  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.
