The Langevin Dynamics Solver

Stochastic differential equations

General Langevin dynamics can be described by the following stochastic differential equation (SDE):

\[d c_{i} = -\lambda\partial_{i} f(\mathbf{c}) + \sigma dW_{i}\;\;\text{with}\;\;c_{i}(0)=0\;\;\forall i=1,\ldots,N\]

where \(\lambda\) and \(\sigma\) are hyperparameters controlling the strengths of the drift and diffusion terms, respectively. We have implemented this solver as a classical solver implemented solely on classical computers, that is, there is no optical simulation of Langevin dynamics in our ccvm_simulators package.

Example script for solving a box-constrained quadratic programming problem using the Langevin solver

The demo script, langevin_boxqp.py, in the examples folder simulates the Langevin dynamics for an example problem instance. It can be run from the project’s root directory using the following command in the terminal:

$ python ccvm/examples/langevin_boxqp.py

The script will print the solution, similar to the example output below.

Solution(
   problem_size=20,
   batch_size=1000,
   iterations=15000,
   ...,
   solve_time=5.026,
   optimal_value=152.602,
   best_value=147.960,
   solution_performance={
      'optimal': 0.0,
      'one_percent': 0.0,
      'two_percent': 0.0,
      'three_percent': 0.0,
      'four_percent': 0.999,
      'five_percent': 0.999,
      'ten_percent': 0.999
   }
)