Diffeomorphically Learning Stable Koopman Operators

Abstract

We propose a novel framework for constructing linear time-invariant (LTI) models for datadriven representations of the Koopman operator for a class of stable nonlinear dynamics. The Koopman operator (generator) lifts a finite-dimensional nonlinear system to a possibly infinite-dimensional linear feature space. To utilize it for modeling, one needs to discover finite-dimensional representations of the Koopman operator. Learning suitable features is challenging, as one needs to learn LTI features that are both Koopman-invariant (evolve linearly under the dynamics) as well as relevant (spanning the original state) -a generally unsupervised learning task. For a theoretically well-founded solution to this problem, we propose learning Koopman-invariant coordinates by composing a diffeomorphic learner with a lifted aggregate system of a latent linear model. Using an unconstrained parameterization of stable matrices along with the aforementioned feature construction, we learn the Koopman operator features without assuming a predefined library of functions or knowing the spectrum, while ensuring stability regardless of the operator approximation accuracy. We demonstrate the superior efficacy of the proposed method in comparison to a state-ofthe-art method on the well-known LASA handwriting dataset.

Temporal Forward-Backward Consistency, Not Residual Error, Measures the Prediction Accuracy of Extended Dynamic Mode Decomposition

“…where in the third and fourth equalities we have used the definition of v +s and v −s and their orthogonality. Now, for the function g * (•) = D(•)r * ∈ span(D), one can use (17) to see that RRMSE g * = s. Hence, the equality in (19) holds, and this concludes the proof. Remark V.4: (Working with Consistency Matrix is More Efficient than the Difference of Projections): According to Theorems V.1 and V.3, one can use the consistency matrix M C ∈ R N d ×N d or the difference of projections D(Y )D(Y ) † − D(X)D(X) † ∈ R N ×N interchangeably to compute the relative root mean square error.…”

“…Let v * be such that p * = D(Y )v * . Using (18) for v * instead of v f , and the properties of p * , one can write (17) to see that RRMSE f * = 1 = s. Hence, equality holds in (19).…”

“…The works in [16], [17] use methods based on neural networks for this task, while [18] directly learns the Koopman eigenfunctions spanning invariant subspaces. Moreover, the works in [19], [20] approximate finitedimensional Koopman models relying on knowledge about the system's attractors and their stability. In our previous work, we have provided efficient algebraic algorithms to identify exact Koopman-invariant subspaces [21], [22] or approximate them with tunable predefined accuracy [23].…”

Recognition Models to Learn Dynamics from Partial Observations with Neural ODEs

“…In deep learning, an encoder can be trained to map the observations to the latent state that follows an ODE (Rubanova et al, 2019;Doyeon et al, 2021;de Brouwer et al, 2019). The particular case in which this ODE is linear and evolves according to the Koopman operator (that can be jointly approximated) is investigated in Lusch et al (2018); Bevanda et al (2021). However, little insight on the desired latent representation is usually provided.…”

Deep Koopman Representation of Nonlinear Time Varying Systems

“…In order to address this, some recent work [7][8][9][10][11][12][13][14] have introduced deep neural networks (DNN) into EDMD to approximate time-invariant systems, which utilize DNN as the observable function for Koopman operator. In these papers, the DNN observable function is tuned with respect to the collected data of state-control pairs by minimizing a properly defined loss function.…”