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

2026-04-06

AI Summary

This study addresses optimal control problems governed by semilinear partial differential equations (PDEs), presenting the first systematic comparison between direct and indirect physics-informed neural network (PINN) approaches. The direct method minimizes the objective functional while enforcing the state equation as a hard constraint, whereas the indirect method leverages Pontryagin-type first-order optimality conditions to jointly solve the state and adjoint equations. Using the Allen–Cahn equation as a benchmark, numerical experiments demonstrate that the indirect PINN formulation yields more accurate approximations of the true solution, produces smoother controls, and adheres more rigorously to both the PDE constraints and the underlying optimality structure. The work further reveals an implicit regularization effect inherent in PINN parameterization, underscoring the advantages of indirect methods in preserving physical consistency and optimality conditions.

Abstract

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.

Problem

• optimal control
• PDE-constrained optimization
• physics-informed neural networks
• semilinear parabolic equations
• Pontryagin optimality conditions

Innovation

• physics-informed neural networks
• PDE-constrained optimal control
• indirect method
• direct method
• adjoint equation