神经流算子可以逼近任意算子：抽象框架与通用逼近
Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approximations
AI总结：
提出神经流抽象框架，涵盖组合与分离结构的连续深度模型，证明其在有限维和无限维空间中的通用逼近性质，并通过时间离散化统一残差与普通架构。
摘要：
我们为神经网络和神经算子引入了一个抽象的神经流框架。该框架包含两种连续深度模型，即具有组合和分离结构的神经流，并涵盖了有限维函数逼近和无限维算子逼近。我们证明了相应神经流的适定性和通用逼近性质，包括据我们所知，首个无限维空间之间基于流的模型的通用逼近结果。我们还获得了卷积神经流模型的通用逼近结果。通过适当的时间离散化，组合结构恢复了ResNet类型的架构，而分离结构通过基于分裂的离散化产生了普通架构。这为具有全连接或卷积线性层的神经网络和神经算子的残差和普通架构提供了一条统一的基于流的路径。
英文摘要：
We introduce an abstract neural flow framework for neural networks and neural operators. The framework contains two continuous-depth models, namely neural flows with composition and separation structures, and covers both finite-dimensional function approximation and infinite-dimensional operator approximation. We prove well-posedness and universal approximation properties for the corresponding neural flows, including, to the best of our knowledge, the first universal approximation result for flow-based models between infinite-dimensional spaces. We also obtain universal approximation results for convolutional neural flow models. Through suitable time discretizations, the composition structure recovers ResNet-type architectures, while the separation structure, via a splitting-based discretization, yields plain architectures. This gives a unified flow-based route to both residual and plain architectures for neural networks and neural operators with fully connected or convolutional linear layers.
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