[Paper Review] Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction

Filipe de Avila Belbute-Peres, Thomas D. Economon+1 more|arXiv (Cornell University)|Jul 8, 2020

Advanced Graph Neural Networks26 references11 citations

TL;DR

This paper proposes a hybrid deep learning framework that integrates a differentiable, coarse-resolution CFD simulator within a graph neural network (GCN) to predict fluid flow with high accuracy and generalization. By jointly training the GCN and the embedded CFD simulator on unstructured meshes, the model achieves faster inference than full CFD simulations while significantly outperforming pure ML models in generalization to unseen physical conditions, such as strong shocks not present in training data.

ABSTRACT

Solving large complex partial differential equations (PDEs), such as those that arise in computational fluid dynamics (CFD), is a computationally expensive process. This has motivated the use of deep learning approaches to approximate the PDE solutions, yet the simulation results predicted from these approaches typically do not generalize well to truly novel scenarios. In this work, we develop a hybrid (graph) neural network that combines a traditional graph convolutional network with an embedded differentiable fluid dynamics simulator inside the network itself. By combining an actual CFD simulator (run on a much coarser resolution representation of the problem) with the graph network, we show that we can both generalize well to new situations and benefit from the substantial speedup of neural network CFD predictions, while also substantially outperforming the coarse CFD simulation alone.

Motivation & Objective

• To address the poor generalization of pure deep learning models in fluid dynamics simulations beyond their training distribution.
• To reduce computational cost of high-fidelity CFD simulations while preserving physical accuracy.
• To integrate a differentiable, low-resolution CFD simulator as a learnable module within a graph neural network for improved simulation fidelity.
• To enable end-to-end training of both the GCN and the coarse mesh input using gradient-based optimization via adjoint methods.
• To demonstrate that the hybrid model generalizes better than pure GCN baselines and outperforms coarse CFD simulations alone on unseen physical behaviors.

Proposed method

• The model uses graph convolutional networks (GCNs) to operate directly on unstructured, non-uniform meshes typical in CFD simulations.
• A differentiable, low-resolution CFD simulator is embedded as a differentiable layer within the GCN, enabling gradient backpropagation through the simulation.
• The coarse mesh is optimized end-to-end via backpropagation using adjoint methods to improve prediction accuracy.
• The GCN processes features on the coarse mesh, and the CFD simulator computes physical dynamics, with predictions upscaled to the original resolution.
• The entire system is trained end-to-end using loss functions that compare predictions to high-fidelity ground truth simulations.
• The method leverages differentiability of the CFD solver to allow joint optimization of both the neural network and the mesh geometry for better generalization.

Experimental results

Research questions

• RQ1 Can a hybrid model combining a differentiable CFD simulator and a GCN generalize better to unseen fluid dynamics scenarios than a pure GCN or a coarse CFD simulation alone?
• RQ2 How does the integration of a differentiable PDE solver within a GCN architecture affect prediction accuracy and inference speed compared to full CFD simulations?
• RQ3 To what extent can end-to-end training of the GCN and the coarse mesh input improve generalization to novel physical conditions such as strong shocks?
• RQ4 Does the model benefit from learning beyond simple upsampling of coarse simulation results, or is it merely interpolating?
• RQ5 How does the computational cost of the hybrid model compare to full CFD simulations and pure neural network baselines?

Key findings

• The CFD-GCN model generalizes significantly better than the pure GCN baseline on test cases with large shocks, which were not present in the training data.
• The model outperforms the coarse CFD simulation alone, achieving lower error (1.4·10⁻²) than the coarse simulation (9.5·10⁻²) on the generalization task.
• The CFD-GCN model achieves a test error of 1.4·10⁻², substantially lower than the pure GCN baseline (1.9·10⁻²), demonstrating improved accuracy.
• The model is able to predict complex flow features such as shocks and vortices more accurately than the GCN-only model, even when those features were absent in training.
• The training process optimizes the coarse mesh geometry in a way that is visually perceptible and improves prediction quality, as shown in Figure 12 (Appendix).
• The CFD-GCN model is approximately 20x faster than full CFD simulations, with 85% of inference time consumed by the CFD simulation, 10% by mesh upsampling, and 5% by graph convolutions.