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

TL;DR

This paper compares mesh-free differentiable programming, physics-informed neural networks, and traditional numerical schemes for optimal control under PDE constraints, highlighting the strengths and limitations of each method.

Contribution

It introduces a comprehensive comparison using a mesh-free PDE solver and demonstrates the effectiveness of differentiable programming in challenging scenarios.

Findings

DP produces the most accurate gradients, even when DAL fails.

PINNs struggle under certain conditions, limiting their applicability.

The paper provides a detailed benchmark for method selection in optimal control.

Abstract

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.