The rapid development of AI for Science is often hindered by the "discretization", where learned representations remain restricted to the specific grids or resolutions used during training. We propose the Neural Proper Orthogonal Decomposition (Neural-POD), a plug-and-play neural operator framework that constructs nonlinear, orthogonal basis functions in infinite-dimensional space using neural networks. Unlike the classical Proper Orthogonal Decomposition (POD), which is limited to linear subspace approximations obtained through singular value decomposition (SVD), Neural-POD formulates basis construction as a sequence of residual minimization problems solved through neural network training. Each basis function is obtained by learning to represent the remaining structure in the data, following a process analogous to Gram--Schmidt orthogonalization. This neural formulation introduces several key advantages over classical POD: it enables optimization in arbitrary norms (e.g., $L^2$, $L^1$), learns mappings between infinite-dimensional function spaces that is resolution-invariant, generalizes effectively to unseen parameter regimes, and inherently captures nonlinear structures in complex spatiotemporal systems. The resulting basis functions are interpretable, reusable, and enabling integration into both reduced order modeling (ROM) and operator learning frameworks such as deep operator learning (DeepONet). We demonstrate the robustness of Neural-POD with different complex spatiotemporal systems, including the Burgers' and Navier-Stokes equations. We further show that Neural-POD serves as a high performance, plug-and-play bridge between classical Galerkin projection and operator learning that enables consistent integration with both projection-based reduced order models and DeepONet frameworks.

翻译：人工智能在科学领域的快速发展常受限于“离散化”问题，即学习到的表示仍局限于训练时使用的特定网格或分辨率。我们提出神经本征正交分解（Neural-POD），一种即插即用的神经算子框架，利用神经网络在无限维空间中构建非线性正交基函数。与经典本征正交分解（POD）仅限于通过奇异值分解（SVD）获得的线性子空间逼近不同，Neural-POD将基函数构建表述为一系列通过神经网络训练求解的残差最小化问题。每个基函数通过学习表示数据中的剩余结构获得，其过程类似于格拉姆-施密特正交化。相较于经典POD，这种神经表述引入若干关键优势：支持任意范数（如$L^2$、$L^1$）下的优化，学习分辨率无关的无限维函数空间映射，对未见参数体系具有良好泛化能力，并能固有地捕捉复杂时空系统中的非线性结构。所得基函数具备可解释性、可复用性，并能集成至降阶建模（ROM）与算子学习框架（如深度算子网络DeepONet）。我们通过伯格斯方程和纳维-斯托克斯方程等复杂时空系统验证了Neural-POD的鲁棒性。进一步研究表明，Neural-POD可作为经典伽辽金投影与算子学习之间的高性能即插即用桥梁，实现与基于投影的降阶模型及DeepONet框架的无缝集成。