Effective coupling of Koopman methods with neural operators

Establish an effective way to couple the Koopman perspective with neural operators for unsteady-flow prediction.

Background

The paper explains that Koopman theory represents nonlinear dynamics through linear evolution in an infinite-dimensional observable space, whereas practical neural-operator implementations use finite grids and finite-dimensional latent representations. This truncation introduces a state-dependent residual that can accumulate during autoregressive rollout.

The authors identify effective integration of Koopman-based temporal propagation with neural-operator architectures as unresolved. Their proposed CoKo-UNO addresses this issue through explicit residual compensation using a selective state-space model, but the broader problem of how to couple the Koopman perspective effectively with neural operators is stated as open.

References

These observations suggest that the Koopman perspective is attractive, but how to couple it effectively with neural operators remains an open problem.

— A Compensated Koopman Neural Operator with Selective State-Space Dynamics for Unsteady Flows  (2608.25879 - Lv et al., 26 Aug 2026) in Introduction

Generating training data that covers the distribution induced by coupling, rather than a distribution chosen beforehand, seems to us a central unsolved problem for this class of method.

— Time Without Timesteps: Simulating Coupled Dynamical Systems via Self-Consistency  (2609.03358 - Zerihun et al., 3 Sep 2026) in Section 6, Limitations and Future Work