Sobol Indices for Parameter-Conditioned Neural Networks.

Networks conditioned on a continuous parameter, such as surrogates for parametric partial differential equations, replace many trained models with one. Two questions follow: what has such a network learned about its parameter, and which simpler architectures could match it? We address both with the Sobol (functional ANOVA) decomposition of the trained network over inputs and parameter, estimated by standard Monte-Carlo sampling. On a loss-conditional physics-informed network for parametric 2D Helmholtz, the indices match those of the exact solution to 0.002 (0.005 across three training seeds) and show that 43% of the output depends jointly on space and the parameter, which first-order attribution methods cannot express. The indices of one pilot network also predict the lowest error that any restricted architecture can reach. Networks trained by regression reach the bound read from the pilot to within 0.003 on the headline target, and the exact bound to within 0.001 on two further targets. Networks trained with the physics-informed loss stay above it. Finally, a neural HDMR estimator that fits the decomposition directly learns a nearly correct function (three-way index within 0.02), but its standard readout at the default resolution overstates that index by 0.14, six times the seed spread. The reported indices still sum to one, so this error does not appear in them. Computing the indices from the fitted function removes the error; at this problem size Monte-Carlo estimation is also about 500 times cheaper.