Shape optimization is at the heart of many industrial applications, such as aerodynamics, heat transfer, and structural analysis. It has recently been shown that Graph Neural Networks (GNNs) can predict the performance of a shape quickly and accurately and be used to optimize more effectively than traditional techniques that rely on response-surfaces obtained by Kriging.
However, GNNs suffer from the fact that they do not evaluate their own accuracy, which is something Bayesian Optimization methods require. Therefore, estimating confidence in generated predictions is necessary to go beyond straight deterministic optimization, which is less effective.
In this paper, we demonstrate that we can use Ensembles-based technique to overcome this limitation and outperform the state-of-the-art. Our experiments on diverse aerodynamics and structural analysis tasks prove that adding uncertainty to shape optimization significantly improves the quality of resulting shapes and reduces the time required for the optimization.
Graph Neural Networks
Graph Convolutional Neural Network (GCNN) is a type of Neural Network that directly operates on graph structures. Given a set of generic surfaces parametrized as meshes (and therefore as graphs), GCCN model is able to accurately emulate various types of simulators used for aerodynamics, hydrodynamic and many other physical applications.
Strong advantages of this approach are:
1) Differentiability of the model with respect to the vertex coordinates, therefore we are able to directly optimimize input shapes.
2) Significant reduction in time required for a simulation comparing to modern CFD simulators.
The goal of Uncertainty Estimation (UE) is to produce a measure of confidence for model predictions. In practice, Deep Ensembles and MC-Dropout approaches are considered to be one of the most popular and effective methods addressing uncertainty of neural networks.
Bayesian Optimization (BO) is an exploration-exploitation procedure that minimizes given black-box function. Uncertainty lies at the core of BO that allows to perform optimization in more efficient and optimal way comparing to deterministic and greedy approaches.
Example 1 (Minimizing Car Drag): Given a training set of car shapes and simulated air pressure (via CFD), we train a GCNN model to predict air pressure fields and MeshSDF model to embed complex car meshes in high-dimensional latent space. MeshSDF's latent vectors and accurate GCNN emulation model both allow to efficiently optimize car bodies in order to reduce pressure drag.
Example 2 (2D Airfoil Optimization): 2D airfoil profile optimization is one of standard benchmarks for shape optimization in the CFD community. In our experiments, we use popular XFoil simulator to compute the pressure distribution over surface and utilize GCNN to precisely model it. Paired with standard NACA parameters, our method finds the best form in terms of lift-to-drag ratio.
If you are interested in more details, you may read the whole paper.