Beyond One-Step Accuracy: Trajectory, Invariant, and Dissipative Fidelity in Modelling of PDE Dynamics

Popular approaches for modelling PDE dynamics in a data-driven way include direct one-step predictors like neural operators. Recently, continuous-time generator-based methods like Trajectory Flow models have also shown strong results in this setting. These approaches typically have good one-step accuracy. However, long-horizon trajectory fidelity under autoregressive composition and physical consistency are also important metrics for physical systems. Low one-step prediction error alone does not determine long-horizon trajectory fidelity or physical consistency. We analyse why continuous-time generators achieve stronger long-horizon fidelity than the direct one-step predictors. We also introduce GENERIC structure into the Trajectory Flow model to better preserve known physical invariants while retaining a continuous-time generator formulation. Our model reduces invariant violations by orders of magnitude. This substantial gain in invariant preservation comes at the cost of some trajectory accuracy, although the structured model retains strong long-horizon fidelity. Moreover, invariant preservation alone does not ensure quantitatively correct dissipation. The structured model can evolve dissipative functionals in the correct direction while misestimating their rate. We therefore extend GENERIC Flow with explicit functional-change supervision, which partially recovers dissipative-rate fidelity.