Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

AW-EL-PINNs: A Multi-Task Learning Physics-Informed Neural Network for Euler-Lagrange Systems in Optimal Control Problems

Chuandong Li, Runtian Zeng

TL;DR

The paper introduces AW-EL-PINNs, a multi-task, physics-informed neural network framework that leverages the Euler-Lagrange theorem to convert optimal control problems into solvable TPBVPs. It integrates three neural networks for state, control, and adjoint variables and employs an adaptive loss weighting scheme to balance multiple residuals, initial/terminal conditions, and control constraints. Across six numerical examples, AW-EL-PINNs consistently deliver higher accuracy and stability than baseline PINN methods, often outperforming EL-PINNs with fixed loss weights. The approach offers a scalable, data-free tool for solving Euler-Lagrange systems in physics- and engineering-driven optimal control tasks, with potential extensions to differential games and constrained/impulsive scenarios.

Abstract

This paper presents adaptive weighted Euler-Lagrange theorem combined physics-informed neural networks (AW-EL-PINNs) for solving Euler-Lagrange systems in optimal control problems. The framework systematically converts optimal control frameworks into two-point boundary value problems (TPBVPs) while establishing a multi-task learning paradigm through innovative integration of the Euler-Lagrange theorem with deep learning architecture. An adaptive loss weighting mechanism dynamically balances loss function components during training, decreasing tedious manual tuning of weighting the loss functions compared to the conventional physics-informed neural networks (PINNs). Based on six numerical examples, it's clear that AW-EL-PINNs achieve enhanced solution accuracy compared to baseline methods while maintaining stability throughout the optimization process. These results highlight the framework's capability to improve precision and ensure stability in solving Euler-Lagrange systems in optimal control problems, offering potential strategies for problems under physical applications.

AW-EL-PINNs: A Multi-Task Learning Physics-Informed Neural Network for Euler-Lagrange Systems in Optimal Control Problems

TL;DR

The paper introduces AW-EL-PINNs, a multi-task, physics-informed neural network framework that leverages the Euler-Lagrange theorem to convert optimal control problems into solvable TPBVPs. It integrates three neural networks for state, control, and adjoint variables and employs an adaptive loss weighting scheme to balance multiple residuals, initial/terminal conditions, and control constraints. Across six numerical examples, AW-EL-PINNs consistently deliver higher accuracy and stability than baseline PINN methods, often outperforming EL-PINNs with fixed loss weights. The approach offers a scalable, data-free tool for solving Euler-Lagrange systems in physics- and engineering-driven optimal control tasks, with potential extensions to differential games and constrained/impulsive scenarios.

Abstract

This paper presents adaptive weighted Euler-Lagrange theorem combined physics-informed neural networks (AW-EL-PINNs) for solving Euler-Lagrange systems in optimal control problems. The framework systematically converts optimal control frameworks into two-point boundary value problems (TPBVPs) while establishing a multi-task learning paradigm through innovative integration of the Euler-Lagrange theorem with deep learning architecture. An adaptive loss weighting mechanism dynamically balances loss function components during training, decreasing tedious manual tuning of weighting the loss functions compared to the conventional physics-informed neural networks (PINNs). Based on six numerical examples, it's clear that AW-EL-PINNs achieve enhanced solution accuracy compared to baseline methods while maintaining stability throughout the optimization process. These results highlight the framework's capability to improve precision and ensure stability in solving Euler-Lagrange systems in optimal control problems, offering potential strategies for problems under physical applications.

Paper Structure

This paper contains 10 sections, 2 theorems, 39 equations, 45 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Fundamental Conceptions and Necessary Conditions
  3. Optimal control
  4. Pontryagin minimum principle
  5. Fundamental model of PINNs
  6. Multi-task adaptive loss weights
  7. Existing approach review
  8. Guideline of AW-EL-PINNs
  9. Numerical Illustration and Performance Comparison
  10. Conclusion

Key Result

Theorem 1

(Pontryagin minimum principle, engwerda2005lq) Let $u^*(t)$ be an optimal control and $x^*(t)$ be the corresponding state. Then, there exists an adjoint variable $p^*(t)\neq 0$ and following equations holding:

Figures (45)

  • Figure 1: Structure of AW-EL-PINNs.
  • Figure 2: $x(t)$ and $u(t)$ in Example \ref{['Linear quadratic (LQ) optimal control problem']} solved by AW-EL-PINNs.
  • Figure 3: $e^{-s_k}$ of AW-EL-PINNs in Example \ref{['Linear quadratic (LQ) optimal control problem']}.
  • Figure 4: $x(t)$ and $u(t)$ in Example \ref{['Linear quadratic (LQ) optimal control problem']} solved by EL-PINNs.
  • Figure 5: $x(t)$ and $u(t)$ in Example \ref{['Linear quadratic (LQ) optimal control problem']} solved by AW-PINNs.
  • ...and 40 more figures

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Example 1
  • Remark 2
  • Example 2
  • Remark 3
  • Example 3
  • Remark 4
  • Example 4
  • ...and 3 more