Climate emulators stop relying on luck: dissipation built into the generative network keeps long rollouts and strong forcing bounded
Synopsis
Partially observed dynamical systems expose only coarse-grained slow variables, while unresolved fast variables enter as random effects, so data-driven climate emulators must be probabilistic, yet generative models such as diffusion models lack stability guarantees over long autoregressive rollouts. Forced-dissipative stochastic interpolants write the energy injection and dissipation structure of turbulence into the drift network as a hard constraint, and a Foster–Lyapunov drift condition for the Markov chain proves geometric ergodicity of the emulator. On the two-scale Lorenz-96 prototype, the emulator stays stable over long integrations and under finite-amplitude external forcing, and accurately predicts the forced response of both mean and variance.
Interpretation
Probabilistic climate emulators can now obtain long-term stability at the architecture level rather than through empirical tuning after training. Earlier generative emulators improved long-run behavior with corrective procedures, multi-step training, Fourier regularizations or soft energy constraints, but these strategies carried no formal mathematical stability guarantee, especially for probabilistic models. On the two-scale Lorenz-96 system, the unconstrained stochastic interpolant ultimately diverges over long integrations, whereas the forced-dissipative interpolant remains stable and accurately reproduces the stationary statistics.
Stability comes from a fixed energy flow inside the drift network, not from a preset energy level for the system. Existing energy-constraint methods must specify an energy level in advance, such as the mean energy of the unforced system, and are therefore not directly compatible with external forcing that changes energy. The drift is parameterized as a bounded internal forcing term, linear and nonlinear energy-preserving terms, and a dissipative term with a fixed dissipation rate, so energy can enter only through forcing and noise; the resulting Foster–Lyapunov drift condition yields geometric ergodicity, and ergodicity of the perturbed chain under steady external forcing.
In the two-scale setting that observes only slow variables, the emulator's reliability of forced response does not degrade with network capacity. In the same experiment, the conditional diffusion model is unstable below 10k parameters in free rollouts and below 32k parameters in the forced regime, while the 500k-parameter diffusion model remains unstable despite having the smallest validation loss. In a capacity study spanning two orders of magnitude in parameter count, the forced-dissipative interpolant remains stable at any capacity; under oscillating forcing, the diffusion model starts to go unstable as the forcing first reaches full amplitude, whereas the emulator tracks the true mean response and stays bounded.
Perspective
The demonstrated scope is the two-scale Lorenz-96 prototype of partially observed multiscale turbulence, covering the fully observed single-layer setting and the two-layer setting that observes only slow variables, with stationary distributions and statistical responses of mean and variance to external forcing as the evaluation criteria. The authors propose next applying the method to more complex systems such as quasi-geostrophic turbulence.
The stability guarantees hold for the continuous-time generative process and do not yet cover the discrete-time sampler; the authors note that the Foster–Lyapunov criterion could be extended to discrete-time sampling. The matrix products in the parameterization become costly in high dimension, so the method is best suited to reduced-order modeling in a low-dimensional latent space. In the oscillating-forcing experiment, accuracy in variance is initially good and then decreases.