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

Changhong Mou; Binghang Lu; Guang Lin

physics.comp-phFeb 17, 2026

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

Authors: Changhong Mou, Binghang Lu, Guang Lin

Organizations: Department of Mathematics and Statistics, Utah State University, 3900 Old Main Hill, Logan, UT, 84322, USA. · School of Electrical and Computer Engineering, Purdue University, 610 Purdue Mall, West Lafayette, IN, 47907, USA. · Department of Mathematics, Purdue University, 610 Purdue Mall, West Lafayette, IN, 47907, USA. · School of Mechanical Engineering, Purdue University, 610 Purdue Mall, West Lafayette, IN, 47907, USA.

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1: Neural-POD for resolution-independent model reduction and operator learning.

Panel of Figure 2 — Figure 2: Neural-POD integrations for reduced order modeling and operator learning.

Panel of Figure 3 — Figure 3: Burgers equations: (a) solution snapshots for different viscosities $\nu$ ; (b) traditional POD modes versus Neural-POD bases learned with $L^{1}$ and $L^{2}$ losses.

Panel of Figure 4 — Figure 4: Summary of Burgers Neural-POD results: (a) training residual decay versus mode index for $L^{2}$ and $L^{1}$ losses; (b–c) reconstruction residuals under $L^{2}$ and $L^{1}$ metrics across representative viscosities; (d–e) in-distribution and out-of-distribution comparisons of POD (with extra snapshots) and Neural-POD for an unseen viscosity $\nu$ .

Panel of Figure 6 — Figure 6: (a–b) Geometry and sampling points of the two-dimensional flow past a cylinder problem. (c–d) First two modes of POD and Neural-POD for two-dimensional flow past the cylinder with $Re=50$ .

$\nu$	Model	$r=2$	$r=4$	$r=6$
0.002	POD–ROM	2.10e-01	1.66e-01	1.76e-01
0.002	Neural–POD–ROM ( $L^{2}$ )	2.11e-01	1.67e-01	1.77e-01
0.0025	POD–ROM	No in distribution ability
0.0025	Neural–POD–ROM ( $L^{2}$ )	9.61e-02	1.18e-01	1.35e-01
0.005	POD–ROM	No out of distribution ability
0.005	Neural–POD–ROM ( $L^{2}$ )	1.91e-01	1.21e-01	8.46e-02

Table 1:

L^{2}

errors of POD–ROM and Neural–POD–ROM for the one-dimensional Burgers equation. At

\nu=0.002

, training snapshots are available for both models. At

\nu=0.005

, Neural–POD–ROM is evaluated in a prediction setting without training snapshots.

Figure 1: First three Neural-POD modes for different viscosity values

Metric	Neural-POD-DeepONet	POD-DeepONet
Training Loss	$3.47\times 10^{-3}$	$4.10\times 10^{-3}$
Testing $L_{2}$ Error	$3.40\times 10^{-3}$	$4.37\times 10^{-3}$
Testing $L_{1}$ Error	$3.16\times 10^{-2}$	$3.59\times 10^{-2}$

Table 1:

L^{2}

and

L^{1}

error comparison between Neural-POD-DeepONet and POD-DeepONet models trained on 64 spatial points with 100,000 training iterations.

Proper Orthogonal Decomposition

Proper Orthogonal Decomposition (POD) extracts dominant orthogonal modes from data to build compact representations and reduced-order models of complex systems.

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Neural operators (NOs) are designed to learn maps between infinite-dimensional function spaces. We propose a novel reframing of their use. By introducing an auxiliary base-space, any finite-dimensional function can be viewed as an operator acting by composition on functions of the base-space. Through a range of benchmarks on analytic functions of increasing complexity and dimensionality, we demonstrate that NOs can match or outperform standard multilayer perceptrons and Kolmogorov--Arnold Networks in accuracy while requiring significantly fewer parameters and training time. As a real-world application, we apply a two-dimensional Tensorized Fourier Neural Operator (TFNO) to the nuclear chart, learning a correction to state-of-the-art nuclear mass models as a partially observed residual field. A TFNO ensemble reaches a held-out root-mean-square error of 198.2 keV, placing it among the best recent neural-network approaches while retaining high parameter efficiency and short training times. More broadly, these results introduce NOs as a scalable framework for finite-dimensional function interpolation, from analytic benchmarks to structured scientific data.

Abstract

Proper Orthogonal Decomposition

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