标题： 通过狄利克雷到诺依曼映射实现逆椭圆偏微分方程的机器学习和有限元框架 显示英文标题

标题： A Machine Learning and Finite Element Framework for Inverse Elliptic PDEs via Dirichlet-to-Neumann Mapping

摘要： 偏微分方程 (PDE) 的反问题在许多应用领域至关重要，例如地球物理学、生物医学成像和材料科学，这些领域需要从间接测量推断未知的物理属性。 在这项工作中，我们通过利用 Dirichlet-to-Neumann (DtN) 映射来解决椭圆型 PDE 的反问题，该映射捕捉了边界输入与通量响应之间的关系。 因此，这种方法能够通过利用边界数据求解反问题，以确定域内的材料属性。 我们的框架采用了一种无监督机器学习算法，在内层循环中结合有限元方法 (FEM) 来求解正向问题，从而确保高精度。 此外，我们的方法可以灵活地使用边界数据的部分观测值，这在实际场景中通常是情况。 通过引入精心设计的损失函数以适应不连续性，该方法迭代地优化系数重构。 这种结合 FEM 和机器学习的方法为广泛的反问题提供了一种稳健且准确的解决方案策略，从而提高了从医学诊断到地下勘探等应用中关键参数的估计精度。 显示英文摘要

摘要： Inverse problems for partial differential equations (PDEs) are crucial in numerous applications such as geophysics, biomedical imaging, and material science, where unknown physical properties must be inferred from indirect measurements. In this work, we address the inverse problem for elliptic PDEs by leveraging the Dirichlet-to-Neumann (DtN) map, which captures the relationship between boundary inputs and flux responses. Thus, this approach enables to solve the inverse problem that seeks the material properties inside the domain by utilizing the boundary data. Our framework employs an unsupervised machine learning algorithm that integrates a finite element method (FEM) in the inner loop for the forward problem, ensuring high accuracy. Moreover our approach is flexible to utilize partial observations of the boundary data, which is often the case in real-world scenarios. By incorporating carefully designed loss functions that accommodate discontinuities, the method refines coefficient reconstructions iteratively. This combined FEM and machine learning approach offers a robust, accurate solution strategy for a broad range of inverse problems, enabling improved estimation of critical parameters in applications from medical diagnostics to subsurface exploration.