Kernel and neural-network solvers for HJB equations on unbounded domains: convergence and error bounds from empirical residuals
For two non-monotone, mesh-free approximations of Hamilton–Jacobi–Bellman equations on unbounded spatial domains—Wendland kernel collocation under a native space norm constraint and physics-informed neural networks with smooth activations under a Sobolev-norm constraint—the work derives error bounds from computable empirical squared residuals: norm constraints, an interpolation inequality and a controlled-diffusion truncation estimate convert the empirical loss into the pointwise residual control needed for viscosity-solution error estimates, yielding uniform convergence on compact sets plus a posteriori and high-probability quantitative bounds governed by the fill distance for kernels and by network and sample sizes for neural networks.