Predicting Unseen Dynamics with Neural Emulators

Can neural emulators predict dynamical regimes that are absent from their training data? Our main results concern two-dimensional Kolmogorov flow: an emulator trained only on chaotic aspect ratios predicts laminar dynamics beyond an $\alpha$-bifurcation, provided the architecture does not carry a symmetry the equations lack. A standard convolution is equivariant to every $y$-shift, an invariance the forced equations do not possess, and drifts unphysically along that unconstrained direction; an ${\mathcal{S}}^2$-aware convolution, equivariant only to the discrete $y$-shifts the shift-reflect symmetry permits, does not. As supporting multi-system evidence, a Kuramoto–Sivashinsky emulator recovers excluded relaminarisation, warm-up, and kink transitions and extrapolates to domains twice as large as those seen in training, while a partially observed beta-plane emulator recovers coalescence, nucleation, and stationary four-jet states excluded from training. Our failures are as informative as our successes: the chaotic transient approaching the Kolmogorov attractor is not recovered, and the learned map retains a dependence on the emulator timestep that a converged evolution map would not have. Both point to what matters being whether the target is represented locally in the training measure, at the scale the architecture sees, rather than whether the whole state has been seen before.