Neural augmented Lagrangian method for approximating viscosity solution of the eikonal equation

Finding the viscosity solution of the eikonal equation with physics-informed neural networks (PINNs) is challenging, as methods that rely solely on the classical eikonal loss can converge to non-viscosity solutions rather than the viscosity solution. We propose a novel mesh-free PINN framework specifically designed to overcome this limitation. Our approach, which uses an implicit neural representation (INR) to model the solution, recasts the problem as a constrained optimization task rooted in the fundamental properties of the eikonal equation. To ensure convergence to the viscosity solution, our framework augments the standard PINN loss with a term derived from the equation’s variational form: maximizing the solution’s volume subject to a 1-Lipschitz gradient constraint. To robustly handle complex geometries with non-convex boundaries or holes, we further incorporate the Soner boundary constraint as an additional constraint. This constrained optimization problem is then transformed into an unconstrained objective using an augmented Lagrangian method (ALM), which provides rigorous and stable enforcement of all constraints during training. We demonstrate through experiments on a variety of challenging problems that our method successfully avoids spurious local minima and consistently converges to the viscosity solution, achieving relative L 2 errors on the order of 10 − 2 or less. Another advantage of the trained model is its ability to instantly recover the shortest time path from any point to the source, highlighting its practical value for applications in navigation and path planning.