# Staying the course: Locating equilibria of dynamical systems on Riemannian manifolds defined by point-clouds

arXiv: https://arxiv.org/abs/2204.10413 · HTML: https://mlsanity.com/paper/2204.10413/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 88 references that are in the map.
- First posted on arXiv 21 Apr 2022.

## Builds on (most cited first, 9 of 9)

- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [On the saddle point problem for non-convex optimization](https://mlsanity.com/paper/1405.4604/) (2014)
- [The Geometric Foundations of Hamiltonian Monte Carlo](https://mlsanity.com/paper/1410.5110/) (2014)
- [Non-linear dimensionality reduction: Riemannian metric estimation and the problem of geometric discovery](https://mlsanity.com/paper/1305.7255/) (2013)
- [Nonlinear Discovery of Slow Molecular Modes using State-Free Reversible VAMPnets](https://mlsanity.com/paper/1902.03336/) (2019)
- [Learning Weighted Submanifolds with Variational Autoencoders and Riemannian Variational Autoencoders](https://mlsanity.com/paper/1911.08147/) (2019)
- [Molecular Latent Space Simulators](https://mlsanity.com/paper/2007.00728/) (2020)
- [Convergence and Cycling in Walker-type Saddle Search Algorithms](https://mlsanity.com/paper/1607.08875/) (2016)

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

- [Gentlest ascent dynamics on manifolds defined by adaptively sampled point-clouds](https://mlsanity.com/paper/2302.04426/) (2023)

## Abstract (arXiv)

We introduce a method to successively locate equilibria (steady states) of dynamical systems on Riemannian manifolds. The manifolds need not be characterized by an a priori known atlas or by the zeros of a smooth map. Instead, they can be defined by point-clouds and sampled as needed through an iterative process. If the manifold is an Euclidean space, our method follows isoclines, curves along which the direction of the vector field X is constant. For a generic vector field X, isoclines are smooth curves and every equilibrium lies on isoclines. We generalize the definition of isoclines to Riemannian manifolds through the use of parallel transport: generalized isoclines are curves along which the directions of X are parallel transports of each other. As in the Euclidean case, generalized isoclines of generic vector fields X are smooth curves that connect equilibria of X. Our algorithm can be regarded as an extension of the method of Newton trajectories to the manifold setting when the manifold is unknown. This work is motivated by computational statistical mechanics, specifically high dimensional (stochastic) differential equations that model the dynamics of molecular systems. Often, these dynamics concentrate near low-dimensional manifolds and have transitions (saddle points with a single unstable direction) between metastable equilibria. We employ iteratively sampled data and isoclines to locate these saddle points. Coupling a black-box sampling scheme (e.g., Markov chain Monte Carlo) with manifold learning techniques (diffusion maps in the case presented here), we show that our method reliably locates equilibria of X.