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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 networks (PINNs) have 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 an improved fully connected neural network structure, 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 and time-step sizes of the Radau IIA method affect the accuracy of neural network solutions. For different time-step sizes, the experimental results indicate that utilizing a 5th-order Radau IIA method in the PINN achieves a high level of system accuracy and stability. Specifically, the absolute errors for all differential variables remain as low as 10 − 6 , and the absolute errors for algebraic variables are maintained at 10 − 5 . 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. [ABSTRACT FROM AUTHOR]

The single-component colistin E2, with superior antibacterial activity and lower toxicity, was being developed as the latest generation of polymyxin drugs. However, colistin E2 has not been tested quantitatively in biological matrices. In this study, based on the quantitative detection of colistin methanesulphonate (CMS) and colistin by Zhao et al., 15N-labeled colistin E2 was used as an internal standard (IS) for a more accurate quantitative detection of CMS E2 in human plasma. A rapid ultra-high-performance liquid chromatography-tandem mass spectrometry (UHPLC–MS/MS) assay method was developed for determination of CMS E2 and colistin E2 in human plasma. After pretreatment of plasma samples by 96-well SPE Supra-Clean Weak Cation Exchange (WCX) plate, the formed colistin E2 was detected and quantified by UHPLC–MS/MS system. All plasma lots were found to be free of interferences with the analyte. The matrix has no effect on the quantitation of the analyte. No significant effect of the carryover was observed. The dilution integrity was demonstrated in plasma samples without the loss of accuracy and precision. The lower limit of quantification (LLOQ) was 0.0300 mg/L for colistin E2 in plasma with accuracy (relative error, 5.1–12.7%) and precision (relative standard deviation, − 5.7–9.3%). Stability of CMS E2 and colistin E2 was demonstrated in biological samples before and during sample treatment, and in the extract. Furthermore, this method was successfully applied to the analysis of plasma samples obtained from Chinese healthy volunteers receiving a single intravenous CMS E2 dose of 5 mg/kg. In conclusion, the detection method was characterized by speed and high accuracy, which laid a solid foundation for the subsequent development of CMS E2 drug. [ABSTRACT FROM AUTHOR]