A Comparison of Mesh-Free Differentiable Programming and Data-Driven
  Strategies for Optimal Control under PDE Constraints
Roussel Desmond Nzoyem David A. W. Barton Tom Deakin
The field of Optimal Control under Partial Differential Equations (PDE)
constraints is rapidly changing under the influence of Deep Learning and the
accompanying automatic differentiation libraries. Novel techniques like
Physics-Informed Neural Networks (PINNs) and Differentiable Programming (DP)
are to be contrasted with established numerical schemes like Direct-Adjoint
Looping (DAL). We present a comprehensive comparison of DAL, PINN, and DP using
a general-purpose mesh-free differentiable PDE solver based on Radial Basis
Functions. Under Laplace and Navier-Stokes equations, we found DP to be
extremely effective as it produces the most accurate gradients; thriving even
when DAL fails and PINNs struggle. Additionally, we provide a detailed
benchmark highlighting the limited conditions under which any of those methods
can be efficiently used. Our work provides a guide to Optimal Control
practitioners and connects them further to the Deep Learning community.