NIGO: Lyapunov-Stable Continuous-Time Neural Operators for Long-Horizon PDE Dynamics

Neural operators have become a promising framework for learning solution operators of partial differential equations, yet their long-horizon use remains limited by the stability of autoregressive rollout. In many spatiotemporal settings, learned time-advancement maps must be composed repeatedly, causing small local errors to accumulate into trajectory drift, spectral artifacts, or non-physical energy growth. We propose NIGO, a continuous-time neural operator that replaces unconstrained discrete rollout maps with a dissipative infinitesimal generator in latent space. Given an initial PDE state, NIGO encodes the field into a compact latent representation and evolves it analytically through a matrix exponential. The generator is parameterized as a skew-symmetric transport component plus a negative-semidefinite dissipative component, which enforces non-increasing latent energy by construction. A physics-inference network predicts the generator coefficients from the initial state and conditioning variables, while temporal refinement and Lipschitz-controlled decoding map stable latent trajectories back to physical fields. We evaluate NIGO on Burgers, cylinder flow, Kuramoto--Sivashinsky, shallow-water, and incompressible Navier--Stokes systems, covering decaying, periodic, and chaotic regimes. Across extended rollouts, NIGO avoids the blowup and freezing modes observed in standard neural-operator and physics-informed baselines. Ablations show that the dissipative generator is necessary for stable long-horizon evolution, and mechanistic analyses reveal learned structure consistent with viscous scaling, operator locality, and advection--dissipation balance. These results indicate that hard-constrained continuous-time latent dynamics provide a principled foundation for stable neural operators for PDE forecasting.