A Comparison of Mesh-Free Differentiable Programming and Data-Driven Strategies for Optimal Control under PDE Constraints

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.