# From Euler to Dormand-Prince: ODE Solvers for Flow Matching Generative Models

arXiv: https://arxiv.org/abs/2605.00836 · HTML: https://mlsanity.com/paper/2605.00836/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 18 of its 21 references that are in the map.
- First posted on arXiv 4 Apr 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 18 of 18)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [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)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)
- ["Hey, that's not an ODE": Faster ODE Adjoints via Seminorms](https://mlsanity.com/paper/2009.09457/) (2020)

## Abstract (arXiv)

Sampling from Flow Matching generative models requires solving an ordinary differential equation (ODE) whose computational cost is dominated by neural network forward passes. We derive four classical ODE solvers -- Euler, Explicit Midpoint, Classical Runge-Kutta (RK4), and Dormand-Prince 5(4) -- from first principles via Taylor expansion, implement them from scratch in PyTorch, and systematically benchmark their efficiency on Conditional Flow Matching tasks ranging from 2D toy distributions to MNIST digits. On the quantitative side, we use sliced Wasserstein distance to construct NFE-quality Pareto frontiers,finding that RK4 at 80 function evaluations achieves sample quality comparable to Euler at 200. Beyond reproducing known convergence rates, we report two empirical observations: (1) the Jacobian eigenvalue spectrum of the learned velocity field stiffens sharply near t=1, explaining why the adaptive Dormand-Prince solver automatically concentrates its step budget at the end of the trajectory; (2) the quality gap between low-order and high-order solvers widens for undertrained and smaller models, indicating that solver choice matters most when the model is imperfect. Code and all experiment scripts are publicly available.