Invariant Manifold

Invariant manifolds represent lower-dimensional subsets within high-dimensional dynamical systems where the long-term behavior of the system resides, offering a powerful tool for simplifying complex models. Current research focuses on learning these manifolds from data using techniques like physics-informed neural networks, autoencoders (including variations like constrained and implicit rank minimizing autoencoders), and diffusion models, often incorporating knowledge of system symmetries or exploiting foliations to better capture the underlying dynamics. This work is significant for enabling reduced-order modeling of complex systems, improving the efficiency of simulations and control algorithms, and facilitating a deeper understanding of the underlying dynamics in diverse fields such as robotics, fluid dynamics, and chemical kinetics.

Papers

October 3, 2024

Learning Optimal Control and Dynamical Structure of Global Trajectory Search Problems with Diffusion Models
Jannik Graebner, Anjian Li, Amlan Sinha, Ryne Beeson
Diffusion Model Optimal Policy Trajectory Tracking Trajectory Design Invariant Manifold Restricted Three Body Problem

September 11, 2024

Manifold Learning via Foliations and Knowledge Transfer
E. Tron, E. Fioresi
Knowledge Transfer Manifold Learning Fisher Information High Dimensional Space Geometric Structure Invariant Manifold

March 21, 2024

Machine-learning invariant foliations in forced systems for reduced order modelling
Robert Szalai
System Description Order Model Invariant Manifold Invariant Torus

March 18, 2024

A physics-informed neural network method for the approximation of slow invariant manifolds for the general class of stiff systems of ODEs
Dimitrios G. Patsatzis, Lucia Russo, Constantinos Siettos
Physic Informed Neural Network Average Approximation Invariant Manifold Stiff Scalar Ordinary Differential Equation Non Equilibrium Steady State

December 15, 2023

Building symmetries into data-driven manifold dynamics models for complex flows
Carlos E. Pérez De Jesús, Alec J. Linot, Michael D. Graham
Long Term Manifold Learning Approximate Symmetry Navier Stokes Symmetry Breaking Invariant Manifold Kolmogorov Flow

September 14, 2023

Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learning
Dimitrios G. Patsatzis, Gianluca Fabiani, Lucia Russo, Constantinos Siettos
Physic Informed Machine Learning Invariant Manifold Numerical Approximation

July 28, 2023

Learning Nonlinear Projections for Reduced-Order Modeling of Dynamical Systems using Constrained Autoencoders
Samuel E. Otto, Gregory R. Macchio, Clarence W. Rowley
Dynamical System Reduced Order Low Dimensional Manifold Reduced Basis Invariant Manifold

March 28, 2023

Diffusion Maps for Group-Invariant Manifolds
Paulina Hoyos, Joe Kileel
Manifold Learning Graph Laplacian Diffusion Map Invariant Manifold Laplacian Operator Single Laplacian Method

November 1, 2022

Augmentation Invariant Manifold Learning
Shulei Wang
Data Augmentation Self Supervised Representation Learning Invariant Manifold Manifold Augmentation

October 13, 2022

Invariance-adapted decomposition and Lasso-type contrastive learning
Masanori Koyama, Takeru Miyato, Kenji Fukumizu
Contrastive Learning Latent Space Microbial Decomposition Invariant Manifold Lasso Condition Number

June 24, 2022

Data-driven reduced order models using invariant foliations, manifolds and autoencoders
Robert Szalai
Supervised Autoencoder Data Driven Closed Loop Model Free Class Manifold Order Model Invariant Manifold

May 9, 2022

Learning effective dynamics from data-driven stochastic systems
Lingyu Feng, Ting Gao, Min Dai, Jinqiao Duan
Data Driven Stochastic Dynamical System Invariant Manifold Effective Dynamic

February 20, 2022

Redundancy Resolution at Position Level
Alin Albu-Schäffer, Arne Sachtler
Robotic System Product Specific Position Information Degree of Freedom Robot Invariant Manifold Robot Kinematics