A new class of physics-based methods related to Koopman theory has been introduced, offering an alternative for processing nonlinear dynamics systems. Koopman theory is based on the insight that a nonlinear dynamical system can be fully encoded using an operator that describes how scalar functions propagate in time. The Koopman operator is linear, and thus preferable to work with in practice, as tools from linear algebra can be directly applied. The Koopman operator maps between function spaces and thus it is infinite-dimensional and can not be represented on a computer. However, most machine learning approaches hypothesize that there exists a data transformation under which an approximate finite-dimensional Koopman operator is available. Typically, this map is represented via an autoencoder network, embedding the input onto a low-dimensional latent space.

Concurrently, inspired by the Kolmogorov-Arnold representation theorem, Kolmogorov-Arnold Networks (KANs) have been proposed as a promising alternative to Multi-Layer Perceptrons (MLPs). While MLps have fixed activation functions on nodes ("neurons"), KANs have learnable activation functions on edges ("weights"). KANs have no linear weights at all -- every weight parameter is replaced by a univariate function parameterized as a spline. Here, we bring together these two worlds by using KANs as the backbone for the Koopman autoencoder.