The field of time series modeling and simulation is witnessing a significant shift towards operator-theoretic approaches, which offer a promising alternative to traditional methods. These approaches, based on the concept of linear maps on Hilbert spaces, enable efficient and accurate modeling of complex dynamics. Recent developments have focused on leveraging spectral decomposition, flow matching, and mean flows to improve the representation and learning of nonlinear and probabilistic state dynamics. Notably, these methods have been applied to various domains, including computational fluid dynamics, thermal simulation, and neural networks, demonstrating their potential for advancing the field.

Some noteworthy papers in this area include: Operator Flow Matching for Timeseries Forecasting, which proposes a novel approach to time series forecasting using flow matching, achieving state-of-the-art results on several benchmark datasets. Sequence Modeling with Spectral Mean Flows, which introduces a new approach to sequence modeling based on spectral mean flows, demonstrating competitive results on time-series modeling datasets. MNO: Multiscale Neural Operator for Computational Fluid Dynamics, which presents a novel architecture for computational fluid dynamics on 3D unstructured point clouds, outperforming state-of-the-art baselines on several tasks.