A Multi-Fidelity Graph U-Net Model for Accelerated Physics Simulations

TL;DR

This work tackles the data-cost bottleneck of high-fidelity PDE surrogates by introducing Multi-Fidelity U-Net (MF-UNet) and MF-UNet Lite, which enable bi-directional information exchange across high-, medium-, and low-resolution graphs within a single Graph U-Network. Both models share encoders/decoders and GN blocks, and employ k-nearest neighbor up/down sampling to couple fidelity levels, optimized via a multilevel loss L = sum_i λ_i L_i. Across 2D cantilever, 2D plate, and 3D Ahmed body CFD tasks, MF-UNet consistently surpasses single-fidelity and transfer-learning baselines, with MF-UNet-3 delivering the best accuracy and MF-UNet Lite offering roughly 35% faster training at modest accuracy costs. These results demonstrate a scalable pathway to accurate high-resolution PDE predictions while substantially reducing data-generation costs, with potential for real-world engineering applications and further extension to time-dependent problems. $L = \sum_{i=1}^n \lambda_i L_i$

Abstract

Physics-based deep learning frameworks have shown to be effective in accurately modeling the dynamics of complex physical systems with generalization capability across problem inputs. Data-driven networks like GNN, Neural Operators have proved to be very effective in generalizing the model across unseen domain and resolutions. But one of the most critical issues in these data-based models is the computational cost of generating training datasets. Complex phenomena can only be captured accurately using deep networks with large training datasets. Furthermore, numerical error of training samples is propagated in the model errors, thus requiring the need for accurate data, i.e. FEM solutions on high-resolution meshes. Multi-fidelity methods offer a potential solution to reduce the training data requirements. To this end, we propose a novel GNN architecture, Multi-Fidelity U-Net, that utilizes the advantages of the multi-fidelity methods for enhancing the performance of the GNN model. The proposed architecture utilizes the capability of GNNs to manage complex geometries across different fidelity levels, while enabling flow of information between these levels for improved prediction accuracy for high-fidelity graphs. We show that the proposed approach performs significantly better in accuracy and data requirement and only requires training of a single network compared to other benchmark multi-fidelity approaches like transfer learning. We also present Multi-Fidelity U-Net Lite, a faster version of the proposed architecture, with 35% faster training, with 2 to 5% reduction in accuracy. We carry out extensive validation to show that the proposed models surpass traditional single-fidelity GNN models in their performance, thus providing feasible alternative for addressing computational and accuracy requirements where traditional high-fidelity simulations can be time-consuming.

Figures (11)

• Figure 1: An overview of (a) a single-fidelity GNN architecture; and (b) our proposed multi-fidelity U-Net architecture, where information flows in two directions: both from higher- to lower-fidelity levels and also from lower- to higher-fidelity. The downward flow from high- to low-resolution level is carried out by adding updated node attributes from the higher-fidelity level to the encoded node attributes of the lower-fidelity level. The upward flow involves adding updated node attributes (from the last GN block) of the lower-fidelity to the updated node attributes of the next higher-fidelity, before it is passed through another set of GN blocks. As the number of nodes are different for graphs of different fidelity, down-sampling and up-sampling operations are carried out based on the nodal distance to calculate nodal information between different levels. Graphs from all the levels of fidelity share the same encoder, decoder and GN blocks.
• Figure 2: An overview of the Multi-Fidelity U-Net Lite architecture, which is similar to the Multi-Fidelity U-Net architecture, with the information flowing in one direction - from lower- to higher-fidelity levels. Similarly to Multi-Fidelity U-Net, the upward flow of information is through the coupling between the updated node attributes from the last GN block on the lower resolution graph and the updated node attributes of the next higher resolution graph, before its passed through another set of GN blocks. We use up-sampling operation based on nodal distance, to pass nodal information between the fidelity levels. Moreover, encoder, decoder and GN blocks share parameters across different levels.
• Figure 3: Cantilever beam with a fixed end considered for the multi-fidelity problem. (a) shows the domain along with the boundary, $\delta D$ as well as the force, $P$, applied as the boundary condition. (b) shows the triangular meshing of the domain.
• Figure 4: A comparison of relative L1-error in the prediction of $u_x$ for individual high-resolution test samples between the benchmark GNN models and the proposed multi-fidelity models, for the cantilever beam problem. Here, 3 levels of fidelity are considered for training Transfer Learning, MF-UNet and MF-UNet Lite.
• Figure 5: A histogram showing the distribution of the relative L1-error in the prediction of displacement for graphs whose resolutions are 3-times that of the high-fidelity graphs used for training transfer learning GNN and MF-UNet model. This shows the generalization capability of MF-UNet to predict the responses on graphs with unseen resolutions.