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.
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., L2, L1), 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
Figure 1: Neural-POD for resolution-independent model reduction and operator learning.
Figure 2: Neural-POD integrations for reduced order modeling and operator learning.
Figure 3: Burgers equations: (a) solution snapshots for different viscosities ν ; (b) traditional POD modes versus Neural-POD bases learned with L1 and L2 losses.
Figure 4: Summary of Burgers Neural-POD results: (a) training residual decay versus mode index for L2 and L1 losses; (b–c) reconstruction residuals under L2 and L1 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 ν .
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 .
ν
Model
r=2
r=4
r=6
0.002
POD–ROM
2.10e-01
1.66e-01
1.76e-01
Neural–POD–ROM ( L2 )
2.11e-01
1.67e-01
1.77e-01
0.0025
POD–ROM
No in distribution ability
Neural–POD–ROM ( L2 )
9.61e-02
1.18e-01
1.35e-01
0.005
POD–ROM
No out of distribution ability
Neural–POD–ROM ( L2 )
1.91e-01
1.21e-01
8.46e-02
Table 1: L2 errors of POD–ROM and Neural–POD–ROM for the one-dimensional Burgers equation. At ν=0.002 , training snapshots are available for both models. At ν=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
Figure 2: Reconstruction of spatiotemporal fields for the one-dimensional Burgers’ equation at an unseen viscosity parameter using Neural-POD with different latent dimensions (2 and 4).
Figure 3: Comparison of POD and Neural-POD on the 1D Burgers equation for different viscosity values: (a) classical POD , (b) Neural-POD trained with L2 loss, (c) Neural-POD trained with L1 loss.
Metric
Neural-POD-DeepONet
POD-DeepONet
Training Loss
3.47×10−3
4.10×10−3
Testing L2 Error
3.40×10−3
4.37×10−3
Testing L1 Error
3.16×10−2
3.59×10−2
Table 1: L2 and L1 error comparison between Neural-POD-DeepONet and POD-DeepONet models trained on 64 spatial points with 100,000 training iterations.
Figure 4: Resolution-independent Neural-POD in DeepONet: A comparison of L1 and L2 errors for Neural-POD-DeepOnet versus POD-DeepOnet for different initial conditions and training resolutions.
Proper Orthogonal Decomposition (POD) extracts dominant orthogonal modes from data to build compact representations and reduced-order models of complex systems.
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.
Department of Physics, CCTP and ITCP, University of Crete, 71303, Greece · Theoretical Physics Department, CERN, Geneva, Switzerland · Physik-Institut, Universität Zürich, 8057 Zürich, Switzerland
Recent advances in scientific machine learning (SciML) have enabled neural operators (NOs) to serve as powerful surrogates for modeling the dynamic evolution of physical systems governed by partial differential equations (PDEs). While existing approaches focus primarily on learning simulations from the target PDE, they often overlook more fundamental physical principles underlying these equations. Inspired by how numerical solvers are compatible with simulations of different settings of PDEs, we propose a multiphysics training framework that jointly learns from both the original PDEs and their simplified basic forms. Our framework enhances data efficiency, reduces predictive errors, and improves out-of-distribution (OOD) generalization, particularly in scenarios involving shifts of physical parameters and synthetic-to-real transfer. Our method is architecture-agnostic and demonstrates consistent improvements in normalized root mean square error (nRMSE) across a wide range of 1D/2D/3D PDE problems. Through extensive experiments, we show that explicit incorporation of fundamental physics knowledge significantly strengthens the generalization ability of neural operators. We will release models and codes at https://sites.google.com/view/sciml-fundemental-pde.
Siying Ma, Mehrdad M. Zadeh, Mauricio Soroco +3
Simon Fraser University · Georgia Institute of Technology
Neural operator learning directly constructs the mapping relationship from the equation parameter space to the solution space, enabling efficient direct inference in practical applications without the need for repeated solution of partial differential equations (PDEs) -- an advantage that is difficult to achieve with traditional numerical methods. In this work, we find that explicitly decoupling linear and nonlinear effects within such operator mappings leads to improved learning efficiency. This yields a novel network structure, namely the Linear-Nonlinear Fusion Neural Operator (LNF-NO), which models operator mappings via the multiplicative fusion of a linear component and a nonlinear component, thus achieving a lightweight and interpretable representation. This linear-nonlinear decoupling enables efficient capture of complex solution features at the operator level while maintaining stability and generality. LNF-NO naturally supports multiple functional inputs and is applicable to both regular grids and irregular geometries. Across a diverse suite of PDE operator-learning benchmarks, including nonlinear Poisson-Boltzmann equations and multi-physics coupled systems, LNF-NO is typically substantially faster to train than several representative neural operator baselines, while achieving comparable or improved accuracy across most tested cases. On the tested 3D Poisson-Boltzmann case, LNF-NO achieves strong accuracy while requiring substantially less training time than the three-dimensional Fourier Neural Operator and Transolver baselines.
Heng Wu, Junjie Wang, Benzhuo Lu
SKLMS, ICMSEC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China · School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China