Sobolev Training Out of Distribution: Closed-Loop Evaluation of a Kolmogorov PDE Surrogate

A neural surrogate for a parametric PDE can match the solution to within a few percent while its second derivative is wrong by forty percent, and neither the training loss nor a validation score on the trainer's own grid reports it. Downstream, a controller consumes the surrogate's automatic-differentiation first derivative, and the error is paid at the states it visits. Does derivative supervision keep those fields accurate outside the training regime, and does that accuracy reach the decision? The instance: a learned Heston Kolmogorov solution feeding delta hedging, endpoints pre-registered and every evaluated coefficient regime excised from training and validation. At held-out coefficients, supervision cuts second-derivative relative RMSE from $0.410$ to $0.045$, and the first-derivative field the controller uses improves alongside. The registered downstream endpoint still fails: every arm carrying second-derivative labels already sits within $1.3\%$ of the exact-Heston-delta benchmark, leaving little remaining improvement from reproducing it more accurately; the registered Bates sweep finds no decisive regime and a separate Merton evaluation separates no arms, every arm sharing one structural misspecification. In one exploratory cell, error under the decision-time state-visitation distribution shows a higher rank association with realised loss than grid error. Evaluate accuracy where it is consumed.