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Double Coupling Architecture and Training Method for Optimization Problems of Differential Algebraic Equations with Parameters

Wenqiang Yang, Wenyuan Wu, Yong Feng, Changbo Chen

Abstract

Simulation and modeling are essential in product development, integrated into the design and manufacturing process to enhance efficiency and quality. They are typically represented as complex nonlinear differential algebraic equations. The growing diversity of product requirements demands multi-task optimization, a key challenge in simulation modeling research. A dual physics-informed neural network architecture has been proposed to decouple constraints and objective functions in parametric differential algebraic equation optimization problems. Theoretical analysis shows that introducing a relaxation variable with a global error bound ensures solution equivalence between the network and optimization problem. A genetic algorithm-enhanced training framework for physics-informed neural networks improves training precision and efficiency, avoiding redundant solving of differential algebraic equations. This approach enables generalization for multi-task objectives with a single, training maintaining real-time responsiveness to product requirements.

Double Coupling Architecture and Training Method for Optimization Problems of Differential Algebraic Equations with Parameters

Abstract

Simulation and modeling are essential in product development, integrated into the design and manufacturing process to enhance efficiency and quality. They are typically represented as complex nonlinear differential algebraic equations. The growing diversity of product requirements demands multi-task optimization, a key challenge in simulation modeling research. A dual physics-informed neural network architecture has been proposed to decouple constraints and objective functions in parametric differential algebraic equation optimization problems. Theoretical analysis shows that introducing a relaxation variable with a global error bound ensures solution equivalence between the network and optimization problem. A genetic algorithm-enhanced training framework for physics-informed neural networks improves training precision and efficiency, avoiding redundant solving of differential algebraic equations. This approach enables generalization for multi-task objectives with a single, training maintaining real-time responsiveness to product requirements.
Paper Structure (19 sections, 3 theorems, 23 equations, 8 figures, 3 tables)

This paper contains 19 sections, 3 theorems, 23 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Optimization and Learning Methods for Parametric DAEs
  3. Contributions
  4. Preliminaries
  5. PINNs for Optimization
  6. Structural Analysis
  7. Global Error Estimation
  8. A Dual Physics-Informed Neural Network Architecture
  9. Genetic Algorithm-Based Training of the Constraint Network
  10. Adaptive Training Strategy with Refinement
  11. Illustrative Example
  12. Error Bound
  13. Experiments
  14. Experimental Setup
  15. Case Study: Biochemical Reaction System
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $\bm{F}$ be a DAE system with rank-deficient Jacobian. Then there exists an embedded system $\bm{G}$ such that where $\pi$ is a projection operator. Moreover, the differentiation index satisfies $\delta(\bm{G}) \leq \delta(\bm{F}) - (n-r)$.

Figures (8)

  • Figure 1: PINNs for Optimization
  • Figure 2: Comparison of supervised and unsupervised training.
  • Figure 3: Dual physics-informed neural network architecture for multi-task optimization
  • Figure 4: Training progression for Example \ref{['ex:song2022']}.
  • Figure 5: Exact and predicted solutions for Example \ref{['ex:song2022']}.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Definition 1.1
  • Example 2.1
  • Theorem 2.1: Yang2024
  • Theorem 2.2: Yang2021
  • Example 4.1
  • Theorem 4.1
  • Example 5.1