Data-driven Multi-Fidelity Modelling for Time-dependent Partial Differential Equations using Convolutional Neural Networks

We present a general multi-fidelity (MF) framework which is applied through utilizing flexible-order explicit finite difference numerical schemes using convolutional neural networks (CNNs) by combining low-order simulation data with higher order simulation data obtained from numerical simulations based on partial differential equations (PDEs). This allows for improving the performance of low-order numerical simulation through learning from the data how to correct the numerical schemes to achieve improved accuracy. Through the lens of numerical analysis we evaluate the accuracy, efficiency and generalizability of constructed data-driven MF-models. To illustrate the concept, the construction of the MF models uses CNNs and is evaluated for numerical schemes designed for solving linear PDEs; the heat, the linear advection equation and linearized 1D shallow water equations. The numerical schemes allow for a high level of explainability of data-driven correction terms obtained via CNNs through numerical analysis of truncation errors. It is demonstrated that data-driven MF models is a means to improve the accuracy of LF models through operator correction.