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Feedback control systems are categorized in various ways based on their design, analysis, and signal types.
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AW-EL-PINNs: A multi-task learning physics-informed neural network for Euler-Lagrange systems in optimal control

Chuandong Li1, Runtian Zeng1

  • 1College of Electronic and Information Engineering, Southwest University, Chongqing, 400715, China.

Neural Networks : the Official Journal of the International Neural Network Society
|February 13, 2026
PubMed
Summary
This summary is machine-generated.

This study introduces adaptive weighted Euler-Lagrange theorem combined physics-informed neural networks (AW-EL-PINNs) for optimal control. AW-EL-PINNs enhance solution accuracy and stability for Euler-Lagrange systems compared to traditional methods.

Keywords:
Adaptive loss weightingEuler-Lagrange theoremOptimal control problemsPhysics-informed neural networksTwo-point boundary value problems

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Area of Science:

  • Computational Mathematics
  • Machine Learning
  • Optimal Control Theory

Background:

  • Euler-Lagrange systems are fundamental in classical mechanics and optimal control.
  • Solving these systems often involves complex numerical methods and manual parameter tuning.
  • Existing physics-informed neural networks (PINNs) require significant effort in balancing loss functions.

Purpose of the Study:

  • To present a novel framework, adaptive weighted Euler-Lagrange theorem combined physics-informed neural networks (AW-EL-PINNs), for solving Euler-Lagrange systems in optimal control.
  • To improve the efficiency and accuracy of solving optimal control problems.
  • To reduce the need for manual tuning of loss function weights in deep learning approaches.

Main Methods:

  • The proposed AW-EL-PINNs framework integrates the Euler-Lagrange theorem with a deep learning architecture.
  • Optimal control problems are systematically transformed into two-point boundary value problems (TPBVPs).
  • An adaptive loss weighting mechanism dynamically balances loss components during training, reducing manual intervention.

Main Results:

  • AW-EL-PINNs demonstrated enhanced solution accuracy across five numerical examples.
  • The framework maintained stability throughout the optimization process.
  • Performance was superior to baseline methods in terms of precision and stability.

Conclusions:

  • AW-EL-PINNs offer a robust and accurate method for solving Euler-Lagrange systems in optimal control.
  • The adaptive loss weighting mechanism significantly improves upon conventional PINNs by reducing manual tuning.
  • This framework holds promise for applications in various physical systems requiring precise optimal control solutions.