Conservation-Constrained Fourier Neural Operators for Physically Consistent PDE Simulation

Neural operators have enabled rapid surrogate modeling of partial differential equations but systematically violate conservation laws, limiting their physical fidelity and long-horizon stability. We introduce CC-FNO, a modified Fourier Neural Operator that enforces mass conservation via a differentiable Leray projection applied to the output velocity field, projecting it onto the divergence-free subspace in Fourier space at negligible computational cost ($<2\%$ overhead). On 2D incompressible Navier--Stokes at $Re \in \{100, 500, 1000, 2000\}$, CC-FNO reduces divergence by five orders of magnitude ($\sim 10^{-5}$ vs.\ $\sim 3.0$ for FNO, $\sim 10^{-2}$ for PINO) while matching baseline accuracy across seven model comparisons including FNO, PINO, Stream-FNO, U-Net, and FNO with pushforward training. We additionally propose CC-FNO-E, which stabilizes autoregressive rollouts through a per-step energy bound that acts as a stability regularizer, preventing catastrophic amplitude amplification. This extends stable rollout horizons by $17$--$52\%$ over FNO across all regimes, though as a stability regularizer rather than a physical conservation law, it systematically underestimates energy growth by $67$--$84\%$ at the final rollout step in forced flow. A systematic penalty weight sweep confirms that soft constraints cannot match hard projection regardless of tuning, and an ablation of forcing-aware energy bounds reveals that simple one-sided capping outperforms physically motivated alternatives. Our results establish a clear constraint hierarchy (none $<$ soft $<$ hard $<$ hard+energy stabilizer) and demonstrate that hard conservation constraints provide ``free'' physical consistency without sacrificing accuracy.