# Double Diffusion Maps and their Latent Harmonics for Scientific Computations in Latent Space

arXiv: https://arxiv.org/abs/2204.12536 · HTML: https://mlsanity.com/paper/2204.12536/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 45 references that are in the map.
- First posted on arXiv 26 Apr 2022.

## Builds on (most cited first, 3 of 3)

- [Selecting the independent coordinates of manifolds with large aspect ratios](https://mlsanity.com/paper/1907.01651/) (2019)
- [Manifold Coordinates with Physical Meaning](https://mlsanity.com/paper/1811.11891/) (2018)
- [Time Series Forecasting Using Manifold Learning](https://mlsanity.com/paper/2110.03625/) (2021)

## Built on by (most cited first, 16 of 16)

- [Data-driven Control of Agent-based Models: an Equation/Variable-free Machine Learning Approach](https://mlsanity.com/paper/2207.05779/) (2022)
- [Nonlinear dimensionality reduction then and now: AIMs for dissipative PDEs in the ML era](https://mlsanity.com/paper/2310.15816/) (2023)
- [Data-driven modelling of brain activity using neural networks, Diffusion Maps, and the Koopman operator](https://mlsanity.com/paper/2304.11925/) (2023)
- [Generative Learning of Densities on Manifolds](https://mlsanity.com/paper/2503.03963/) (2025)
- [Early warning indicators via latent stochastic dynamical systems](https://mlsanity.com/paper/2309.03842/) (2023)
- [Tasks Makyth Models: Machine Learning Assisted Surrogates for Tipping Points](https://mlsanity.com/paper/2309.14334/) (2023)
- [Generative Learning for Slow Manifolds and Bifurcation Diagrams](https://mlsanity.com/paper/2504.20375/) (2025)
- [A physics-informed neural network method for the approximation of slow invariant manifolds for the general class of stiff systems of ODEs](https://mlsanity.com/paper/2403.11591/) (2024)
- [Machine Learning of Slow Collective Variables and Enhanced Sampling via Spatial Techniques](https://mlsanity.com/paper/2412.20868/) (2024)
- [A Mechanistic Analysis of Transformers for Dynamical Systems](https://mlsanity.com/paper/2512.21113/) (2025)
- [Geometric regularization of autoencoders via observed stochastic dynamics](https://mlsanity.com/paper/2604.16282/) (2026)
- [Tipping Points of Evolving Epidemiological Networks: Machine Learning-Assisted, Data-Driven Effective Modeling](https://mlsanity.com/paper/2311.00797/) (2023)
- [Machine Learning for the identification of phase-transitions in interacting agent-based systems: a Desai-Zwanzig example](https://mlsanity.com/paper/2310.19039/) (2023)
- [Enabling Probabilistic Learning on Manifolds through Double Diffusion Maps](https://mlsanity.com/paper/2506.02254/) (2025)
- [Invariant Manifolds of Discrete-time Dynamical Systems with Nonlinear Exosystems via Hybrid Physics-Informed Neural Networks](https://mlsanity.com/paper/2506.13950/) (2025)
- [RANDSMAPs: Random-Feature/multi-Scale Neural Decoders with Mass Preservation](https://mlsanity.com/paper/2601.14794/) (2026)

## Abstract (arXiv)

We introduce a data-driven approach to building reduced dynamical models through manifold learning; the reduced latent space is discovered using Diffusion Maps (a manifold learning technique) on time series data. A second round of Diffusion Maps on those latent coordinates allows the approximation of the reduced dynamical models. This second round enables mapping the latent space coordinates back to the full ambient space (what is called lifting); it also enables the approximation of full state functions of interest in terms of the reduced coordinates. In our work, we develop and test three different reduced numerical simulation methodologies, either through pre-tabulation in the latent space and integration on the fly or by going back and forth between the ambient space and the latent space. The data-driven latent space simulation results, based on the three different approaches, are validated through (a) the latent space observation of the full simulation through the Nystr"om Extension formula, or through (b) lifting the reduced trajectory back to the full ambient space, via Latent Harmonics. Latent space modeling often involves additional regularization to favor certain properties of the space over others, and the mapping back to the ambient space is then constructed mostly independently from these properties; here, we use the same data-driven approach to construct the latent space and then map back to the ambient space.