Physical-Admissibility Projection and Split-Conformal Calibration for Burgers Neural Operators

Accurate PDE prediction does not ensure physical admissibility or calibrated uncertainty. For periodic viscous Burgers dynamics, we combine an input-dependent Euclidean projection, determined by the initial condition, onto discrete mean and centered-energy constraints with split-conformal bands based on one maximum-coordinate residual per function. For fixed input, the projection is nonexpansive and cannot increase Euclidean error to an admissible target. Under exchangeability, the bands provide finite-sample marginal coverage simultaneously over the 64-point terminal grid. The nonzero-mean Cole–Hopf reference agrees with an independent dealiased Fourier–RK4 solver, with maximum relative L2 and scaled L∞ discrepancies of 3.62×10−8 and 5.83×10−8. Projected FNO gives the lowest ID mean relative L2 error, 0.0190, among the four implementations at N=64 and T=0.5 over five initializations. Full projection eliminates measured constraint violations and lowers mean error across models and regimes from 0.1984 to 0.1904. At nominal 90%, constant-width ID simultaneous coverage ranges from 0.904 to 0.930; normalized FNO bands reduce mean width from 0.108 to 0.085 with coverage 0.901. Under distribution shift, undercoverage and occasional conservative width inflation restrict conclusions to empirical diagnostics. The framework separates admissibility with respect to the two selected discrete constraints, predictive accuracy, and statistical coverage without a new architecture.