Uploaded September 2021 | Updated September 2026, 17 hours ago
Dynamic Mode Decomposition is an operator theoretic approach to the study of dynamical systems. The way it got its start was by looking at high dimensional systems, and stacking the snapshots into a matrix that was decomposed later. The eigendecomposition of this matrix provided Dynamic Modes, which gave the dominant modes of the system.
However, this approach is extremely limiting, where you can only have as many samples of your space as you have dimensions before you saturate the rank of the matrix. The led to the use of extended DMD methods, where the states were thrown into a high dimensional feature space. Inevitably, this gave rise to infinite dimensional feature spaces and Koopman operators, which is how the connections with reproducing Kernel Hilbert Spaces first came about.
Our work has been towards expanding the scope of Dynamic Mode Decompositions, where our first manuscripts in the field removed the Koopman operator from the analysis, and instead used Liouville operators. By directly accessing these operators, we are removing the requirements that a system is discretizable (forward complete/invariant). But this still leverages operators of functions of the state.
To further extend these ideas to include nonlocal operators within their scope, we need to operate on Hilbert Spaces composed of functions that send continuous signals or trajectories to signals and trajectories. By introducing spaces like these, we can now look at operators that are nonlocal, which includes second order dynamical systems as well as fractional order dynamical systems.
Have a watch! Please subscribe. And I look forward to a lively discussion.
Music:
Dream Away by Consolate
Nobody Knows by Duckmaw
Before the Dawn by Wild Beaches
Clouds & Rainbows by Giants' Nest
Numb to It by Nbhd Nick
Like a Blind Girl's Dog by Daxten
Overthinking by Dylan Sitts
Guardians + Tek by Craig Hardgrove
Dynamic Mode Decomposition is an operator theoretic approach to the study of dynamical systems. The way it got its start was by looking at high dimensional systems, and stacking the snapshots into a matrix that was decomposed later. The eigendecomposition of this matrix provided Dynamic Modes, which gave the dominant modes of the system.
However, this approach is extremely limiting, where you can only have as many samples of your space as you have dimensions before you saturate the rank of the matrix. The led to the use of extended DMD methods, where the states were thrown into a high dimensional feature space. Inevitably, this gave rise to infinite dimensional feature spaces and Koopman operators, which is how the connections with reproducing Kernel Hilbert Spaces first came about.
Our work has been towards expanding the scope of Dynamic Mode Decompositions, where our first manuscripts in the field removed the Koopman operator from the analysis, and instead used Liouville operators. By directly accessing these operators, we are removing the requirements that a system is discretizable (forward complete/invariant). But this still leverages operators of functions of the state.
To further extend these ideas to include nonlocal operators within their scope, we need to operate on Hilbert Spaces composed of functions that send continuous signals or trajectories to signals and trajectories. By introducing spaces like these, we can now look at operators that are nonlocal, which includes second order dynamical systems as well as fractional order dynamical systems.
Have a watch! Please subscribe. And I look forward to a lively discussion.
Music:
Dream Away by Consolate
Nobody Knows by Duckmaw
Before the Dawn by Wild Beaches
Clouds & Rainbows by Giants' Nest
Numb to It by Nbhd Nick
Like a Blind Girl's Dog by Daxten
Overthinking by Dylan Sitts
Guardians + Tek by Craig Hardgrove
