Machine learning of independent conservation laws through neural deflation

Area of Science:

• Dynamical Systems
• Computational Physics
• Machine Learning

Background:

• Hamiltonian dynamical systems are fundamental in physics.
• Identifying conservation laws is crucial for understanding system integrability.
• Existing methods for finding conservation laws can be computationally intensive.

Purpose of the Study:

• To introduce a novel data-driven methodology for discovering conservation laws.
• To assess the integrability of dynamical systems using computational tools.
• To apply the method to lattice differential-difference equations.

Main Methods:

• Development of "neural deflation," an iterative neural network training approach.
• Minimization of a regularized loss function enforcing involution and functional independence of conserved quantities.
• Application to integrable and nonintegrable lattice differential-difference equations.

Main Results:

• Neural deflation successfully predicts conservation laws in various systems.
• For integrable systems, the number of conservation laws scales with degrees of freedom.
• For nonintegrable systems, conservation laws are found to saturate at a threshold.

Conclusions:

• Neural deflation is an effective tool for identifying conservation laws.
• The method provides insights into the integrability of dynamical systems.
• This data-driven approach aids in model assessment and theoretical analysis.