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

Abstract

We introduce a deep least action method (DLAM) rooted in the principle of least action to solve the trajectory of an evolution problem. DLAM offers an efficient unsupervised solution and can be applied once the action or Lagrangian of the concerned physical system is clear, totally avoiding the differential equations. As required by the least action principle, we incorporate a normalized deep neural network to exactly satisfy the initial-terminal value conditions; thus the evolution problem is transformed into an unconstrained optimization problem. We conduct systematic investigations, initially focusing on Newtonian dynamics modeled by ordinary differential equations. Subsequently, we move on to the wave dynamics modeled by partial differential equations, covering nonlinear, high-order, and high-dimensional cases in detail. Our results showcase the effectiveness of DLAM and illustrate its efficiency and accuracy.