# Nonlinear dimensionality reduction then and now: AIMs for dissipative PDEs in the ML era

arXiv: https://arxiv.org/abs/2310.15816 · HTML: https://mlsanity.com/paper/2310.15816/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 63 references that are in the map.
- First posted on arXiv 24 Oct 2023.

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

- [Deep learning to discover and predict dynamics on an inertial manifold](https://mlsanity.com/paper/2001.04263/) (2019)
- [Data-driven reduced-order models via regularized operator inference for a single-injector combustion process](https://mlsanity.com/paper/2008.02862/) (2020)
- [Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations](https://mlsanity.com/paper/2109.00060/) (2021)
- [Reduced operator inference for nonlinear partial differential equations](https://mlsanity.com/paper/2102.00083/) (2021)
- [Double Diffusion Maps and their Latent Harmonics for Scientific Computations in Latent Space](https://mlsanity.com/paper/2204.12536/) (2022)
- [Improved Surrogates in Inertial Confinement Fusion with Manifold and Cycle Consistencies](https://mlsanity.com/paper/1912.08113/) (2019)
- [Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems](https://mlsanity.com/paper/2305.01090/) (2023)
- [Data-driven low-dimensional dynamic model of Kolmogorov flow](https://mlsanity.com/paper/2210.16708/) (2022)
- [Turbulence control in plane Couette flow using low-dimensional neural ODE-based models and deep reinforcement learning](https://mlsanity.com/paper/2301.12098/) (2023)

## Built on by (most cited first, 7 of 7)

- [Active search for Bifurcations](https://mlsanity.com/paper/2406.11141/) (2024)
- [TRIE: An Evaluation Framework for Stochastic PDE Surrogates](https://mlsanity.com/paper/2607.00196/) (2026)
- [Thinner Latent Spaces: Detecting Dimension and Imposing Invariance with Conformal Autoencoders](https://mlsanity.com/paper/2408.16138/) (2024)
- [Invariant Manifolds of Discrete-time Dynamical Systems with Nonlinear Exosystems via Hybrid Physics-Informed Neural Networks](https://mlsanity.com/paper/2506.13950/) (2025)
- [Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamics](https://mlsanity.com/paper/2507.21299/) (2025)
- [Geometry-Aware Hallucination Detection in Large Language Models](https://mlsanity.com/paper/2601.06196/) (2026)
- [RANDSMAPs: Random-Feature/multi-Scale Neural Decoders with Mass Preservation](https://mlsanity.com/paper/2601.14794/) (2026)

## Abstract (arXiv)

This study presents a collection of purely data-driven workflows for constructing reduced-order models (ROMs) for distributed dynamical systems. The ROMs we focus on, are data-assisted models inspired by, and templated upon, the theory of Approximate Inertial Manifolds (AIMs); the particular motivation is the so-called post-processing Galerkin method of Garcia-Archilla, Novo and Titi. Its applicability can be extended: the need for accurate truncated Galerkin projections and for deriving closed-formed corrections can be circumvented using machine learning tools. When the right latent variables are not a priori known, we illustrate how autoencoders as well as Diffusion Maps (a manifold learning scheme) can be used to discover good sets of latent variables and test their explainability. The proposed methodology can express the ROMs in terms of (a) theoretical (Fourier coefficients), (b) linear data-driven (POD modes) and/or (c) nonlinear data-driven (Diffusion Maps) coordinates. Both Black-Box and (theoretically-informed and data-corrected) Gray-Box models are described; the necessity for the latter arises when truncated Galerkin projections are so inaccurate as to not be amenable to post-processing. We use the Chafee-Infante reaction-diffusion and the Kuramoto-Sivashinsky dissipative partial differential equations to illustrate and successfully test the overall framework.