面向偏微分方程多域与多参数的微分同胚神经算子
Diffeomorphism Neural Operator for various domains and parameters of partial differential equations
AI总结：
针对现有神经算子仅能在固定域求解偏微分方程的局限，提出微分同胚神经算子框架，将多域算子学习转化为通用域学习，在多类流体与力学问题中验证了其稳健学习与跨域跨参数泛化能力。
摘要：
在科学与工程应用中，求解跨不同参数和域的偏微分方程（PDEs）通常依赖计算资源消耗巨大的数值方法。基于深度学习的神经算子通过直接从数据中学习物理规律，为PDE求解提供了一种颇具前景的替代方案。然而，当前的神经算子方法仅能在固定域上求解PDEs。将神经算子拓展至可在不同域上求解PDEs，在几何与参数变化至关重要的医学成像、工程设计和制造等应用领域具有重要价值。本文提出了一种新型神经算子学习框架，用于求解物理系统中定义的、具有不同域和参数的PDEs，名为微分同胚神经算子（diffeomorphism neural operator, DNO）。其核心思想是：神经算子在一个通用域中学习，该通用域由满足同一PDE的各类物理域通过微分同胚映射得到。通过这种方式，多域上的算子学习难题被转化为通用域上的算子学习问题。研究还提出了一种评估方法，通过衡量新域与训练数据集各域在微分同胚变换后的几何相似性，来评估DNO在不同域上的泛化性能。研究在达西流（Darcy flow）、管流、翼型流和力学问题上开展了实验，其中二维和三维域分别采用调和参数化和体积参数化作为微分同胚方式。实验结果表明，DNO框架具备稳健的学习能力，在不同域和参数下均表现出优异的泛化性能。
英文摘要：
In scientific and engineering applications, solving partial differential equations (PDEs) across various parameters and domains normally relies on resource-intensive numerical methods. Neural operators based on deep learning offered a promising alternative to PDEs solving by directly learning physical laws from data. However, the current neural operator methods were limited to solve PDEs on fixed domains. Expanding neural operators to solve PDEs on various domains hold significant promise in medical imaging, engineering design and manufacturing applications, where geometric and parameter changes are essential. This paper presents a novel neural operator learning framework for solving PDEs with various domains and parameters defined for physical systems, named diffeomorphism neural operator (DNO). The main idea is that a neural operator learns in a generic domain which is diffeomorphically mapped from various physics domains expressed by the same PDE. In this way, the challenge of operator learning on various domains is transformed into operator learning on the generic domain. The generalization performance of DNO on different domains can be assessed by a proposed method which evaluates the geometric similarity between a new domain and the domains of training dataset after diffeomorphism. Experiments on Darcy flow, pipe flow, airfoil flow and mechanics were carried out, where harmonic and volume parameterization were used as the diffeomorphism for 2D and 3D domains. The DNO framework demonstrated robust learning capabilities and strong generalization performance across various domains and parameters.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
