Solving initial-terminal value problem of time evolutions by a deep least action method: Newtonian dynamics and wave...

Area of Science:

• Computational Physics
• Machine Learning Applications
• Numerical Methods

Background:

• Solving evolution problems often requires complex differential equations.
• Existing methods may struggle with nonlinear, high-order, or high-dimensional systems.
• The principle of least action offers an alternative formulation for physical systems.

Purpose of the Study:

• Introduce a novel deep least action method (DLAM) for solving trajectory evolution problems.
• Provide an efficient, unsupervised alternative to traditional differential equation solvers.
• Demonstrate DLAM's applicability to various physical dynamics, including Newtonian and wave equations.

Main Methods:

• Formulated the problem using the principle of least action.
• Employed a normalized deep neural network to satisfy initial-terminal value conditions.
• Transformed the problem into an unconstrained optimization task.
• Applied DLAM to Newtonian and wave dynamics, including complex cases.

Main Results:

• DLAM effectively solves trajectory evolution problems.
• The method accurately handles ordinary and partial differential equations.
• Demonstrated success in nonlinear, high-order, and high-dimensional scenarios.
• Achieved efficient and accurate trajectory predictions.

Conclusions:

• DLAM provides a powerful, equation-free approach to solving physical evolution problems.
• The method is versatile, applicable to a wide range of dynamics.
• DLAM offers a promising direction for computational physics and machine learning integration.