Inverted Gaussian Process Optimization for Probabilistic Koopman Operator Discovery
Abhigyan Majumdar, Navid Mojahed, Shima Nazari
TL;DR
The paper tackles uncertainty quantification and the reliance on handcrafted observables in data-driven Koopman learning. It introduces Inverted Gaussian Process optimization for probabilistic Koopman (iGPK), which inverts the GP workflow by treating virtual targets $Z$ and kernel hyperparameters $\Theta$ as optimization variables to jointly learn the lifted space $\Phi$ and the Koopman operator. Gradient-based optimization with automatic differentiation is employed to minimize a reduced functional $\mathcal{L}_1$ and a GP marginal likelihood, yielding $\mathbb{K}^*$, $C^*$, and $\Phi_{Z^*,\Theta^*}$. Empirical results show that iGPK robustly learns nonlinear dynamics from noisy data and provides predictive distributions that consistently enclose the ground truth, outperforming traditional eDMD and SSID-GPK approaches.
Abstract
Koopman Operator Theory has opened the doors to data-driven learning of globally linear representations of complex nonlinear systems. However, current methodologies for Koopman Operator discovery struggle with uncertainty quantification and the dependency on a finite dictionary of heuristically chosen observable functions. We leverage Gaussian Process Regression (GPR) to learn a probabilistic Koopman linear model from data, while removing the need for heuristic observable specification. We present inverted Gaussian Process optimization based Koopman operator learning (iGPK), an automatic differentiation-based approach to simultaneously learn the observable-operator combination. Our numerical results show that the iGPK method is able to learn complex nonlinearities from simulation data while being resilient to measurement noise in the training data and consistently encapsulating the ground truth in the predictive distribution.