Etienne Meunier

Koopman with complex eigenfunctions

In many systems koopman models are going to live in complex space, even if the physical variable is in real space. This impact the way we do our temporal integration and our…

Matrix Exponential vs Numerical Integration

From Euler steps to exp(At), and how to learn linear dynamics

In this tutorial we study the linear ODE

Etienne Meunier

Koopman Eigenfunctions — An Introduction

From the flow of a dynamical system to a linear, infinite-dimensional operator

Consider a (possibly nonlinear) dynamical system on the state \(\Phi(t)\):

Etienne Meunier

Pendulum - Action angle coordinates

\[ \ddot\theta = -\sin\theta \qquad\Longrightarrow\qquad \begin{cases} \dot\theta = \omega \\ \dot\omega = -\sin\theta \end{cases} \]

Etienne Meunier

Koopman with complex eigenfunctions

In many systems Koopman models live in complex space, even when the physical variables are real. This impacts the way we do our temporal integration and our reconstruction.

Etienne Meunier

Koopman unrolling — learning eigenfunctions from data

In the previous tutorials we found Koopman eigenfunctions by hand. Here everything is learned from data: encoder, dynamic and decoder are trained jointly with a single…

Etienne Meunier

Koopman Eigenfunctions — A Nonlinear Example

Testing, decomposing and forecasting a nonlinear system with closed-form eigenfunctions

In the introduction we applied the Koopman decomposition to a scalar linear ODE. Here we do the full loop — test eigenfunctions, decompose an observable, forecast — on a…

Etienne Meunier

Representation learning for action-angle coordinates

In this post we show that we can learn the invariant of a dynamic from trajectories alone; those invariants are then useful to infer and learn the dynamic itself. We start…

Etienne Meunier

Koopman Complex-Valued Eigenfunctions

A follow-up Q&A on the introduction post

This is a follow-up to Koopman Eigenfunctions — An Introduction. There we kept things simple by working with real-valued eigenfunctions, but in general the Koopman…

Etienne Meunier

Learning Linear Dynamics

Four ways to recover A from observations of z(t) = exp(At)z₀

In the previous post we studied how to evaluate \(e^{At}z_0\) efficiently. Here we flip the question:

Etienne Meunier

The function difference trick — an exact linearisation between two points

Let’s say we have a function \(D : \mathbb R^n \to \mathbb R^n\), continuously differentiable, and we want to compute

Etienne Meunier