Muti-Fidelity Prediction and Uncertainty Quantification with Laplace Neural Operators for Parametric Partial Differential Equations

TL;DR

This paper introduces multi-fidelity Laplace Neural Operators (MF-LNOs) that efficiently combine low- and high-fidelity data for accurate, uncertainty-aware predictions of parametric PDEs, reducing data requirements and improving model reliability.

Contribution

The paper proposes a novel multi-fidelity framework for Laplace Neural Operators, integrating a correction mechanism and advanced sampling for better uncertainty quantification in parametric PDE modeling.

Findings

Achieved 40-80% reduction in testing loss compared to traditional methods.

Validated effectiveness across four canonical dynamical systems.

Demonstrated improved data efficiency and uncertainty quantification.

Abstract

Laplace Neural Operators (LNOs) have recently emerged as a promising approach in scientific machine learning due to the ability to learn nonlinear maps between functional spaces. However, this framework often requires substantial amounts of high-fidelity (HF) training data, which is often prohibitively expensive to acquire. To address this, we propose multi-fidelity Laplace Neural Operators (MF-LNOs), which combine a low-fidelity (LF) base model with parallel linear/nonlinear HF correctors and dynamic inter-fidelity weighting. This allows us to exploit correlations between LF and HF datasets and achieve accurate inference of quantities of interest even with sparse HF data. We further incorporate a modified replica exchange stochastic gradient Langevin algorithm, which enables a more effective posterior distribution estimation and uncertainty quantification in model predictions. Extensive validation across four canonical dynamical systems (the Lorenz system, Duffing oscillator, Burgers equation, and Brusselator reaction-diffusion system) demonstrates the framework's effectiveness. The results show significant improvements, with testing losses reduced by 40% to 80% compared to traditional approaches. This validates MF-LNO as a versatile tool for surrogate modeling in parametric PDEs, offering significant improvements in data efficiency and uncertainty-aware prediction.