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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 · Apr 6, 2026 · Citations: 0

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Evidence quality

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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.

Abstract-only analysis — low confidence

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Should You Rely On This Paper?

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Background context only

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Trust level

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Human Feedback Signal

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Evaluation Signal

Weak / implicit signal

Usefulness for eval research

Provisional (processing)

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What We Could Verify

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Human Feedback Types

provisional (inferred)

None explicit

No explicit feedback protocol extracted.

"We study physics-informed neural networks (PINNs) as numerical tools for the optimal control of semilinear partial differential equations."

Evaluation Modes

provisional (inferred)

None explicit

Validate eval design from full paper text.

"We study physics-informed neural networks (PINNs) as numerical tools for the optimal control of semilinear partial differential equations."

Quality Controls

provisional (inferred)

Not reported

No explicit QC controls found.

"We study physics-informed neural networks (PINNs) as numerical tools for the optimal control of semilinear partial differential equations."

Benchmarks / Datasets

provisional (inferred)

Not extracted

No benchmark anchors detected.

"We study physics-informed neural networks (PINNs) as numerical tools for the optimal control of semilinear partial differential equations."

Reported Metrics

provisional (inferred)

Not extracted

No metric anchors detected.

"We study physics-informed neural networks (PINNs) as numerical tools for the optimal control of semilinear partial differential equations."

Rater Population

provisional (inferred)

Unknown

Rater source not explicitly reported.

"We study physics-informed neural networks (PINNs) as numerical tools for the optimal control of semilinear partial differential equations."

Human Feedback Details

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  • Potential benchmark anchors: No benchmark names detected in abstract.
  • Abstract highlights: 3 key sentence(s) extracted below.

Evaluation Details

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  • Potential metric signals: No metric keywords detected.
  • Confidence: Provisional (metadata-only fallback).

Research Brief

Metadata summary

We study physics-informed neural networks (PINNs) as numerical tools for the optimal control of semilinear partial differential equations.

Based on abstract + metadata only. Check the source paper before making high-confidence protocol decisions.

Key Takeaways

  • 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.

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