Solving physics-based initial value problems with unsupervised machine learning

Area of Science:

• Computational Physics
• Machine Learning Applications
• Classical Mechanics

Background:

• Initial value problems, systems of ordinary differential equations with initial conditions, are fundamental to describing physical phenomena.
• Traditional methods for solving these problems can be computationally intensive, especially for nonlinear, coupled, or chaotic systems.

Purpose of the Study:

• To develop and demonstrate an unsupervised machine learning framework for solving physics-based initial value problems.
• To model the dynamics of various mechanical systems using deep neural networks.
• To assess the framework's ability to handle nonlinear, coupled, and chaotic dynamical systems.

Main Methods:

• Implementation of a deep learning framework utilizing neural networks to model system dynamics.
• Application of probabilistic activation functions for learning solutions in the strictest sense of initial value problems.
• Development of coupled neural networks to address coupled dynamical systems.

Main Results:

• Demonstrated the framework's effectiveness on diverse systems: free particle, particle in a gravitational field, pendulum, and Hénon-Heiles system.
• Deep neural networks successfully approximated solutions, generating trajectories that conserved physical properties like energy and stationary action.
• Probabilistic activation functions and coupled neural networks were shown to be crucial for accurate solutions of specific problem types.

Conclusions:

• Unsupervised deep learning provides a powerful and flexible approach to solving complex initial value problems in physics.
• The proposed framework accurately models mechanical system dynamics and preserves essential physical invariants.
• This method offers a promising alternative for simulating and understanding intricate physical phenomena.