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RONOM: Reduced-Order Neural Operator Modeling

Sven Dummer, Dongwei Ye, Christoph Brune

TL;DR

RONOM addresses the inefficiency of time-dependent PDE surrogates by uniting reduced-order modeling with neural operators. It introduces a discretization-robust, kernel-informed encoder, a neural-ODE latent flow, and a kernel-based decoder to learn infinite-dimensional solution operators while maintaining discretization convergence. The paper proves a discretization error bound and demonstrates, through Burgers', Wave', and Navier–Stokes experiments, that RONOM offers competitive input generalization to existing NOs while delivering superior spatial super-resolution and robustness to discretization changes. These findings illuminate the trade-offs between ROM biases and neural operator flexibility and point to future avenues in physics-informed and structure-preserving extensions.

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.

RONOM: Reduced-Order Neural Operator Modeling

TL;DR

RONOM addresses the inefficiency of time-dependent PDE surrogates by uniting reduced-order modeling with neural operators. It introduces a discretization-robust, kernel-informed encoder, a neural-ODE latent flow, and a kernel-based decoder to learn infinite-dimensional solution operators while maintaining discretization convergence. The paper proves a discretization error bound and demonstrates, through Burgers', Wave', and Navier–Stokes experiments, that RONOM offers competitive input generalization to existing NOs while delivering superior spatial super-resolution and robustness to discretization changes. These findings illuminate the trade-offs between ROM biases and neural operator flexibility and point to future avenues in physics-informed and structure-preserving extensions.

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.

Paper Structure

This paper contains 29 sections, 8 theorems, 56 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The reduced-order neural operator model
  3. Problem statement
  4. Overview of RONOM
  5. Encoder
  6. Discretized projection
  7. Error estimate projection's basis coefficients
  8. Integral discretization error estimates
  9. Neural ODE flow operator
  10. Decoder
  11. RONOM error estimates
  12. An a-posteriori error estimate
  13. Discretization error estimate
  14. Numerical experiments
  15. Data generation
  16. ...and 14 more sections

Key Result

Theorem 2.2

Let $\tilde{\phi}_i$ be an orthonormal basis of the Hilbert Space $(\mathcal{V}, \braket{\cdot}{\cdot}_{\mathcal{V}})$ with $\mathcal{V}\coloneqq \mathop{\mathrm{span}}\nolimits\{\phi_1, \phi_2, \cdots, \phi_{N_b} \}$. Define $K\colon \Omega \times \Omega \to \mathbb{R}$ as $K(\bm{x},\bm{y}) = \sum_

Figures (9)

  • Figure 1: RONOM is a modeling framework that combines neural operators and reduced order modeling. It first maps the initial condition into a latent space. From this initial latent representation, a latent trajectory is obtained over time. The full trajectory of functions is recovered by decoding the latent codes at each time instance.
  • Figure 1: Input generalization (Burgers' equation). Visualizations of FNO, CNO, DeepONet, and RONOM model predictions over time on the Burgers' equation. The ground truth (GT) solution and the corresponding discrepancy between GT and model prediction are also demonstrated for comparison.
  • Figure 2: Robustness and superresolution through RONOM. Traditional finite-dimensional ROM methods for PDE surrogate modeling can generalize to new inputs, provided these input functions use the same fixed discretization as in training (black arrow). In contrast, RONOM can also enhance the resolution beyond the training resolution (blue arrow), and it remains applicable and robust when inputs are given at different discretizations than those used in training (green arrows).
  • Figure 2: Input generalization (wave equation). Visualization of prediction errors for FNO, CNO, DeepONet, and RONOM, shown alongside the ground truth (GT).
  • Figure 3: Input generalization (Navier-Stokes). Visualization of prediction errors FNO, CNO, DeepONet, and RONOM, shown alongside the ground truth (GT).
  • ...and 4 more figures

Theorems & Definitions (24)

  • Remark 2.1
  • Theorem 2.2
  • Proof 1
  • Remark 2.3
  • Theorem 2.4
  • Proof 2
  • Theorem 2.5
  • Remark 2.6
  • Theorem 2.7
  • Proof 3
  • ...and 14 more