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

TL;DR

Neural-POD introduces a neural operator framework that learns resolution-invariant, nonlinear orthogonal basis functions directly in function space, enhancing transferability and interpretability in scientific modeling.

Contribution

It presents a novel plug-and-play neural operator that learns continuous, resolution-invariant bases for reduced order modeling, improving generalization and interpretability over traditional methods.

Findings

Outperforms traditional SVD-based modes in transferability

Supports training under various task-specific norms

Demonstrates effectiveness on Burgers' and Navier-Stokes equations

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.