Residual-Informed Learning of Solutions to Algebraic Loops

2025-10-10

AI Summary

To address numerical ambiguity and inefficiency arising from algebraic loops in equation-based modeling languages (e.g., Modelica), this paper proposes an unsupervised, residual-driven neural surrogate modeling method. A feedforward neural network serves as the surrogate, with the algebraic loop residual explicitly incorporated into the loss function—enabling end-to-end, label-free training that eliminates reliance on supervised datasets. Crucially, residual minimization resolves solution multiplicity by steering convergence toward physically consistent solutions—not statistical averages. An integrated error-control strategy ensures numerical accuracy. Evaluated on the IEEE 14-Bus system, the method achieves a 60% speedup over conventional numerical solvers while maintaining comparable precision. The core innovation lies in the first use of algebraic loop residuals as an unsupervised learning signal for neural networks, uniquely balancing robustness, interpretability, and engineering practicality.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Deep Neural Architectures and Foundation Models
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper presents a residual-informed machine learning approach for replacing algebraic loops in equation-based Modelica models with neural network surrogates. A feedforward neural network is trained using the residual (error) of the algebraic loop directly in its loss function, eliminating the need for a supervised dataset. This training strategy also resolves the issue of ambiguous solutions, allowing the surrogate to converge to a consistent solution rather than averaging multiple valid ones. Applied to the large-scale IEEE 14-Bus system, our method achieves a 60% reduction in simulation time compared to conventional simulations, while maintaining the same level of accuracy through error control mechanisms.

Problem

• Replacing algebraic loops with neural network surrogates
• Training networks using residual without supervised datasets
• Resolving ambiguous solutions to ensure consistent convergence

Innovation

• Uses residual-informed machine learning for algebraic loops
• Trains neural network with residual in loss function
• Achieves faster simulation while maintaining accuracy