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 equation(s) being enforced are closed on the fields being fitted. An example of where this is not valid is on coarse-grained, stochastically forced data: any nonlinear terms in the PDE do not commute with the filter, so the coarse-grained fields do not satisfy the original equations, and attempting to enforce these equations as constraints by driving their residual to zero encodes an implicit closure into the learned operator. Any linear terms remain valid on the filtered grid and survive coarse-graining exactly. For the 2D incompressible Navier--Stokes equations, two linear constraints that remain valid 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 $\max|\nabla\cdot \boldsymbol{u}|=5.0\times10^{-7}$ against $0.51$ for a plain Fourier neural operator and $0.17$ for a physics-informed FNO, and a global momentum balance error of $|\langle \boldsymbol{u} \rangle_\Omega|=8.6\times10^{-11}$ against $7.5\times10^{-4}$ and $8.3\times10^{-4}$. 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 $0.87$. By comparison, test-time optimisation of the same residual via a physics-informed loss costs $3{,}300\times$ more per trajectory and only manages a maximum divergence three orders of magnitude above that of the projection. Nonlinear constraints can also be enforced once their unclosed terms are modelled: closing the (quadratic) 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.