| Contents | The eikonal equation is a fundamental model for distance and travel-time computation in complex geometries, but reliably approximating its viscosity solution on realistic and complex three-dimensional domains remains challenging. I will first present a Laplacian-regularized finite volume approach on polyhedral meshes, where the eikonal equation is interpreted through a vanishing-viscosity viewpoint and discretized by a cell- centered finite volume method with the Soner boundary condition ensuring the correct solution on non-convex domains. This algorithm numerically achieves second-order experimental convergence for smooth test cases, scales efficiently in parallel computing, and significantly reduces computational cost compared with time-relaxed formulations when the region of interest is far from the source set.
Building on this PDE-based foundation, I will then discuss three mesh-free deep learning methods that target the same viscosity solution without relying on a mesh. The first is a neural augmented Lagrangian method, which models the solution as an implicit neural representation and recasts the problem as a constrained optimization: it maximizes a geometric functional subject to a Lipschitz-type gradient constraint and a Soner-type boundary inequality, enforced robustly by an augmented Lagrangian formulation. The second is a viscosity-reduction variational approach for anisotropic eikonal equations, which derives an unconstrained variational problem from the vanishing-viscosity formulation and resolves both nonlinearity and small-viscosity instability via variable splitting and a normalized-output neural network architecture, enabling stable training on discontinuous anisotropic metrics and point-cloud geometry. The third is a stochastic displacement-based method inspired by the Derivative-Free Loss Method: instead of minimizing pointwise residuals, it learns local gradient-aligned transport using a Feynman-Kac type stochastic representation, yielding a transport-aware neural solver whose cost is essentially independent of the diffusion scale and which naturally adapts to non-convex domains with obstacles. |
