Propagating Geometry Uncertainty from a Single Implicit Neural Representation through a Differentiable PDE Solver

Implicit neural representations (INRs) such as neural signed distance functions are increasingly used as the geometry for physics simulations that solve partial differential equations (PDEs). However, in these simulations, the reconstructed geometry is usually treated as deterministic even though it is fitted to noisy data (such as point cloud). The effect of these geometric uncertainties on the solution or quantities of interest (QoIs) of a PDE is unclear. In this work, we propagate this uncertainty from the geometry (a trained INR) to PDE solutions. We freeze the INR's nonlinear features and treat only its final layer as free, which makes the boundary an explicit, low-dimensional function of a small set of deformation modes; both inference and propagation are then closed-form within this span. The noisy scan yields a Gaussian posterior over the mode coefficients, which a Shifted Boundary Method (SBM) solver propagates to PDE QoIs $300-350\times$ faster than Monte Carlo. We validate against two ground truths, exact random-field geometry and a controlled ground truth drawn from the posterior itself, to separate the accuracy of the posterior estimate from the expressive limits of the frozen representation. The result is close to ideal on the controlled test, and against exact geometry the posterior still accounts for most of the variability.