DeepM &Mnet：基于算子逼近的电对流多物理场神经网络推断
DeepM&Mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks
摘要：
电对流是一个涉及流场与电场以及阴阳离子浓度场耦合的多物理场问题。对于较小的德拜长度，会产生非常陡峭的边界层，但标准的数值方法可以相当精确地模拟不同的区域。在这里，我们使用电对流作为基准问题，提出了一个新的数据同化框架，DeepM&Mnet，用于模拟多物理场和多尺度问题，速度比使用预训练神经网络（NN）的标准数值方法快得多。我们首先预训练DeepONets，它可以独立地预测每个字段，给定耦合系统其余字段的一般输入。DeepONets可以近似非线性算子，由两个子网络组成，一个用于输入字段的分支网络和一个用于输出字段位置的主干网络。DeepONets速度极快，被用作DeepM&Mnet中的构建块，并形成多物理场解决方案的约束，以及任何场的一些稀疏可用测量。我们展示了新的方法，并记录了每个单独的DeepONet的准确性，随后我们提出了两种不同的DeepM&Mnet架构，可以准确有效地推断看不见的电势的二维电对流场。DeepM&Mnet框架是通用的，可以应用于以即插即用模式使用预训练的DeepONets基于很少的测量来构建任何复杂的多物理场和多尺度模型。
英文摘要：
Electroconvection is a multiphysics problem involving coupling of the flow field with the electric field as well as the cation and anion concentration fields. For small Debye lengths, very steep boundary layers are developed, but standard numerical methods can simulate the different regimes quite accurately. Here, we use electroconvection as a benchmark problem to put forward a new data assimilation framework, the DeepM&Mnet, for simulating multiphysics and multiscale problems at speeds much faster than standard numerical methods using pre-trained neural networks (NNs). We first pre-train DeepONets that can predict independently each field, given general inputs from the rest of the fields of the coupled system. DeepONets can approximate nonlinear operators and are composed of two sub-networks, a branch net for the input fields and a trunk net for the locations of the output field. DeepONets, which are extremely fast, are used as building blocks in the DeepM&Mnet and form constraints for the multiphysics solution along with some sparse available measurements of any of the fields. We demonstrate the new methodology and document the accuracy of each individual DeepONet, and subsequently we present two different DeepM&Mnet architectures that infer accurately and efficiently 2D electroconvection fields for unseen electric potentials. The DeepM&Mnet framework is general and can be applied for building any complex multiphysics and multiscale models based on very few measurements using pre-trained DeepONets in a plug-and-play mode.
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