提出神经流算子框架，统一构建残差与普通网络结构。
Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approximations
- 设计连续深度的神经流模型，支持组合与分离结构。
- 首次证明无限维空间间流模型的通用逼近性，涵盖卷积型结构。
- 离散化后可还原ResNet与普通网络，适用于神经网络和算子学习。
我们提出一种抽象的神经流框架，用于神经网络和神经算子。该框架包含两种连续深度模型：具有组合与分离结构的神经流，覆盖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，包括目前已知首个在无限维空间之间基于流模型的通用逼近结果。同时，还获得了卷积神经流模型的通用逼近性。通过合适的时域离散化，组合结构恢复为类似ResNet的架构，而分离结构经分裂离散化后得到标准前馈架构。这为全连接或卷积线性层的神经网络和神经算子提供了一条统一的流基路径，涵盖残差与普通架构。
原文摘要 · Abstract (English)
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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