Technology Category
Constraint Satisfaction and Optimization: Distributed CSP/OptimizationMachine Learning: Deep Neural Architectures and Foundation ModelsMultiagent Systems: Distributed Problem Solving
DAE-KAN: A Kolmogorov-Arnold Network Model for High-Index Differential-Algebraic Equations
🤖 AI Summary
To address low accuracy, severe integration drift, and poor generalization in solving high-index differential-algebraic equations (DAEs), this paper proposes DAE-KAN: a novel physics-informed solver that integrates Kolmogorov–Arnold Networks (KANs) with Physics-Informed Neural Networks (PINNs), thereby combining superior function approximation capability with rigorous physical constraint embedding. Methodologically, it jointly optimizes differential and algebraic variables and introduces a coupled differential-algebraic residual loss function to effectively suppress long-term integration drift. Experiments on benchmark index-1 to index-3 DAE systems demonstrate that DAE-KAN reduces absolute errors in both differential and algebraic variables by one to two orders of magnitude compared to standard PINNs; achieves superior drift control relative to classical numerical solvers; and exhibits enhanced generalization and computational robustness.
Technology Category
Application Category
📝 Abstract
Kolmogorov-Arnold Networks (KANs) have emerged as a promising alternative to Multi-layer Perceptrons (MLPs) due to their superior function-fitting abilities in data-driven modeling. In this paper, we propose a novel framework, DAE-KAN, for solving high-index differential-algebraic equations (DAEs) by integrating KANs with Physics-Informed Neural Networks (PINNs). This framework not only preserves the ability of traditional PINNs to model complex systems governed by physical laws but also enhances their performance by leveraging the function-fitting strengths of KANs. Numerical experiments demonstrate that for DAE systems ranging from index-1 to index-3, DAE-KAN reduces the absolute errors of both differential and algebraic variables by 1 to 2 orders of magnitude compared to traditional PINNs. To assess the effectiveness of this approach, we analyze the drift-off error and find that both PINNs and DAE-KAN outperform classical numerical methods in controlling this phenomenon. Our results highlight the potential of neural network methods, particularly DAE-KAN, in solving high-index DAEs with substantial computational accuracy and generalization, offering a promising solution for challenging partial differential-algebraic equations.
Problem
Research questions and friction points this paper is trying to address.
Solving high-index differential-algebraic equations (DAEs) using KANs and PINNs
Improving accuracy of DAEs solutions by reducing errors significantly
Addressing drift-off error in DAEs with neural network methods
Innovation
Methods, ideas, or system contributions that make the work stand out.
Integrates KANs with PINNs for DAEs
Reduces errors by 1-2 orders magnitude
Enhances accuracy and generalization
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