergence; for that, we must compute time-integrals of BM such as intst Wr , dr. With the aim of using high order SDE solvers adaptively, we extend the VBT to generate these integrals of BM in addition to the Brownian increments. A JAX-based implementation of our construction is included in the popular Diffrax library (https://github.com/patrick-kidger/diffrax). Since the entire Brownian path produced by VBT is uniquely determined by a single PRNG seed, previously generated samples need not be stored, which results in a constant memory footprint and enables experiment repeatability and strong error estimation. Based on binary search, the VBT's time complexity is logarithmic in the tolerance parameter varepsilon. Unlike the original VBT algorithm, which was only precise at some dyadic times, we prove that our construction exactly matches the joint distribution of the Brownian motion and its time integrals at any query times, provided they are at least varepsilon apart. We present two applications of adaptive high order solvers enabled by our new VBT. Using adaptive solvers to simulate a high-volatility CIR model, we achieve more than twice the convergence order of constant stepping. We apply an adaptive third order underdamped or kinetic Langevin solver to an MCMC problem, where our approach outperforms the No U-Turn Sampler, while using only a tenth of its function evaluations.