Neural-POD: A Plug-and-Play Neural Operator Framework for Infinite-Dimensional Functional Nonlinear Proper Orthogonal Decomposition

TL;DR

Neural-POD generalizes classical POD by learning nonlinear, resolution-invariant basis functions in infinite-dimensional function spaces through residual-minimization with neural networks. It delivers a plug-and-play component for both Galerkin projection–based ROMs and operator learning frameworks (e.g., DeepONet), enabling accurate, parameter-robust, and resolution-flexible reduced representations. The approach supports optimization in norms such as $L^2$ and $L^1$, and demonstrates strong generalization to unseen parameter regimes (e.g., viscosity in Burgers and Navier–Stokes flows) while preserving interpretability and reusability. Numerical experiments on Burgers’ equation and 2D Navier–Stokes validate robustness, with Neural-POD–DeepONet showing improved performance at coarser resolutions and convergence to POD behavior at higher resolutions, highlighting its practical impact for real-time simulation, multi-query tasks, and open-source basis libraries.

Abstract

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.

Figures (16)

• Figure 1: Neural-POD connects classical projection-based model reduction and modern operator learning. By representing basis functions as neural networks, Neural-POD enables resolution-independent, parameter-aware ROMs and can be integrated with DeepONet operator learning frameworks.
• Figure 2: Overview of Neural-POD training. Snapshots $u(x,t,\kappa)$ are collected over time for varying parameter values $\kappa$ (upper left). The training architecture decomposes each snapshot into products of spatial mode functions $\Phi_i(x;\kappa)$ and temporal coefficients $\Psi_i(t;\kappa)$, which guarantees the orthogonality for all modes (bottom left). Once trained, the Neural-POD basis provides plug-and-play reduced representations that can be evaluated on any discretization $\hat{x}$ and parameter choice $\hat{\kappa}$ (right, $\hat{\kappa}$ may differ from the training snapshots’ parameter $\kappa$).
• Figure 3: Schematic diagram of the proposed Neural-POD-ROMs and traditional POD-based reduced order models (POD-ROMs). Top portion: the Neural-POD extension that enables plug-and-play operation on arbitrary meshes and at arbitrary resolution, informed by pretrained neural representations of POD bases and PDE parameters. Bottom portion: the traditional POD flowchart, including snapshot generation for parameter $\kappa$, computation of full POD modes via SVD, construction of the reduced basis, and Galerkin projection of the governing PDE.
• Figure 4: Schematic of the proposed Neural-POD Deep Operator Network (NP-DeepONet). The branch network takes initial condition fields as input, while the trunk network evaluates functions at spatial query points. A pretrained Neural-POD module provides reduced basis information ${\phi_1, \phi_2, \ldots, \phi_P}$ obtained from snapshot data. The outputs are combined to produce the operator evaluation $G_{\theta}(u_m)(x)$, enabling efficient and data-driven surrogate modeling of PDE solutions in reduced coordinates.
• Figure 5: Solution snapshots of the Burgers equation at different sampled viscosity values $\nu$ used for Neural-POD training.

Theorems & Definitions (1)

• Remark 1