Only Linear Constraints Survive Coarse-Graining: Evaluating Physics-Constrained Neural Operators on Stochastically Forced Turbulence

A physics-informed loss adds a discretised PDE residual to the data term, but is only strictly valid if the equations being enforced are closed on the fields being fitted. This fails on coarse-grained, stochastically forced data: nonlinear terms in the PDE generally do not commute with the filter, so the coarse-grained fields do not satisfy the original equations, and driving their residual to zero encodes an implicit closure into the learned operator. Under normalised, translation-invariant filtering on a periodic domain, constant-coefficient linear constraints remain valid on the filtered grid. For the 2D incompressible Navier-Stokes equations, two such constraints are continuity and global momentum balance. We enforce these exactly by applying a closed-form projection on the output of a Fourier neural operator (FNO) at $O(N\log N)$ cost. On two-dimensional isotropic turbulence, truncated so that its forcing lies entirely beyond the cutoff wavenumber, the projection achieves a continuity error of $5.0\times {10}^{-7}$ against 0.51 for a plain FNO and 0.17 for a physics-informed FNO, and a global momentum balance error of $8.6\times {10}^{-11}$ against $7.5\times {10}^{-4}$ and $8.3\times {10}^{-4}$. The projection only costs an extra 2% of training time, and takes the fraction of rollouts whose energy remains bounded at 448 steps from 0.18 to 0.87. On resolved Kolmogorov flow, test-time optimisation costs approximately 3,300$\times$ as much per trajectory as the projection, while leaving maximum divergence three orders of magnitude higher. Nonlinear constraints can also be enforced once their unclosed terms are modelled: closing the global energy balance with a constant subgrid flux, fitted from the coarse training data alone, removes the 20.9% energy deficit that enforcing the unclosed balance produces and keeps every free-running rollout bounded over 2,000 steps.