EchoKoop: Koopman Forecasting with Learned Lifting and Frozen Multiscale Dynamics

Neural Koopman forecasters typically learn both nonlinear observables and a finite-dimensional latent transition operator. While expressive, the learned operator can be parameter-intensive and difficult to stabilize under repeated long-horizon rollout. We introduce EchoKoop, a forecasting framework based on the principle of learning the lift while freezing the flow. Instead of optimizing Koopman transition matrices, EchoKoop learns nonlinear observables that are propagated by fixed multiscale normal operators. These operators generate flows with different memory ranges, while the learned encoder, modal projections, selectors, and decoders adapt observed histories to this dynamical basis. Removing trainable transition matrices eliminates their parameter cost and spectral drift during training. Across ten long-term forecasting benchmarks, EchoKoop attains the lowest dataset-averaged MSE on six; controlled ablations show no consistent gain from training its transitions.