Event-Timing Priors: Temporal Point Process Likelihoods Correct the Long-Horizon Extreme-Event Statistics of Generative Surrogates

Generative surrogates of chaotic dynamical systems — including modern diffusion-basedweather and climate emulators — are routinely evaluated on extreme events but never trained ontheir timing. We show that this omission is consequential and correctable. Working with a twoscale Lorenz-96 testbed observed under realistic conditions (coarse emulator timestep, observationnoise), we find that a conditional flow-matching surrogate reproduces marginal statistics wellwhile systematically corrupting the temporal statistics of threshold-crossing extreme events in longrollouts: event rates inflate, inter-event-time distributions distort, and return periods drift. Weintroduce an event-timing prior: a neural temporal point process (TPP) fit to the event timingsof a curated catalog, whose likelihood — evaluated on soft, differentiable events extracted fromshort surrogate rollouts — serves as an auxiliary fine-tuning signal. Naive likelihood maximizationis mode-seeking and collapses the event rate; we instead use a likelihood-ratio objective against aco-trained “self-TPP” (the point-process analog of variational score distillation), whose gradientvanishes exactly when the surrogate’s event process matches the catalog’s. Across seeds, TPP-auxfine-tuning recovers inter-event-time distributions, burstiness, and return-period curves thatinvariant-measure matching [7] provably leaves unconstrained, while a rate-matched Poissoncontrol shows the gains require the prior’s temporal structure, not merely its rate. A secondsystem (Kuramoto–Sivashinsky) maps the method’s boundary: when rollout dynamics are deeplycorrupted, a timing-only prior collapses the event rate rather than repairing it — a characteristic,easily monitored failure signature whose mechanism we analyze. Event-timing structure is thusa distinct, directly controllable axis of surrogate quality — one that complements, and does notsubstitute for, dynamics-level training signals.