# PINNs in PDE Constrained Optimal Control Problems: Direct vs Indirect Methods

arXiv: https://arxiv.org/abs/2604.04920 · HTML: https://mlsanity.com/paper/2604.04920/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 17 references that are in the map.
- First posted on arXiv 6 Apr 2026.

## Builds on (most cited first, 6 of 6)

- [Optimal control of PDEs using physics-informed neural networks](https://mlsanity.com/paper/2111.09880/) (2021)
- [Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2501.16371/) (2025)
- [SympOCnet: Solving optimal control problems with applications to high-dimensional multi-agent path planning problems](https://mlsanity.com/paper/2201.05475/) (2022)
- [Sequential-in-time training of nonlinear parametrizations for solving time-dependent partial differential equations](https://mlsanity.com/paper/2404.01145/) (2024)
- [A PINN approach for the online identification and control of unknown PDEs](https://mlsanity.com/paper/2408.03456/) (2024)
- [State-dependent Riccati equation feedback stabilization for nonlinear PDEs](https://mlsanity.com/paper/2106.07163/) (2021)

## Built on by (most cited first, 3 of 3)

- [Universal Approximation of Nonlinear Operators and Their Derivatives](https://mlsanity.com/paper/2605.15285/) (2026)
- [Adjoint Method versus Physics-Informed Neural Networks in PDE-Constrained Inverse Problems](https://mlsanity.com/paper/2606.12337/) (2026)
- [Dual-Network PINNs for Optimal Control: A Reproducible Benchmark on the Mass-Spring-Damper System](https://mlsanity.com/paper/2606.15271/) (2026)

## Abstract (arXiv)

We study physics-informed neural networks (PINNs) as numerical tools for the optimal control of semilinear partial differential equations. We first recall the classical direct and indirect viewpoints for optimal control of PDEs, and then present two PINN formulations: a direct formulation based on minimizing the objective under the state constraint, and an indirect formulation based on the first-order optimality system. For a class of semilinear parabolic equations, we derive the state equation, the adjoint equation, and the stationarity condition in a form consistent with continuous-time Pontryagin-type optimality conditions. We then specialize the framework to an Allen-Cahn control problem and compare three numerical approaches: (i) a discretize-then-optimize adjoint method, (ii) a direct PINN, and (iii) an indirect PINN. Numerical results show that the PINN parameterization has an implicit regularizing effect, in the sense that it tends to produce smoother control profiles. They also indicate that the indirect PINN more faithfully preserves the PDE contraint and optimality structure and yields a more accurate neural approximation than the direct PINN.