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

Zhen Zhang, Shanqing Liu, Alessandro Alla, Jerome Darbon, George Em Karniadakis

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.

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

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.

Paper Structure

This paper contains 28 sections, 2 theorems, 30 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. Preliminary
  3. PDE-constrained optimal control
  4. Direct and indirect numerical viewpoints
  5. Physics-informed neural networks
  6. PINN FORMULATIONS FOR OPTIMAL CONTROL OF SEMILINEAR PDES
  7. Problem formulation
  8. First-order optimality system and maximum principle
  9. Direct PINN formulation
  10. Indirect PINN formulation
  11. Numerical results
  12. Problem formulation
  13. Method 1: Adjoint method (discretize-then-optimize)
  14. Method 2: PINN -- direct formulation
  15. Neural network architecture
  16. ...and 13 more sections

Key Result

Proposition B.1

Under Assumption ass:nonlinearity, for every $u\in\mathcal{U}$, the state equation eq:abstract_state admits a unique weak solution $y=S(u)\in\mathcal{Y}$. Moreover, the control-to-state map $S:\mathcal{U}\to\mathcal{Y}$ is well defined and locally Fréchet differentiable. $\blacktriangleleft$$\blackt

Figures (3)

  • Figure D1: Loss histories for the three approaches: adjoint optimization starting from scratch (top), direct PINN (middle), and indirect PINN (bottom). In the adjoint panel, the legend shows the total objective $\mathcal{J}$, the terminal tracking term $\mathcal{J}_T$, the control regularization term $\mathcal{J}_u$, and the relative $L_2$ error of the control, $\mathrm{rel}\,L^2(u)$. In the direct PINN panel, the legend additionally separates the PDE, boundary-condition (BC), and initial-condition (IC) residual losses, together with $\mathcal{J}_T$, $\mathcal{J}_u$, and $\mathrm{rel}\,L^2(u)$. In the indirect PINN panel, the legend further includes the residual losses associated with the adjoint equation ($\mathrm{PDE}\ \lambda$), adjoint boundary condition ($\mathrm{BC}\ \lambda$), and terminal optimality condition ($\mathcal{T}(\lambda)$), in addition to the state PDE/BC/IC residuals, the total loss, and $\mathrm{rel}\,L^2(u)$. For the PINN cases, only the SSBroyden phase is shown after the short Adam warm-up, and the total wall-clock time of each method is indicated in the panel title.
  • Figure D2: Control profiles, shown from top to bottom: adjoint starting from scratch, direct PINN, indirect PINN, and adjoint initialized from the direct PINN solution, which is treated as the reference value for this optimal control problem. The relative $L_2$ errors of control are shown in titles. Note that according to the optimality condition, the terminal solution is $y(T)=-\beta_{Q}u(T), \beta_{Q}=1\times10^{-3}$.
  • Figure D3: State trajectories corresponding to the converged controls, shown from top to bottom: adjoint starting from scratch, direct PINN, indirect PINN, and adjoint initialized from the direct PINN solution. For the PINN-based methods, each panel also compares the state predicted directly by the neural network with the state recomputed by the numerical solver using the converged PINN control. The relative $L_2$ error of each time snapshot is marked in the legend.

Theorems & Definitions (4)

  • Proposition B.1
  • Remark B.1
  • Remark B.2
  • Proposition C.1