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Machine-learning (ML) based discretization has been developed to simulatecomplex partial differential equations (PDEs) with tremendous success acrossvarious fields. These learned PDE solvers can effectively resolve theunderlying solution structures of interest and achieve a level of accuracywhich often requires an order-of-magnitude finer grid for a conventionalnumerical method using polynomial-based approximations. In a previous work in[13], we introduced a learned finite volume discretization that furtherincorporates the semi-Lagrangian (SL) mechanism, enabling larger CFL numbersfor stability. However, the efficiency and effectiveness of such methodologyheavily rely on the availability of abundant high-resolution training data,which can be prohibitively expensive to obtain. To address this challenge, inthis paper, we propose a novel multi-fidelity ML-based SL method for transportequations. This method leverages a combination of a small amount ofhigh-fidelity data and sufficient but cheaper low-fidelity data. The approachis designed based on a composite convolutional neural network architecture thatexplore the inherent correlation between high-fidelity and low-fidelity data.The proposed method demonstrates the capability to achieve a reasonable levelof accuracy, particularly in scenarios where a single-fidelity model fails togeneralize effectively. We further extend the method to the nonlinearVlasov-Poisson system by employing high order Runge-Kutta exponentialintegrators. A collection of numerical tests are provided to validate theefficiency and accuracy of the proposed method.

A multi-fidelity machine learning based semi-Lagrangian finite volume scheme for linear transport equations and the nonlinear Vlasov-Poisson system