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The field of Optimal Control under Partial Differential Equations (PDE)constraints is rapidly changing under the influence of Deep Learning and theaccompanying automatic differentiation libraries. Novel techniques likePhysics-Informed Neural Networks (PINNs) and Differentiable Programming (DP)are to be contrasted with established numerical schemes like Direct-AdjointLooping (DAL). We present a comprehensive comparison of DAL, PINN, and DP usinga general-purpose mesh-free differentiable PDE solver based on Radial BasisFunctions. Under Laplace and Navier-Stokes equations, we found DP to beextremely effective as it produces the most accurate gradients; thriving evenwhen DAL fails and PINNs struggle. Additionally, we provide a detailedbenchmark highlighting the limited conditions under which any of those methodscan be efficiently used. Our work provides a guide to Optimal Controlpractitioners and connects them further to the Deep Learning community.

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