ShapeUQ: Propagating 3D Reconstruction Uncertainty Through Scientific PDE Simulations via Shape Calculus

Kaustubh S. Bukkapatnam, Soham Batra, Siddharth Karuturi, Tanush Ajay Shastry, Akshath Sharma, Aarav Lala, Andrew Bae, Laksh Patel

Keywords: uncertainty quantification; neural implicit SDFs; scientific PDE simulations

TL;DR: ShapeUQ uses shape calculus and adjoint-based sensitivity of neural SDF reconstructions to efficiently propagate geometric uncertainty into rigorous confidence intervals for scientific PDE simulations without expensive Monte Carlo reruns.

Abstract: Neural implicit representations have transformed 3D reconstruction of scientific geometries such as glaciers, blood vessels, and protein surfaces, by providing continuous, differentiable signed distance functions (SDFs) from multi-view observations. Yet a critical gap persists: geometric reconstruction error propagates silently into downstream physical simulations, inflating PDE solution uncertainty with no principled quantification. We introduce ShapeUQ, the first framework, to our knowledge, to formally propagate SDF reconstruction uncertainty into rigorous confidence intervals on PDE solutions, without requiring full simulation reruns per geometric sample. Our theoretical contribution rests on three results from shape calculus. Theorem 1 establishes a computable linear sensitivity operator mapping SDF perturbations to PDE solution changes via the Hadamard shape derivative, with a closed-form expression derivable from automatic differentiation through the SDF network. Theorem 2 shows that, for Gaussian-process reconstruction uncertainty with covariance K, the induced distribution over PDE solutions has an expected squared error bounded by a trace-norm of K scaled by the solution’s normal derivative on the boundary. Theorem 3 proves that the adjoint method computes the full sensitivity field in one additional PDE solve, the same cost as one forward simulation, regardless of the dimensionality of the geometric uncertainty. We instantiate ShapeUQ on three scientific benchmarks: surface heat diffusion on Jakobshavn Glacier (glaciology), Poisson–Boltzmann electrostatics on a protein surface (biophysics), and viscous flow over a coral reef geometry (fluid dynamics). ShapeUQ produces confidence intervals that cover the true error at the 90% level across all settings, while reducing runtime versus Monte Carlo sampling by 14×–31×.

Submission Number: 23