English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

As is well known, differential algebraic equations (DAEs), which are able todescribe dynamic changes and underlying constraints, have been widely appliedin engineering fields such as fluid dynamics, multi-body dynamics, mechanicalsystems and control theory. In practical physical modeling within thesedomains, the systems often generate high-index DAEs. Classical implicitnumerical methods typically result in varying order reduction of numericalaccuracy when solving high-index systems.~Recently, the physics-informed neuralnetwork (PINN) has gained attention for solving DAE systems. However, it faceschallenges like the inability to directly solve high-index systems, lowerpredictive accuracy, and weaker generalization capabilities. In this paper, wepropose a PINN computational framework, combined Radau IIA numerical methodwith a neural network structure via the attention mechanisms, to directly solvehigh-index DAEs. Furthermore, we employ a domain decomposition strategy toenhance solution accuracy. We conduct numerical experiments with two classicalhigh-index systems as illustrative examples, investigating how different ordersof the Radau IIA method affect the accuracy of neural network solutions. Theexperimental results demonstrate that the PINN based on a 5th-order Radau IIAmethod achieves the highest level of system accuracy. Specifically, theabsolute errors for all differential variables remains as low as $10^{-6}$, andthe absolute errors for algebraic variables is maintained at $10^{-5}$,surpassing the results found in existing literature. Therefore, our methodexhibits excellent computational accuracy and strong generalizationcapabilities, providing a feasible approach for the high-precision solution oflarger-scale DAEs with higher indices or challenging high-dimensional partialdifferential algebraic equation systems.

Physical Information Neural Networks for Solving High-index Differential-algebraic Equation Systems Based on Radau Methods