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

Wenqiang Yang; Wenyuan Wu; Yong Feng; Changbo Chen

cs.LGMar 24, 2026

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

Authors: Wenqiang Yang, Wenyuan Wu, Yong Feng, Changbo Chen

Organizations: Chongqing Key Laboratory of Automated Reasoning and Cognition, Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, 266 Fangzheng Avenue, Liangjiang District, 400714, Chongqing, China. · Chongqing School, University of Chinese Academy of Sciences, 266 Fangzheng Avenue, Liangjiang District, 400714, Chongqing, China.

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.

Figures & tables

Panel of Figure 2 — Figure 2: Comparison of supervised and unsupervised training.

Figure 3: Dual physics-informed neural network architecture for multi-task optimization

Panel of Figure 4 — Figure 4: Training progression for Example 4.1 .

Panel of Figure 5 — Figure 5: Exact and predicted solutions for Example 4.1 .

Method	Minimum Objective Value	Iteration Count
New OB relaxation Song and Khan (2022)	$-0.0607$	23
SBM relaxation Scott and Barton (2012)	$-0.0607$	33
Our method	$-0.0607$	6

Table 1: Comparison of solving efficiency for Example 4.1 .

Figure 6: Maximum absolute error of the constraint network for the E. coli model.

Figure 7: Traversal process of the $\epsilon$ -global method in Scott and Barton (2015) .

Figure 8: Objective function value distribution predicted by the dual network architecture.

Method	Parameters $(k_{1},k_{3})$	$Obj$
$\epsilon$ -global method Scott and Barton (2015)	$(4.31348\times 10^{-3},162.9297)$	$0.029096$
Method in Kremling et al. (2004)	$(2.9\times 10^{-3},90)$	$0.0712$
Our method (predicted)	$(4.4881\times 10^{-3},1.7231\times 10^{2})$	$0.0296$
Our method (corrected)	$(4.32191\times 10^{-3},1.632459\times 10^{2})$	$0.029089$

Table 2: Comparison of optimization results for the E. coli model.

$n$	Training Cost (s)	Prediction Cost (s)	Correction Cost (s)	B $\&$ B Cost (s) Bajaj and Hasan (2020)	Predicted $J(\bm{p})$	Predicted $\bm{p}$	Optimal $\bm{p}^{*}$
$2$	$96.1$	$0.6$	$0.003$	$1.4$	$6.50e-08$	$\begin{array}[]{c}[0.5714,\\ 0.4541]\end{array}$	$\begin{array}[]{c}[0.5721,\\ 0.4544\end{array}$
$3$	$435.2$	$1.2$	$0.003$	$8.1$	$1.17e-07$	$\begin{array}[]{c}[0.6325,\\ 0.5137,\\ 0.4474]\end{array}$	$\begin{array}[]{c}[0.6330,\\ 0.5141,\\ 0.4481]\end{array}$
$4$	$1452.3$	$1.3$	$0.011$	$19.4$	$1.28e-07$	$\begin{array}[]{c}[0.6751,\\ 0.5564,\\ 0.4902,\\ 0.4437]\end{array}$	$\begin{array}[]{c}[0.6771,\\ 0.5572,\\ 0.4904,\\ 0.4444]\end{array}$
$6$	$5991.9$	$1.5$	$0.005$	$90.6$	$8.07e-07$	$\begin{array}[]{c}[0.7396,\\ 0.6170,\\ 0.5484,\\ 0.5014,\\ 0.4667,\\ 0.4392]\end{array}$	$\begin{array}[]{c}[0.7404,\\ 0.6187,\\ 0.5506,\\ 0.5038,\\ 0.4685,\\ 0.4403]\end{array}$
$8$	$18424.8$	$2.7$	$0.013$	$934.6$	$1.60e-07$	$\begin{array}[]{c}[0.7852,\\ 0.6633,\\ 0.5949,\\ 0.5473,\\ 0.5109,\\ 0.4823,\\ 0.4578,\\ 0.4372]\end{array}$	$\begin{array}[]{c}[0.7859,\\ 0.6636,\\ 0.5949,\\ 0.5474,\\ 0.5113,\\ 0.4825,\\ 0.4582,\\ 0.4377]\end{array}$
$10$	$53472.4$	$4.8$	$0.008$	$6083.5$	$2.92e-07$	$\begin{array}[]{c}[0.8241,\\ 0.6981,\\ 0.6285,\\ 0.5809,\\ 0.5443,\\ 0.5148,\\ 0.4902,\\ 0.4692,\\ 0.4513,\\ 0.4348]\end{array}$	$\begin{array}[]{c}[0.8241,\\ 0.6984,\\ 0.6287,\\ 0.5811,\\ 0.5445,\\ 0.5150,\\ 0.4904,\\ 0.4694,\\ 0.4515,\\ 0.4350]\end{array}$

Table 3: Performance Comparison for Example

Constrained Optimization

Constrained optimization finds the best decisions while satisfying equality, inequality, resource, safety, or fairness requirements.

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Neural surrogates for stiff differential-algebraic equations (DAEs) face two barriers: soft-constraint methods leave algebraic residuals that stiffness amplifies into errors, and hard-constraint methods require trajectory data from stiff integrators. We introduce an extended Newton implicit layer that enforces algebraic constraints exactly and reduces fast dynamics to their quasi-steady-state values in a single differentiable solve. Embedded in a physics-informed DeepONet, the layer recovers all fast and algebraic states exactly from slow-state predictions, removes the per-window stiffness-amplification pathway, and yields a stiffness-scaled Implicit Function Theorem gradient absent from penalty methods. Cascaded implicit layers extend this to multi-component systems with provable convergence. On a grid-forming inverter (stiffness ratio of about 4712), extended Newton attains 1.42% error versus 39.3% (penalty) and 57.0% (standard Newton); augmented Lagrangian and feedback linearization diverged. Two independently trained models compose without retraining (0.72% to 1.16% error, exact constraint satisfaction). Cross-domain validation on the Robertson stiff DAE (stiffness ratio up to

10^5

) confirms generalization. Conformal prediction provides 90% coverage with automatic out-of-distribution detection.

Abstract

Constrained Optimization

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