A Physics-Augmented GraphGPS Framework for the Reconstruction of 3D Riemann Problems from Sparse Data

TL;DR

The paper tackles reconstructing 3D compressible flow fields from sparse measurements by introducing GraphGPS, a physics-informed, graph-based framework that fuses local message-passing with global context and positional encodings. It introduces a shock-aware MP variant (SA-GATConv), a guided message-passing scheme that restricts information flow to known-to-unknown paths, and a masked projection to keep unknown nodes agnostic to initialization, all evaluated on 3D Riemann problems under the Euler equations. The approach achieves sharper shock reconstructions and outperforms several ML benchmarks, at the expense of higher memory usage, and demonstrates robust performance under substantial data sparsity. The work suggests strong potential for sparse-flow reconstruction in engineering settings and outlines avenues to reduce memory requirements and generalize across configurations and grids.

Abstract

In compressible fluid flow, reconstructing shocks, discontinuities, rarefactions, and their interactions from sparse measurements is an important inverse problem with practical applications. Moreover, physics-informed machine learning has recently become an increasingly popular approach for performing reconstructions tasks. In this work we explore a machine learning recipe, known as GraphGPS, for reconstructing canonical compressible flows known as 3D Riemann problems from sparse observations, in a physics-informed manner. The GraphGPS framework combines the benefits of positional encodings, local message-passing of graphs, and global contextual awareness, and we explore the latter two components through an ablation study. Furthermore, we modify the aggregation step of message-passing such that it is aware of shocks and discontinuities, resulting in sharper reconstructions of these features. Additionally, we modify message-passing such that information flows strictly from known nodes only, which results in computational savings, better training convergence, and no degradation of reconstruction accuracy. We also show that the GraphGPS framework outperforms numerous machine learning benchmarks.

Figures (13)

• Figure 1: Density fields of the 3D Riemann configurations that we explore in this study.
• Figure 1: The three-wave model assumed by the HLLC solver. The left and right characteristic lines correspond to the fastest and slowest signals, $s_l$ and $s_r$, emerging from the cell interface at $x=0$. The middle characteristic corresponds to the wave of speed $s_*$ which accounts for contact and shear waves.
• Figure 2: Overview of GraphGPS framework. $\mathcal{F}_{\mathcal{MP}}^{\mathbf{E}^{(n)}}$ represents a message-passing layer acting on edge indices $\mathbf{E}^{(n)}$ for layer $n$. $\mathcal{F}_{\mathcal{GC}}$ denotes a layer that captures global dependencies of its input features. $pe$ denotes positional encodings. Residual connections and normalization layers are omitted for clarity.
• Figure 3: Illustration of guided message-passing. Red squares indicate known nodes. Blue circles indicate unknown nodes.
• Figure 4: Mamba-2 module.