Partial observations, coarse graining and equivariance in Koopman
operator theory for large-scale dynamical systems
- URL: http://arxiv.org/abs/2307.15325v1
- Date: Fri, 28 Jul 2023 06:03:19 GMT
- Title: Partial observations, coarse graining and equivariance in Koopman
operator theory for large-scale dynamical systems
- Authors: Sebastian Peitz, Hans Harder, Feliks N\"uske, Friedrich Philipp,
Manuel Schaller, Karl Worthmann
- Abstract summary: The Koopman operator has become an essential tool for data-driven analysis, prediction and control of complex systems.
We show that symmetries in the system dynamics can be carried over to the Koopman operator, which allows us to massively increase the model efficiency.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: The Koopman operator has become an essential tool for data-driven analysis,
prediction and control of complex systems, the main reason being the enormous
potential of identifying linear function space representations of nonlinear
dynamics from measurements. Until now, the situation where for large-scale
systems, we (i) only have access to partial observations (i.e., measurements,
as is very common for experimental data) or (ii) deliberately perform coarse
graining (for efficiency reasons) has not been treated to its full extent. In
this paper, we address the pitfall associated with this situation, that the
classical EDMD algorithm does not automatically provide a Koopman operator
approximation for the underlying system if we do not carefully select the
number of observables. Moreover, we show that symmetries in the system dynamics
can be carried over to the Koopman operator, which allows us to massively
increase the model efficiency. We also briefly draw a connection to domain
decomposition techniques for partial differential equations and present
numerical evidence using the Kuramoto--Sivashinsky equation.
Related papers
- Nonparametric Control-Koopman Operator Learning: Flexible and Scalable Models for Prediction and Control [2.7784144651669704]
We present a nonparametric framework for learning Koopman operator representations of nonlinear control-affine systems.
We also enhance the scalability of control-Koopman operator estimators by leveraging random projections.
The efficacy of our novel cKOR approach is demonstrated on both forecasting and control tasks.
arXiv Detail & Related papers (2024-05-12T15:46:52Z) - Mori-Zwanzig latent space Koopman closure for nonlinear autoencoder [0.0]
This study presents a novel approach termed Mori-Zwanzig autoencoder (MZ-AE) to robustly approximate the Koopman operator in low-dimensional spaces.
The proposed method leverages a nonlinear autoencoder to extract key observables for approximating a finite invariant Koopman subspace.
It provides a low dimensional approximation for Kuramoto-Sivashinsky with promising short-term predictability and robust long-term statistical performance.
arXiv Detail & Related papers (2023-10-16T18:22:02Z) - Koopman operator learning using invertible neural networks [0.6846628460229516]
In Koopman operator theory, a finite-dimensional nonlinear system is transformed into an infinite but linear system using a set of observable functions.
Current methodologies tend to disregard the importance of the invertibility of observable functions, which leads to inaccurate results.
We propose FlowDMD, aka Flow-based Dynamic Mode Decomposition, that utilizes the Coupling Flow Invertible Neural Network (CF-INN) framework.
arXiv Detail & Related papers (2023-06-30T04:26:46Z) - Estimating Koopman operators with sketching to provably learn large
scale dynamical systems [37.18243295790146]
The theory of Koopman operators allows to deploy non-parametric machine learning algorithms to predict and analyze complex dynamical systems.
We boost the efficiency of different kernel-based Koopman operator estimators using random projections.
We establish non error bounds giving a sharp characterization of the trade-offs between statistical learning rates and computational efficiency.
arXiv Detail & Related papers (2023-06-07T15:30:03Z) - Learning Linear Causal Representations from Interventions under General
Nonlinear Mixing [52.66151568785088]
We prove strong identifiability results given unknown single-node interventions without access to the intervention targets.
This is the first instance of causal identifiability from non-paired interventions for deep neural network embeddings.
arXiv Detail & Related papers (2023-06-04T02:32:12Z) - Capturing dynamical correlations using implicit neural representations [85.66456606776552]
We develop an artificial intelligence framework which combines a neural network trained to mimic simulated data from a model Hamiltonian with automatic differentiation to recover unknown parameters from experimental data.
In doing so, we illustrate the ability to build and train a differentiable model only once, which then can be applied in real-time to multi-dimensional scattering data.
arXiv Detail & Related papers (2023-04-08T07:55:36Z) - A Causality-Based Learning Approach for Discovering the Underlying
Dynamics of Complex Systems from Partial Observations with Stochastic
Parameterization [1.2882319878552302]
This paper develops a new iterative learning algorithm for complex turbulent systems with partial observations.
It alternates between identifying model structures, recovering unobserved variables, and estimating parameters.
Numerical experiments show that the new algorithm succeeds in identifying the model structure and providing suitable parameterizations for many complex nonlinear systems.
arXiv Detail & Related papers (2022-08-19T00:35:03Z) - Supervised DKRC with Images for Offline System Identification [77.34726150561087]
Modern dynamical systems are becoming increasingly non-linear and complex.
There is a need for a framework to model these systems in a compact and comprehensive representation for prediction and control.
Our approach learns these basis functions using a supervised learning approach.
arXiv Detail & Related papers (2021-09-06T04:39:06Z) - Estimating Koopman operators for nonlinear dynamical systems: a
nonparametric approach [77.77696851397539]
The Koopman operator is a mathematical tool that allows for a linear description of non-linear systems.
In this paper we capture their core essence as a dual version of the same framework, incorporating them into the Kernel framework.
We establish a strong link between kernel methods and Koopman operators, leading to the estimation of the latter through Kernel functions.
arXiv Detail & Related papers (2021-03-25T11:08:26Z) - On dissipative symplectic integration with applications to
gradient-based optimization [77.34726150561087]
We propose a geometric framework in which discretizations can be realized systematically.
We show that a generalization of symplectic to nonconservative and in particular dissipative Hamiltonian systems is able to preserve rates of convergence up to a controlled error.
arXiv Detail & Related papers (2020-04-15T00:36:49Z) - Forecasting Sequential Data using Consistent Koopman Autoencoders [52.209416711500005]
A new class of physics-based methods related to Koopman theory has been introduced, offering an alternative for processing nonlinear dynamical systems.
We propose a novel Consistent Koopman Autoencoder model which, unlike the majority of existing work, leverages the forward and backward dynamics.
Key to our approach is a new analysis which explores the interplay between consistent dynamics and their associated Koopman operators.
arXiv Detail & Related papers (2020-03-04T18:24:30Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.