{
  "id": "2310.12846",
  "version": "v1",
  "published": "2023-10-19T15:57:10.000Z",
  "updated": "2023-10-19T15:57:10.000Z",
  "title": "Physical Information Neural Networks for Solving High-index Differential-algebraic Equation Systems Based on Radau Methods",
  "authors": [
    "Jiasheng Chen",
    "Juan Tang",
    "Ming Yan",
    "Shuai Lai",
    "Kun Liang",
    "Jianguang Lu",
    "Wenqiang Yang"
  ],
  "categories": [
    "math.NA",
    "cs.AI",
    "cs.NA"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-19T15:57:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "solving high-index differential-algebraic equation systems",
      "physical information neural networks",
      "differential algebraic equation",
      "partial differential algebraic",
      "radau iia method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}