RONOM: Reduced-Order Neural Operator Modeling

TL;DR

RONOM combines reduced-order modeling and neural operators to create a flexible, low-dimensional surrogate for PDEs, offering error bounds, improved discretization robustness, and superior super-resolution capabilities.

Contribution

It introduces RONOM, a novel framework that integrates ROM and neural operator concepts, providing error bounds and enhanced discretization robustness for PDE modeling.

Findings

RONOM achieves comparable input generalization to existing neural operators.

RONOM outperforms in spatial super-resolution tasks.

RONOM demonstrates superior discretization robustness.

Abstract

Time-dependent partial differential equations are ubiquitous in physics-based modeling, but they remain computationally intensive in many-query scenarios, such as real-time forecasting, optimal control, and uncertainty quantification. Reduced-order modeling (ROM) addresses these challenges by constructing a low-dimensional surrogate model but relies on a fixed discretization, which limits flexibility across varying meshes during evaluation. Operator learning approaches, such as neural operators, offer an alternative by parameterizing mappings between infinite-dimensional function spaces, enabling adaptation to data across different resolutions. Whereas ROM provides rigorous numerical error estimates, neural operator learning largely focuses on discretization convergence and invariance without quantifying the error between the infinite-dimensional and the discretized operators. This work introduces the reduced-order neural operator modeling (RONOM) framework, which bridges concepts from ROM and operator learning. We establish a discretization error bound analogous to those in ROM, and get insights into RONOM's discretization convergence and discretization robustness. Moreover, three numerical examples are presented that compare RONOM to existing neural operators for solving partial differential equations. The results demonstrate that RONOM using standard vector-to-vector neural networks can achieve comparable performance in input generalization and achieves superior performance in both spatial super-resolution and discretization robustness, while also offering novel insights into temporal super-resolution scenarios and ROM-based approaches for learning on time-dependent data.