Learning Large-Step Fluid Dynamics with Navier–Stokes Structure

While the Navier–Stokes equations provide a continuous-time description of fluid motion, learning discrete-time fluid dynamics directly from observations remains highly challenging. Over large time intervals, fluid evolution becomes strongly nonlinear, making it difficult for purely data-driven neural networks to recover the underlying dynamical mechanisms. Consequently, existing methods may fit observed trajectories well with low per-step errors, yet still suffer from rapidly accumulating errors during autoregressive rollout and often generalize poorly beyond the training distribution. We introduce NeNS${}^2$, a neural surrogate for incompressible flow that learns large-step dynamics with Navier–Stokes structure. On rectangular domains with periodic or free-slip boundaries, NeNS${}^2$ represents the flow using analytical eigenfunctions of the Stokes operator, thereby satisfying incompressibility and boundary conditions by construction. In this modal basis, the viscous term is naturally diagonalized. Guided by the algebraic structure of the equations, we parameterize the remaining nonlinear dynamics with a low-rank skew-symmetric bilinear operator. This operator is learned directly from finite-time state pairs, without requiring physics-informed losses, while inherently preserving the kinetic-energy conservation property of the nonlinear term. On in-domain benchmarks, NeNS${}^2$ reduces held-out relative vorticity error by approximately $30\%$, compared with the strongest evaluated baselines. Moreover, although trained only on randomized initial conditions, NeNS${}^2$ exhibits strong out-of-distribution generalization, achieving the lowest errors on independently generated leapfrogging-vortex and Taylor-vortex dynamics.