Second-Order Neural Contact Potentials

While conventional contact models are often defined over separate discrete contact cases, continuous surface-integral potentials provide a unified alternative. However, approximating these potentials with neural networks places new demands on smoothness and derivative quality. We introduce Second-Order Neural Contact Potentials (SNCP), a smooth neural representation for sampling-free contact handling in physics-based simulation. Building on Neural Collision Fields (NCF), which replace explicit contact sampling with a learned integral over pairs of mesh primitives, we address two key limitations that prevent their use in second-order solvers: nonsmooth derivatives and difficulties representing energies near zero. Our formulation replaces the Gaussian kernel of NCF with a $C^2$-continuous Wendland polynomial and introduces a neural parameterization that enforces a smooth outer cutoff and supports consistent differentiation, yielding stable and well-behaved first- and second-order derivatives. The resulting contact potential generalizes to different meshes without retraining and supports both position-based dynamics and Newton-based finite-element methods. Across challenging simulations involving rigid bodies, volumetric deformable bodies, and thin shells, our method produces accurate and stable contact responses while enabling robust second-order optimization.