Physics-Residual Verification for Test-Time Selection in Stochastic Surrogates for PDEs

Partial differential equation (PDE) surrogates that are stochastic can produce multiple candidate solution trajectories for the same initial condition, but selecting an accurate candidate at test time is difficult because the reference solution is unavailable. We study whether governing-equation residuals can provide a reference-free test-time ranking signal. We introduce a weak-form residual verifier that avoids direct differentiation through sharp gradients and discontinuities, making it suitable for shock-forming systems, and consider a strong finite-difference residual verifier for smoother solutions. A stochastic surrogate generates multiple candidates, which are ranked using either verifier, and the lowest-scoring candidate is selected without retraining or modifying the surrogate. We evaluate this procedure using candidate pools generated by U-Net surrogates with dropout active at inference time on one-dimensional inviscid Burgers, two-dimensional scalar Burgers, and two-dimensional viscous incompressible Navier--Stokes. At a candidate budget of 32, the best residual verifier recovers 70.6\%, 94.8\%, and 68.4\% of the improvement available to a reference-informed oracle, respectively. Between the two residual formulations, weak-form verification yields lower selected-trajectory errors for the tested shock-forming conservation laws, whereas strong finite-difference verification yields lower error for the smoother viscous flow. These experiments indicate that physics residuals can provide useful test-time ranking signals even when they do not perfectly predict trajectory error and that the most effective residual formulation is problem- and regularity-dependent.