Parametric Koopman Neural Operators for Condition-Dependent PDE Dynamics

Neural operators have emerged as powerful surrogates for learning evolution operators underlying PDE dynamics in complex dynamical systems. However, time-varying physical conditions, such as forcing and background fields, induce condition-dependent dynamics, posing the challenge of adapting evolution operators to changing conditions while preserving temporal composition across intervals. We introduce the **Parametric Koopman Neural Operator (PKNO)**, which learns condition-dependent Koopman dynamics in shared observable coordinates. We construct shared hybrid observables over finite field histories, combining prescribed physical quantities with learned supplementary observables to capture temporal context in condition-independent coordinates. On these shared coordinates, we learn a condition-to-propagator map that generates mode-wise spectral Koopman propagators, enabling ordered composition under time-varying physical conditions. We further condition propagator generation on the rolling predicted history and derive a rollout-error bound characterizing how rolling-history sensitivity amplifies local prediction errors. Across four PDE benchmarks, PKNO is competitive under standard fixed conditions and demonstrates substantial gains over conditional baselines in challenging time-varying regimes.