Graph neural network recovers missing boundary data for field reconstruction
An Attention-Guided Graph Neural Network Surrogate for Boundary Integral Computation in Physical Field Reconstruction
Machine Learning
Summary
Sometimes, engineers need to know what happens inside an object based on what happens on its edges, but parts of this edge information might be missing. The authors present a method that uses a special type of artificial intelligence called a graph neural network combined with an attention system to guess the missing edge information. After recovering this missing data, traditional calculations can rebuild the inside details of the physical field. They tested this approach on math problems common in physics and engineering and found it could reconstruct the inside fields accurately and reliably.
What this means in practice
- •For engineering simulation teams: Recover missing boundary information to enable faster and more accurate physical field reconstructions in complex simulations.
- •For computational physics developers: Incorporate learned boundary data prediction to improve solution quality for partial differential equation solvers under incomplete data conditions.
Authors
RongZe Gao, YuWei Wu, XinRan Ji, XingYue Ren
Abstract
In practical engineering and scientific computing, a real physical system can be reduced to a boundary value problem, and the adequate boundary information to determine the interior spatial distribution and evolutionary state of the physical field can be solved with linear equations, alternatively, the the real physical problem can be treated with boundary problems with in adequate information problems in field of machine learning. This paper proposes a deep learning framework for the recovery of missing boundary information using an attention-guided graph neural network (GNN) surrogate. The framework extracts features from the physical information on the discretized boundary into a latent space, and the missing boundary quantities are then predicted by the network. Thus, the nonlinear mapping from known boundary quantities to missing boundary quantities is learned using a variational autoencoder (VAE) architecture, while an attention mechanism is incorporated to better model the correlations among different types of boundary information. Furthermore, the interior physical field can be reconstructed using the conventional boundary element method (BEM) integral formulation. The effectiveness of the GNN-BEM framework is evaluated using numerical examples of the two-dimensional Laplace and Poisson equations, which demonstrate good performance in reconstruction accuracy, local detail preservation, and model stability. The results show that GNN-BEM provides a feasible computational framework for boundary information recovery and fast interior physical field reconstruction for partial differential equations under complex boundary conditions.