Yin Minglang, Charon Nicolas, Brody Ryan, Lu Lu, Trayanova Natalia, Maggioni Mauro
Department of Biomedical Engineering, Johns Hopkins University, Baltimore, MD, USA.
Alliance for Cardiovascular Diagnostic and Treatment Innovation, Johns Hopkins University, Baltimore, MD, USA.
Nat Comput Sci. 2024 Dec;4(12):928-940. doi: 10.1038/s43588-024-00732-2. Epub 2024 Dec 9.
Solving partial differential equations (PDEs) using numerical methods is a ubiquitous task in engineering and medicine. However, the computational costs can be prohibitively high when many-query evaluations of PDE solutions on multiple geometries are needed. Here we aim to address this challenge by introducing Diffeomorphic Mapping Operator Learning (DIMON), a generic artificial intelligence framework that learns geometry-dependent solution operators of different types of PDE on a variety of geometries. We present several examples to demonstrate the performance, efficiency and scalability of the framework in learning both static and time-dependent PDEs on parameterized and non-parameterized domains; these include solving the Laplace equations, reaction-diffusion equations and a system of multiscale PDEs that characterize the electrical propagation on thousands of personalized heart digital twins. DIMON can reduce the computational costs of solution approximations on multiple geometries from hours to seconds with substantially less computational resources.
使用数值方法求解偏微分方程（PDEs）在工程和医学中是一项普遍存在的任务。然而，当需要对多个几何形状上的PDE解进行多次查询评估时，计算成本可能高得令人望而却步。在这里，我们旨在通过引入微分同胚映射算子学习（DIMON）来应对这一挑战，DIMON是一个通用的人工智能框架，它可以在各种几何形状上学习不同类型PDE的几何相关解算子。我们给出了几个例子，以展示该框架在学习参数化和非参数化域上的静态和时间相关PDE方面的性能、效率和可扩展性；这些例子包括求解拉普拉斯方程、反应扩散方程以及一个多尺度PDE系统，该系统描述了数千个个性化心脏数字双胞胎上的电传播。DIMON可以用少得多的计算资源将多个几何形状上解近似的计算成本从数小时降低到数秒。
