Interpretable A-posteriori Error Indication for Graph Neural Network Surrogate Models

TL;DR

The end result is an interpretable GNN model that isolates regions in physical space, corresponding to sub-graphs, that are intrinsically linked to the forecasting task while retaining the predictive capability of the baseline.

Abstract

Data-driven surrogate modeling has surged in capability in recent years with the emergence of graph neural networks (GNNs), which can operate directly on mesh-based representations of data. The goal of this work is to introduce an interpretability enhancement procedure for GNNs, with application to unstructured mesh-based fluid dynamics modeling. Given a black-box baseline GNN model, the end result is an interpretable GNN model that isolates regions in physical space, corresponding to sub-graphs, that are intrinsically linked to the forecasting task while retaining the predictive capability of the baseline. These structures identified by the interpretable GNNs are adaptively produced in the forward pass and serve as explainable links between the baseline model architecture, the optimization goal, and known problem-specific physics. Additionally, through a regularization procedure, the interpretable GNNs can also be used to identify, during inference, graph nodes that correspond to a majority of the anticipated forecasting error, adding a novel interpretable error-tagging capability to baseline models. Demonstrations are performed using unstructured flow field data sourced from flow over a backward-facing step at high Reynolds numbers, with geometry extrapolations demonstrated for ramp and wall-mounted cube configurations.

Figures (18)

• Figure 1: (Top, blue box) Schematic of baseline GNN model architecture. (Bottom, red box) Schematic of interpretable sub-sampling module. Module is implemented via skip connection between first and last message passing layers in baseline processor block. Interpretability comes from adaptive pooling layer, which generates sub-graphs via sub-sampled nodes that can be visualized as masked fields. Using a regularization strategy, masked fields can then be used to identify regions of forecasting approximation errors during model evaluation. Details provided in Sec. \ref{['sec:results:fine_tuning']}. (Bottom, green box) Error tagging procedure by means of a budget regularization strategy that encourages prediction errors to be contained within the extracted nodes in sub-graphs (the masked fields). Details provided in Sec. \ref{['sec:results:budget']}.
• Figure 2: Schematic of Top-K reduction procedure. ${\bf X} \in \mathbb{R}^{|V| \times N_F}$ denotes an input node attribute matrix, and ${\bf y} = \sigma({\bf X}{{\bf p}}/\lVert{{\bf p}}\rVert) \in \mathbb{R}^{|V| \times 1}$ is the result of the node-wise projection operation, where ${\bf p} \in \mathbb{R}^{N_F \times 1}$ is learned during training and $\sigma$ is a hyperbolic tangent activation function.
• Figure 3: Mean-squared error objective (Eq. \ref{['eq:mse']}) versus training epochs during interpretability enhancement phase.
• Figure 4: Comparison of GNN prediction workflows between baseline and interpretable GNNs. interpretable workflow includes masked fields generated in forward pass. Input flowfield (top) sourced from Re=27,233 trajectory of the testing set. Bottom-most plots show baseline absolute error fields ($\widetilde{\bf X}_{m+1} - {\bf X}_{m+1}$) for streamwise (left) and vertical (right) velocity components.
• Figure 5: Single-step prediction outputs generated by interpretable GNN (RF=16) using input fields at time-ordered instants in the Re=27233 trajectory. Each frame shows predicted y-component of velocity (top) alongside masked field (bottom), with inlays showing subgraph representations.