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

arXiv: https://arxiv.org/abs/2310.02286 · HTML: https://mlsanity.com/paper/2310.02286/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 55 references that are in the map.
- First posted on arXiv 2 Oct 2023.

## Builds on (most cited first, 11 of 11)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Learning to Control PDEs with Differentiable Physics](https://mlsanity.com/paper/2001.07457/) (2020)
- [Optimal control of PDEs using physics-informed neural networks](https://mlsanity.com/paper/2111.09880/) (2021)
- [Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data](https://mlsanity.com/paper/1808.04327/) (2018)
- [JAX-FLUIDS: A fully-differentiable high-order computational fluid dynamics solver for compressible two-phase flows](https://mlsanity.com/paper/2203.13760/) (2022)
- [JAX, M.D.: A Framework for Differentiable Physics](https://mlsanity.com/paper/1912.04232/) (2019)
- [Modeling, state estimation, and optimal control for the US COVID-19 outbreak](https://mlsanity.com/paper/2004.06291/) (2020)

## Built on by (most cited first, 6 of 6)

- [Neural Context Flows for Meta-Learning of Dynamical Systems](https://mlsanity.com/paper/2405.02154/) (2024)
- [Towards Foundational Models for Dynamical System Reconstruction: Hierarchical Meta-Learning via Mixture of Experts](https://mlsanity.com/paper/2502.05335/) (2025)
- [PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problems](https://mlsanity.com/paper/2409.01899/) (2024)
- [KANtrol: A Physics-Informed Kolmogorov-Arnold Network Framework for Solving Multi-Dimensional and Fractional Optimal Control Problems](https://mlsanity.com/paper/2409.06649/) (2024)
- [Reevaluating Meta-Learning Optimization Algorithms Through Contextual Self-Modulation](https://mlsanity.com/paper/2410.01655/) (2024)
- [Adjoint Method versus Physics-Informed Neural Networks in PDE-Constrained Inverse Problems](https://mlsanity.com/paper/2606.12337/) (2026)

## Abstract (arXiv)

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.