Machine-learning (ML) based discretization has been developed to simulate complex partial differential equations (PDEs) with tremendous success across various fields. These learned PDE solvers can effectively resolve the underlying solution structures of interest and achieve a level of accuracy which often requires an order-of-magnitude finer grid for a conventional numerical method using polynomial-based approximations. In a previous work in [13], we introduced a learned finite volume discretization that further incorporates the semi-Lagrangian (SL) mechanism, enabling larger CFL numbers for stability. However, the efficiency and effectiveness of such methodology heavily rely on the availability of abundant high-resolution training data, which can be prohibitively expensive to obtain. To address this challenge, in this paper, we propose a novel multi-fidelity ML-based SL method for transport equations. This method leverages a combination of a small amount of high-fidelity data and sufficient but cheaper low-fidelity data. The approach is designed based on a composite convolutional neural network architecture that explore the inherent correlation between high-fidelity and low-fidelity data. The proposed method demonstrates the capability to achieve a reasonable level of accuracy, particularly in scenarios where a single-fidelity model fails to generalize effectively. We further extend the method to the nonlinear Vlasov-Poisson system by employing high order Runge-Kutta exponential integrators. A collection of numerical tests are provided to validate the efficiency and accuracy of the proposed method.

翻译：基于机器学习的离散化方法已成功应用于模拟各类复杂偏微分方程，在诸多领域取得了显著成效。这类学习型求解器能够有效捕捉解的结构特征，在达到同等精度时，其所需的网格分辨率往往比传统多项式近似数值方法低一个量级。在先前工作[13]中，我们提出了一种融合半拉格朗日机制的卷积有限体积离散格式，该格式允许更大的CFL数以保障稳定性。然而，该方法的高效性与有效性高度依赖充足的高分辨率训练数据，而获取此类数据的成本往往难以承受。针对这一挑战，本文提出了一种新颖的基于多保真机器学习的半拉格朗日输运方程求解方法。该方法结合了少量高保真数据与充足但低价的低保真数据，基于复合卷积神经网络架构设计，以挖掘高保真与低保真数据间的内在关联。所提方法在单保真模型难以有效泛化的场景下，仍能实现合理的精度水平。我们进一步通过高阶龙格-库塔指数积分器将该方法拓展至非线性Vlasov-Poisson系统。一系列数值实验验证了所提方法的效率与精度。