## 1 Introduction  
Nonlinear partial differential equations are often used to describe mathematical models of physical, mechanical and engineering problems. However, the complexity of the real world and the limitations of people's cognition, the mathematical models and numerical calculation processes of partial differential equations have various sources of uncertainties, such as uncertain parameters, initial and boundary conditions and a variety of model forms in the same process [1]. In recent years, uncertainty quantification [2, 3] has become an emerging discipline in the aerospace and defense security industries, and it has attracted great attention in many cutting-edge areas, including academia and engineering applications. Among the common uncertainty quantification methods, the Monte Carlo (MC) method is simple, easy to implement and robust [4]. However, the convergence speed of MC method is slow, especially in the application for the uncertainty analysis of high-dimensional problems of numerical methods for partial differential equations (e.g., finite element, finite difference and the Laplace adomain decomposition method [5]), which limits the application in some uncertainty quantification problems. [PERSON] et al. [6] proposed a type of MC variance reduction technique, which ameliorates the error partially. However, there is no comprehensive consideration to reduce error, improve convergence speed and optimize the computational cost. [PERSON] and [PERSON] [7] first proposed a multi-fidelity Monte Carlo method (MFMC) to estimate the mean using the control variate technique. They used efficient low-fidelity models to reduce the computational cost of high-fidelity models in uncertain optimization. Moreover, [PERSON] et al. [8, 9, 10, 11] extended the multi-fidelity method by using a large number of low-fidelity models and optimized the model allocation according to its corresponding authenticity and cost. In addition, [PERSON] et al. [12] proposed a hybrid multi-fidelity method under a reasonable calculation cost, which improved the accuracy of the uncertainty quantification of cardiovascular dynamics. Recently, uncertainty quantification based on multi-fidelity Monte Carlo method has been successfully applied in many fields, as can be seen in [13; 14; 15; 16; 17; 18; 19]. In this work, we aim to extend the application of MFMC to the field of nonlinear partial differential equations, such as the nonlinear Schrodinger (NLS) equation and Burges' equation, using some of the latest data-driven low-fidelity models, to improve the computational efficiency. As we know, in the framework of MFMC method, the high-fidelity models are selected according to the type and parameter characteristics of the actual problems, and the low-fidelity models are computationally cheap approximations of the underlying high-fidelity models. In recent years, several different types of low-fidelity models, for example, projection-based models [20; 21; 22], surrogate models [23], up-scaled models [24], have been applied to the MFMC framework. Here, for the NLS equation, inspired by Ref. [19], the proper orthogonal decomposition (POD) method [22; 25; 26] coupled with discrete empirical interpolation method (DEIM) [27; 28; 29; 30] is used to construct an optimal low fidelity model, which can deal with the complexity of higher order nonlinear terms of the NLS equation. Otherwise, for Burgers' equation, we attempt to construct another kind of the low-fidelity model by using the operator reduction (OR) method [31; 32; 33], combined with DEIM for the parameterized Burgers' equation. In particular, the quality of MFMC approach based on data-driven low-fidelity models is determined by its Pearson correlation coefficient relative to the high-fidelity model. Even if the accuracy of the data-driven low-fidelity models cannot be guaranteed, our MFMC estimator is unbiased for the output statistics of the high-fidelity models. This paper is organized as follows. In Section 2, the MFMC algorithm and how to construct low-fidelity models are introduced.