A stochastic vanishing viscosity approach for eikonal equations

We propose a stochastic vanishing viscosity approach to solve the eikonal equations. This framework yields a stochastic gradient-aligned displacement formulation that captures the directional transport structure of the viscosity solution. Building on the Derivative-Free Loss Method (DFLM), our approach trains a neural network to capture directional transport along gradient trajectories, aligning closely with the underlying physical propagation mechanism rather than enforcing pointwise partial differential equation (PDE) residuals. The proposed method demonstrates robustness across a variety of source configurations, anisotropic or spatially varying speeds. The computational cost remains independent of the viscosity scale, achieved by adjusting the time step of stochastic walkers while maintaining consistent displacement. Furthermore, the method handles non-convex domains with internal obstacles by adapting the stochastic process to the local geometry, enabling the network to learn appropriate boundary behavior without explicitly imposing constraints such as the Soner constraint.