CASL-HJX: A comprehensive guide to solving deterministic and stochastic hamilton-Jacobi equations

CASL-HJX is a high-performance C++ framework for solving deterministic and stochastic Hamilton-Jacobi equations in two spatial dimensions. It integrates operator-splitting techniques with implicit treatment of parabolic terms, yielding substantial speedups over explicit methods commonly used for stochastic problems. The solver leverages monotone schemes to ensure convergence to viscosity solutions, for which we provide numerical evidence through systematic validation. The Hamilton-Jacobi-Bellman formulation enables global optimization beyond local methods. This performance advantage opens the door to applications that were previously intractable, including real-time control and rapid design iteration. We demonstrate the framework’s capabilities on benchmark PDEs as well as a neuroscience case study designing energy-efficient controllers for neural populations. The modular architecture allows users to define custom Hamiltonians and boundary conditions, making CASL-HJX broadly applicable to optimal control, front propagation, and uncertainty quantification across finance, engineering, and machine learning. Although currently limited to two spatial dimensions, CASL-HJX addresses critical gaps where gradient-based methods struggle in non-convex landscapes and local optimization yields suboptimal results. Complete source code, documentation, and examples are freely available.