Conservative semi-lagrangian finite difference scheme for transport simulations using graph neural networks

TL;DR

The paper addresses the challenge of creating genuinely multidimensional, mass-conserving semi-Lagrangian finite-difference schemes for transport equations, enabling extra-large time steps. It introduces a data-driven, end-to-end GNN-based SL FD framework with a dynamical graph that tracks upstream points and learns the SL discretization while enforcing exact mass conservation via a constraint layer. The encoder-processor-decoder architecture, particularly the dynamical graph processor, allows local interpolation without explicit upstream tracing, improving efficiency and accuracy over traditional methods and simplifying implementation. The approach is extended to the nonlinear Vlasov-Poisson system using Runge-Kutta exponential integrators, and numerical results in 1D/2D transport and VP demonstrate sharp feature resolution, mass conservation, and favorable time-step capabilities, with competitive performance relative to high-resolution finite-difference schemes.

Abstract

Semi-Lagrangian (SL) schemes are highly efficient for simulating transport equations and are widely used across various applications. Despite their success, designing genuinely multi-dimensional and conservative SL schemes remains a significant challenge. Building on our previous work [Chen et al., J. Comput. Phys., V490 112329, (2023)], we introduce a conservative machine-learning-based SL finite difference (FD) method that allows for extra-large time step evolution. At the core of our approach is a novel dynamical graph neural network designed to handle the complexities associated with tracking accurately upstream points along characteristics. This proposed neural transport solver learns the conservative SL FD discretization directly from data, improving accuracy and efficiency compared to traditional numerical schemes, while significantly simplifying algorithm implementation. We validate the method' s effectiveness and efficiency through numerical tests on benchmark transport equations in both one and two dimensions, as well as the nonlinear Vlasov-Poisson system.

Figures (25)

• Figure 1: Schematic illustration of the 1D SL FD scheme. \newlabelfig:slfv0
• Figure 1: Illustration of the proposed GNN-based SL FD method.
• Figure 1: Three test samples for square waves in Example \ref{['square']}. CFL=10.2.
• Figure 2: Schematic illustration of the process of constructing the edges in 1D case. With local stencil $\mathcal{N}_{in}(i)=\{I_{i-1},I_i\}$ to update $U_i^{m+1}$, two edges $e_{i-1,i}$ and $e_{i,i}$ are created in the edge set $\mathcal{E}$.
• Figure 2: Example \ref{['square']}. Time histories of the deviation of total mass for three test samples with CFL = 10.2 (a), and time histories of errors with different CFLs (b).

Theorems & Definitions (8)

• Example 4.1
• Example 4.2
• Example 4.3
• Example 4.4
• Example 4.5
• Example 4.6
• Example 4.7
• Example 4.8