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

2025-05-12

AI Summary

This work addresses the efficient numerical solution of two-dimensional deterministic and stochastic Hamilton–Jacobi (HJ) equations. Methodologically, it introduces a unified computational framework—applicable to forward HJ evolution, optimal control, and stochastic Hamilton–Jacobi–Bellman (HJB) equations—built upon a novel integration of implicit second-order derivative discretization, operator splitting, hyperbolic PDE techniques, implicit time integration, and mixed-order derivative solvers, all implemented within a high-performance C++ architecture to ensure viscosity solution convergence and global optimality. It constitutes the first systematic unification of deterministic optimal control and stochastic modeling under a single HJ-based paradigm. Experimental validation across diverse domains—including neuromodulation (designing energy-efficient controllers to suppress pathological neural synchrony), finance, engineering, and machine learning—demonstrates high accuracy, strong robustness to problem conditioning, and broad cross-disciplinary applicability.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Solves deterministic and stochastic Hamilton-Jacobi equations
• Models front propagation and optimal control problems
• Handles mixed-order derivatives with time-varying dynamics

Innovation

• Integrates hyperbolic PDEs with operator splitting
• Implements implicit methods for second-order derivatives
• High-performance C++ core for mixed-order systems