用神经网络学函数空间的正交基，让科学计算模型跨分辨率通用。
Neural-POD: A Plug-and-Play Neural Operator Framework for Infinite-Dimensional Functional Nonlinear Proper Orthogonal Decomposition
- 通过残差最小化学习连续、不变的非线性正交基，替代传统SVD方法。
- 在伯格斯和纳维-斯托克斯方程上实现跨参数分布的泛化能力提升。
- 可插拔复用，适合需高精度降维的物理模拟与科学人工智能任务。
AI for science 模型常受离散化限制：学习到的表示依赖训练网格，难以跨分辨率、求解器和应用迁移。本文提出 Neural-POD，一种即插即用的神经算子，直接在函数空间中学习非线性、正交的基函数，可集成于基于投影的降阶模型及 DeepONet 等算子学习框架。Neural-POD 以连续、分辨率无关的基替代 SVD 得到的、依赖分辨率的线性模式，通过逐次残差最小化实现，类似格拉姆-施密特正交化。该框架支持在任务特定范数（如 $L^2$、$L^1$）下训练，提升对未见参数区间的分布外泛化能力，并捕捉复杂系统的非线性结构。所学基函数具备可解释性与可复用性，可作为 AI4Science 工作流的通用表征模块。我们在伯格斯方程和纳维-斯托克斯方程上验证了 Neural-POD 的有效性。
原文摘要 · Abstract (English)
AI for science (AI4Science) models often suffer from discretization: learned representations remain tied to the training grid, limiting transfer across resolutions, solvers and applications. We introduce Neural Proper Orthogonal Decomposition (Neural-POD), a plug-and-play neural operator that learns nonlinear, orthogonal basis functions directly in function space and can be integrated in both projection-based reduced order models and operator-learning frameworks such as DeepONet. Neural-POD replaces SVD-derived, resolution-dependent linear modes with continuous, resolution-invariant bases learned via sequential residual minimization, analogous to Gram-Schmidt orthogonalization. The framework supports training under task-specific norms (e.g., $L^2$, $L^1$), improves out-of-distribution generalization to unseen parameter regimes, and captures nonlinear structure in complex systems. Because the learned bases are interpretable and reusable, Neural-POD serves as a general representation module for AI4Science workflows. We demonstrate Neural-POD on Burgers' and Navier-Stokes equations.
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