Ming Yan

Jiasheng Chen, Juan Tang, Ming Yan et al.

As is well known, differential algebraic equations (DAEs), which are able to describe dynamic changes and underlying constraints, have been widely applied in engineering fields such as fluid dynamics, multi-body dynamics, mechanical systems and control theory. In practical physical modeling within these domains, the systems often generate high-index DAEs. Classical implicit numerical methods typically result in varying order reduction of numerical accuracy when solving high-index systems.~Recently, the physics-informed neural network (PINN) has gained attention for solving DAE systems. However, it faces challenges like the inability to directly solve high-index systems, lower predictive accuracy, and weaker generalization capabilities. In this paper, we propose a PINN computational framework, combined Radau IIA numerical method with a neural network structure via the attention mechanisms, to directly solve high-index DAEs. Furthermore, we employ a domain decomposition strategy to enhance solution accuracy. We conduct numerical experiments with two classical high-index systems as illustrative examples, investigating how different orders of the Radau IIA method affect the accuracy of neural network solutions. The experimental results demonstrate that the PINN based on a 5th-order Radau IIA method achieves the highest level of system accuracy. Specifically, the absolute errors for all differential variables remains as low as $10^{-6}$, and the absolute errors for algebraic variables is maintained at $10^{-5}$, surpassing the results found in existing literature. Therefore, our method exhibits excellent computational accuracy and strong generalization capabilities, providing a feasible approach for the high-precision solution of larger-scale DAEs with higher indices or challenging high-dimensional partial differential algebraic equation systems.

Kai Luo, Juan Tang, Mingchao Cai et al.

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.

Ming Yan, Junjie Chen, Xuejie Cao et al.

Many test coverage metrics have been proposed to measure the Deep Neural Network (DNN) testing effectiveness, including structural coverage and non-structural coverage. These test coverage metrics are proposed based on the fundamental assumption: they are correlated with test effectiveness. However, the fundamental assumption is still not validated sufficiently and reasonably, which brings question on the usefulness of DNN test coverage. This paper conducted a revisiting study on the existing DNN test coverage from the test effectiveness perspective, to effectively validate the fundamental assumption. Here, we carefully considered the diversity of subjects, three test effectiveness criteria, and both typical and state-of-the-art test coverage metrics. Different from all the existing studies that deliver negative conclusions on the usefulness of existing DNN test coverage, we identified some positive conclusions on their usefulness from the test effectiveness perspective. In particular, we found the complementary relationship between structural and non-structural coverage and identified the practical usage scenarios and promising research directions for these existing test coverage metrics.