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This paper investigates the limitations of physics-informed neural operators when applied to coarse-grained, stochastically forced turbulence data, demonstrating that only linear constraints remain strictly valid under such filtering. The authors propose and implement an exact, closed-form projection method for Fourier Neural Operators (FNOs) to enforce continuity and global momentum balance exactly, achieving orders-of-magnitude reduction in errors compared to standard physics-informed or plain FNOs. This projection significantly improves the stability of rollouts and can be extended to nonlinear constraints by modeling unclosed terms, enabling bounded energy evolution over extended time scales.
Enforcing only linear physics constraints exactly via projection dramatically improves the stability and accuracy of neural operators on coarse-grained turbulence data, outperforming complex nonlinear closure models.
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 $5.0\times10^{-7}$ against 0.51 for a plain FNO and 0.17 for a physics-informed FNO, and a global momentum balance error of $8.6\times10^{-11}$ 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. 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.