ViscoReg: Neural Signed Distance Functions via Viscosity Solutions

Abstract: Implicit Neural Representations (INRs) that learn a Signed Distance Function (SDF) are a powerful tool for continuous 3D scene reconstruction. These models are trained by enforcing the Eikonal equation. We demonstrate theoretically that despite the ill-posedness of the Eikonal equation, generalization error estimates may be obtained for Neural SDFs in terms of the training error. However, training with the Eikonal loss can lead to unstable gradient flows, necessitating alternate stabilization techniques. Traditional numerical solvers for the equation have relied on viscosity approaches for regularization. We enhance Neural SDF training using this well-developed theory, and introduce a new loss formulation we call ViscoReg. We theoretically demonstrate the stability of the gradient flow equation of our proposed loss term. Empirically, ViscoReg outperforms state-of-the-art approaches such as SIREN, DiGS, and StEik without adding significant computational cost.