Staying on the Attractor: Supervising Neural Surrogates of 3D Turbulence Where They Leave It

Neural surrogates are trained to predict 3D turbulent flows in place of direct numerical simulation (DNS). For chaotic flows, the goal is short-term pointwise accuracy followed by long-term physical and statistical fidelity. However, small prediction errors can carry a surrogate away from the flow's attractor. Off-attractor states are poorly represented in training data, leaving their evolution weakly constrained. The learned dynamics can then amplify deviations and lead to blow-up, freezing, or statistical drift. We propose off-attractor supervision (OAS) to supervise neural surrogates where they leave the attractor. OAS teaches the model how the true Navier-Stokes dynamics would evolve from these states. Each selected state is paired with its own future computed by DNS. Three generators select a few hundred states for relabeling. The first collects states from the surrogate's own rollouts. The second uses surrogate attacks to target freezing, excessive amplification, and violations of incompressibility and energy balance. The third perturbs training states along an amplified direction and a strongly damped random direction of the dynamics. All attacks run on the surrogate alone, and DNS relabeling is performed offline once per selected state. Experiments on 1283 turbulence show that OAS increases the median time to failure from 21 to 721 steps. The compared baselines achieve medians of at most 110 steps, and the advantage holds across training seeds. OAS also achieves the lowest pointwise error at step 15 and the best long-horizon statistics among the compared methods. OAS integrates physical models into neural simulation by extending supervision from fixed reference trajectories to states where the surrogate is likely to fail. This principle can guide the development of more reliable scientific surrogates when deployment takes models beyond the coverage of their training data.