DAE-KAN: A Kolmogorov-Arnold Network Model for High-Index Differential-Algebraic Equations

TL;DR

This work introduces DAE-KAN, a framework that fuses Kolmogorov-Arnold Networks (KAN) with Physics-Informed Neural Networks (PINNs) to solve high-index differential-algebraic equations (DAEs). By employing dual KAN branches to predict differential and algebraic variables and trainable edge-based activation via B-splines, DAE-KAN achieves superior accuracy and reduced drift-off compared to standard PINNs across index-$1$ to index-$3$ systems. Numerical results on constrained circular motion and a two-link robot arm show 1–2 order-of-magnitude improvements in absolute and relative errors for both differential and algebraic variables, alongside improved constraint satisfaction. The approach offers a promising, direct solver for challenging high-index DAEs with potential for broader applications and integration with traditional numerical methods.

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.

Figures (9)

• Figure 1: Error in the constraints for DOPRI5.
• Figure 2: Conventional physical information neural networks with MLP.
• Figure 3: Illustration of a 3-layer KAN network of the shape [2,3,2,1].
• Figure 4: Diagram of the DAE-KAN network structure.
• Figure 5: The $\text{AE}$ curves of the predicted solution and the exact solution obtained by using the traditional PINNs and DAE-KAN to solve the corresponding index-1, index-2, and index-3 forms of equation \ref{['4.2']}, respectively. The green line represents DAE-KAN, and the blue line represents PINNs.

Theorems & Definitions (1)

• Definition 1: Index of a DAE