{
  "id": "2412.04409",
  "version": "v1",
  "published": "2024-12-05T18:31:14.000Z",
  "updated": "2024-12-05T18:31:14.000Z",
  "title": "Stabilizing and Solving Inverse Problems using Data and Machine Learning",
  "authors": [
    "Erik Burman",
    "Mats G. Larson",
    "Karl Larsson",
    "Carl Lundholm"
  ],
  "categories": [
    "math.NA",
    "cs.LG",
    "cs.NA"
  ],
  "abstract": "We consider an inverse problem involving the reconstruction of the solution to a nonlinear partial differential equation (PDE) with unknown boundary conditions. Instead of direct boundary data, we are provided with a large dataset of boundary observations for typical solutions (collective data) and a bulk measurement of a specific realization. To leverage this collective data, we first compress the boundary data using proper orthogonal decomposition (POD) in a linear expansion. Next, we identify a possible nonlinear low-dimensional structure in the expansion coefficients using an auto-encoder, which provides a parametrization of the dataset in a lower-dimensional latent space. We then train a neural network to map the latent variables representing the boundary data to the solution of the PDE. Finally, we solve the inverse problem by optimizing a data-fitting term over the latent space. We analyze the underlying stabilized finite element method in the linear setting and establish optimal error estimates in the $H^1$ and $L^2$-norms. The nonlinear problem is then studied numerically, demonstrating the effectiveness of our approach.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-05T18:31:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "solving inverse problems",
      "machine learning",
      "nonlinear partial differential equation",
      "unknown boundary conditions",
      "direct boundary data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}