CASL-HJX: A Comprehensive Guide to Solving Deterministic and Stochastic Hamilton-Jacobi Equations

TL;DR

CASL-HJX is a versatile computational framework that efficiently solves deterministic and stochastic Hamilton-Jacobi equations in two dimensions, enabling advanced modeling in various scientific and engineering fields.

Contribution

It introduces a flexible, high-performance C++ framework integrating numerical methods for hyperbolic PDEs and operator splitting, capable of handling mixed derivatives and stochastic elements.

Findings

Successfully solves a range of PDEs with high accuracy

Demonstrates applications in neuroscience for neural control

Bridges deterministic and stochastic Hamilton-Jacobi methods

Abstract

CASL-HJX is a computational framework designed for solving deterministic and stochastic Hamilton-Jacobi equations in two spatial dimensions. It provides a flexible and efficient approach to modeling front propagation problems, optimal control problems, and stochastic Hamilton-Jacobi Bellman equations. The framework integrates numerical methods for hyperbolic PDEs with operator splitting techniques and implements implicit methods for second-order derivative terms, ensuring convergence to viscosity solutions while achieving global rather than local optimization. Built with a high-performance C++ core, CASL-HJX efficiently handles mixed-order derivative systems with time-varying dynamics, making it suitable for real-world applications across multiple domains. We demonstrate the solver's versatility through tutorial examples covering various PDEs and through applications in neuroscience, where it enables the design of energy-efficient controllers for regulating neural populations to mitigate pathological synchrony. While our examples focus on these applications, the mathematical foundation of the solver makes it applicable to problems in finance, engineering, and machine learning. The modular architecture allows researchers to define computational domains, configure problems, and execute simulations with high numerical accuracy. CASL-HJX bridges the gap between deterministic control methods and stochastic models, providing a robust tool for managing uncertainty in complex dynamical systems.