将经典神经网络架构扩展到函数空间，实现对微分方程的高效学习。
Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning
- 提出通用原则，将常见神经网络转化为可处理函数空间的神经算子。
- 在多个物理系统上验证，能统一求解不同边界条件与系数的微分方程。
- 适合从事科学计算、物理建模的研究者快速部署神经算子方法。
大量科学问题，如连续时间动力系统和偏微分方程（PDEs），天然定义在函数空间中。尽管函数空间通常是无限维的，深度学习主要在计算机视觉和自然语言处理等有限维空间映射任务中取得成功。这种数据本质的差异限制了神经网络在科学应用中的表现。神经算子是一种将神经网络推广至函数空间映射的合理方法，为复制深度学习在科学领域的变革性影响提供了路径。例如，神经算子可学习整个类PDE的解算子，如具有不同边界条件、系数函数和几何结构的物理系统。深度学习的成功关键在于通过大量实验优化神经架构。将这些架构转化为神经算子，使算子学习也能享受相同的实证优化优势。然而，以往的神经算子架构多作为独立模型提出，未直接源于现有神经网络架构。本文识别并提炼出构建函数空间映射实用实现的关键原则，提出一套最小修改即可将多种主流神经架构转化为神经算子的方案。本文旨在指导实践者完成该转化过程，并详述具体实施步骤。代码见：https://github.com/neuraloperator/NNs-to-NOs。
原文摘要 · Abstract (English)
A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically infinite-dimensional, deep learning has predominantly advanced through applications in computer vision and natural language processing that focus on mappings between finite-dimensional spaces. Such fundamental disparities in the nature of the data have limited neural networks from achieving a comparable level of success in scientific applications as seen in other fields. Neural operators are a principled way to generalize neural networks to mappings between function spaces, offering a pathway to replicate deep learning's transformative impact on scientific problems. For instance, neural operators can learn solution operators for entire classes of PDEs, e.g., physical systems with different boundary conditions, coefficient functions, and geometries. A key factor in deep learning's success has been the careful engineering of neural architectures through extensive empirical testing. Translating these neural architectures into neural operators allows operator learning to enjoy these same empirical optimizations. However, prior neural operator architectures have often been introduced as standalone models, not directly derived as extensions of existing neural network architectures. In this paper, we identify and distill the key principles for constructing practical implementations of mappings between infinite-dimensional function spaces. Using these principles, we propose a recipe for converting several popular neural architectures into neural operators with minimal modifications. This paper aims to guide practitioners through this process and details the steps to make neural operators work in practice. Our code can be found at https://github.com/neuraloperator/NNs-to-NOs
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