标题： 基于变形的框架用于学习定义在变化域上的PDE的解映射 显示英文标题

标题： A deformation-based framework for learning solution mappings of PDEs defined on varying domains

摘要： 在本工作中，我们建立了一个基于变形的框架，用于学习定义在变化域上的偏微分方程（PDE）的解映射。根据变形，定义在变化域上的函数的并集可以被识别为一个度量空间，然后解映射被视为一个连续的度量到度量的映射，随后可以通过两种不同的策略表示为另一个连续的度量到巴拿赫空间的映射，分别称为D2D框架和D2E框架。我们指出，这种度量到巴拿赫空间的映射可以通过神经网络进行学习，因此解映射也随之被学习。通过这个框架，为学习定义在变化域上的PDE的解映射问题建立了严格的收敛性分析。由于该理论框架基于几个关键假设，这些假设需要针对给定的具体问题进行验证，因此我们以星形域作为典型例子进行研究，其他情况也可以类似地进行验证。该框架有三个重要特征：(1) 所考虑的域不需要是微分同构的，因此只要它们是同胚的，就可以由一个模型覆盖广泛的区域。(2) 变形映射不需要是连续的，因此可以通过结合主要的身份映射和局部变形映射来灵活建立。这种能力有助于解决只有几何局部部分发生变化的大规模系统。(3) 如果采用保持线性的神经算子，如MIONet，该框架仍然在其源项上保持对线性PDE的代理解映射的线性，因此可以应用于混合迭代方法。最后，我们进行了若干数值实验以验证我们的理论结果。 显示英文摘要

摘要： In this work, we establish a deformation-based framework for learning solution mappings of PDEs defined on varying domains. The union of functions defined on varying domains can be identified as a metric space according to the deformation, then the solution mapping is regarded as a continuous metric-to-metric mapping, and subsequently can be represented by another continuous metric-to-Banach mapping using two different strategies, referred to as the D2D framework and the D2E framework, respectively. We point out that such a metric-to-Banach mapping can be learned by neural networks, hence the solution mapping is accordingly learned. With this framework, a rigorous convergence analysis is built for the problem of learning solution mappings of PDEs on varying domains. As the theoretical framework holds based on several pivotal assumptions which need to be verified for a given specific problem, we study the star domains as a typical example, and other situations could be similarly verified. There are three important features of this framework: (1) The domains under consideration are not required to be diffeomorphic, therefore a wide range of regions can be covered by one model provided they are homeomorphic. (2) The deformation mapping is unnecessary to be continuous, thus it can be flexibly established via combining a primary identity mapping and a local deformation mapping. This capability facilitates the resolution of large systems where only local parts of the geometry undergo change. (3) If a linearity-preserving neural operator such as MIONet is adopted, this framework still preserves the linearity of the surrogate solution mapping on its source term for linear PDEs, thus it can be applied to the hybrid iterative method. We finally present several numerical experiments to validate our theoretical results.