# MLPGradientFlow: going with the flow of multilayer perceptrons (and finding minima fast and accurately)

arXiv: https://arxiv.org/abs/2301.10638 · HTML: https://mlsanity.com/paper/2301.10638/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 9 references that are in the map.
- First posted on arXiv 25 Jan 2023.

## Builds on (most cited first, 4 of 4)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Dynamics of stochastic gradient descent for two-layer neural networks in the teacher-student setup](https://mlsanity.com/paper/1906.08632/) (2019)
- [Fashionable Modelling with Flux](https://mlsanity.com/paper/1811.01457/) (2018)
- [A Linearly-Convergent Stochastic L-BFGS Algorithm](https://mlsanity.com/paper/1508.02087/) (2015)

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

- [Should Under-parameterized Student Networks Copy or Average Teacher Weights?](https://mlsanity.com/paper/2311.01644/) (2023)
- [Flat Channels to Infinity in Neural Loss Landscapes](https://mlsanity.com/paper/2506.14951/) (2025)
- [Teacher Geometry Shapes Learnability in Teacher-Student Networks](https://mlsanity.com/paper/2609.09595/) (2026)

## Abstract (arXiv)

MLPGradientFlow is a software package to solve numerically the gradient flow differential equation dot θ = -nabla mathcal L(θ; mathcal D), where θ are the parameters of a multi-layer perceptron, mathcal D is some data set, and nabla mathcal L is the gradient of a loss function. We show numerically that adaptive first- or higher-order integration methods based on Runge-Kutta schemes have better accuracy and convergence speed than gradient descent with the Adam optimizer. However, we find Newton's method and approximations like BFGS preferable to find fixed points (local and global minima of mathcal L) efficiently and accurately. For small networks and data sets, gradients are usually computed faster than in pytorch and Hessian are computed at least 5× faster. Additionally, the package features an integrator for a teacher-student setup with bias-free, two-layer networks trained with standard Gaussian input in the limit of infinite data. The code is accessible at https://github.com/jbrea/MLPGradientFlow.jl.