Amortized Generative Modelling of Invariant Measures Across a Family of Kuramoto-Sivashinsky PDEs

The long-run statistics of chaotic dissipative partial differential equations are described by invariant measures, and when the equation carries a parameter these measures form a family whose members can differ in kind, from steady states to sustained spatiotemporal chaos. We ask whether a single generative model can cover such a family. We train a single conditional diffusion model once, on states of the Kuramoto–Sivashinsky equation at 160 values of its viscosity, and evaluate it against held-out data at interleaved parameter values not seen in training, using a classifier two-sample test supported by spectral, geometric and tail statistics. Agreement improves as the dynamics become more chaotic. We identify a structural cause: the weakly chaotic members have invariant measures concentrated on low-dimensional sets, which a model with full-support output cannot represent, and the target approaches full dimensionality as chaos increases. Restricting training to the strongly chaotic members reduces the classifier's excess over chance on those members by roughly 40%, and this restricted conditional model matches or exceeds models trained at single parameter values. On those members, generated samples reproduce energy spectra to within a resampling floor, with a small systematic deficit at the largest scales, and match the energy balance of held-out data.