{
  "id": "2502.00488",
  "version": "v1",
  "published": "2025-02-01T16:26:53.000Z",
  "updated": "2025-02-01T16:26:53.000Z",
  "title": "Learn Sharp Interface Solution by Homotopy Dynamics",
  "authors": [
    "Chuqi Chen",
    "Yahong Yang",
    "Yang Xiang",
    "Wenrui Hao"
  ],
  "categories": [
    "cs.LG",
    "cs.NA",
    "math.NA"
  ],
  "abstract": "This paper explores challenges in training Physics-Informed Neural Networks (PINNs), emphasizing the role of the loss landscape in the training process. We examine difficulties in minimizing the PINN loss function, particularly due to ill-conditioning caused by differential operators in the residual term. We compare gradient-based optimizers Adam, L-BFGS, and their combination \\al{}, showing the superiority of \\al{}, and introduce a novel second-order optimizer, NysNewton-CG (NNCG), which significantly improves PINN performance. Theoretically, our work elucidates the connection between ill-conditioned differential operators and ill-conditioning in the PINN loss and shows the benefits of combining first- and second-order optimization methods. Our work presents valuable insights and more powerful optimization strategies for training PINNs, which could improve the utility of PINNs for solving difficult partial differential equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-01T16:26:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "learn sharp interface solution",
      "homotopy dynamics",
      "differential operators",
      "partial differential equations",
      "pinn loss function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}