NestyNet. II. Coherent Function-Space Posteriors from Scientific Neural Surrogates (or How to Avoid Expensive MCMC)

Abstract

Scientific analyses increasingly use flexible neural networks, but their thousands of correlated parameters make it challenging to interpret the associated uncertainties. Here we develop a low-dimensional posterior for the fitted function itself, for scientific neural surrogates trained with second-order optimization. Linearizing the fitting procedure with respect to the randomized residual rows gives a measurement-to-function transport, the linear map, assembled from the converged Jacobians and Gauss--Newton curvature, that carries measurement perturbations into the function perturbations that refitting would produce. Its leading singular functions define coherent deformation modes. Independent Gaussian coefficients then generate smooth function draws, so that any derived quantity, including those requiring derivatives or integrals of the draw, inherits the posterior. The construction distinguishes repeated-experiment covariance from the local Gauss--Newton/Laplace posterior and propagates both to correlated quantities of scientific interest. The result is conditional on the fit's declared choices (architecture, hyperparameters, active set, and optimization branch), and every fit is certified as converged by checking that a further optimization step would change the fitted predictions by less than a chosen small fraction of the measurement errors. Our primary example is an 800-parameter phase-space distribution function fit for a mock stellar disk. Four uncertainty coordinates, two orders of magnitude fewer than the fitted parameters and stable under refinement of the force basis, capture 99\% of the vertical-force posterior variance, and 4000 coherent draws propagate through the force, total-density, surface-density, and frequency calculations in 0.8s. The method provides a highly efficient route to uncertainty propagation for derivative-dependent scientific inference.