Verification, seed variability and calibrated error allowance for physics-informed surrogates of nonlinear transient conjugate heat transfer

Physics-informed neural networks are often proposed as fast surrogates for heat transfer problems that are too expensive to solve inside a design or control loop.Two questions about them are rarely answered. How much of the reported accuracy survives a change of random seed, and how large an error allowance does adownstream decision rule need in order to stay safe? We study both on a nonlinear transient conjugate heat transfer problem for a convectively cooled wall withtemperature-dependent conductivity and specific heat. A vertex-centred finite volume reference solver is verified in two ways. It reproduces a closed-form nonlinearsteady solution, obtained by Kirchhoff transform, to 7.3×10−10 K, and the method of manufactured solutions recovers the design orders of accuracy of the scheme (observed 2.00 in space and 0.95 in time). We then train a parametric surrogate over peak heat flux and coolant-side film coefficient in four configurations. The configurations differ only in the loss composition and in how many reference solutions they are given, and each is repeated for three random seeds. Adding the physicsresidual to four reference solutions cuts the mean peak-stress error by a factor of 3.5. It also cuts the seed-to-seed standard deviation of the field error from 12.42 Kto 0.27 K, a factor of 46. A residual-only surrogate trained without any reference solutions still beats a supervised surrogate that was given four. With twenty-fourreference solutions the supervised model is the more accurate of the two inside the envelope, and we report that result as well. Finally we use the surrogate as aninequality constraint in a coolant scheduler. With no error allowance the scheduler violates the true structural constraint on 14.4% of manoeuvres, averaged overfour independently trained surrogates, even though the mean margin residuals are of order 0.03 and conservative in sign. A split-conformal allowance calibrated ona marginal sample brings the measured violation rate to 5.8% against a nominal 5%. An allowance calibrated only on points near the constraint boundary does notwork, and gives 13.2%. The calibrated allowance varies by more than an order of magnitude between surrogates of identical architecture, so it has to be measured forthe model that will be deployed.