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

TL;DR

This work tackles high-fidelity data scarcity in operator learning for parametric PDEs by introducing multi-fidelity Laplace Neural Operators (MF-LNOs) that couple a low-fidelity LF model with parallel HF linear and nonlinear corrections under a dynamic inter-fidelity weighting. It integrates a pole-residue based Laplace neural operator with a two-phase training scheme and employs replica-exchange stochastic gradient Langevin dynamics (reSGLD) to obtain robust uncertainty quantification. Across four canonical dynamical systems—the Lorenz system, Duffing oscillator, Burgers equation, and Brusselator—the MF-LNO framework achieves 40%–80% reductions in testing losses compared with baselines, while providing calibrated predictive intervals. The approach enables data-efficient surrogate modeling for parametric PDEs and holds promise for applications in scientific digital twins and multi-physics design, with future work focusing on active HF data selection and automated hyper-parameter tuning.

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.

Figures (11)

• Figure 1: Schematic overview of the MF-LNO framework (adapted from cao2024laplace), which showcases the integration of LF and HF modeling to achieve efficient MF modeling. The framework begins with an LNO to approximate the mapping from LF inputs to LF outputs. Next, the MF enhancement involves two parallel LNOs. The linear LNO models the linear relationships between LF and HF data without nonlinear activation functions $\sigma(\cdot)$, while the nonlinear LNO incorporates non-linear corrections through activation functions. Both linear and non-linear LNOs utilize similar architecture, with the addition of trainable weight parameters $\alpha$ to dynamically adjust the contributions from linear and nonlinear operators. Each LNO employs a series of Laplace layers, which leverages kernel convolution and pole-residue parametrization to transform inputs into a feature space optimized for MF approximation. Uncertainty estimates are generated by combining predictions from multiple ensemble models.
• Figure 2: Sample Trajectory of replica exchange stochastic gradient Langevin dynamics (adapted from deng2020nonzheng2025exploring). Yellow lines denote the trajectory of a low-temperature chain (exploitation), and red lines represent a high-temperature chain (exploration). The chains exchange states according to a swap mechanism. The empirical distribution formed by the yellow trajectory aids in uncertainty quantification.
• Figure 3: Ground truth (HF data) of the Lorenz system, LF data (with linear correlations), and the approximation yielded by MF-LNO. The x-axis denotes the time stamp, and the y-axis represents the corresponding responses given different forced functions $f(t)$. The red stars are the ground truth, blue dots are LF data, black solid lines are responses approximated by MF-LNO, and the gray shaded areas denote 95% confidence intervals.
• Figure 4: Approximation of the Lorenz system response $u(t)$ from MF data. Red stars are ground truth, black solid lines are approximated by MF-LNO, blue dashed lines are given by a single LNO with HF data, green dashed lines are yielded by a single LNO with LF, and the yellow dashed lines are a single LNO with MF data.
• Figure 5: Ground truth (HF data) of the Lorenz system, LF data (with non-linear correlations), and the approximation yielded by MF-LNO. The x-axis denotes the time stamp, and the y-axis represents the corresponding responses given different forced functions $f(t)$. The red stars are the ground truth, blue dots are LF data, black solid lines are response approximated by MF-LNO, and the gray shaded areas denote 95% confidence intervals.

Theorems & Definitions (2)

• Remark 2.1
• Remark 2.2