G-PARC: Graph-Physics Aware Recurrent Convolutional neural networks for spatiotemporal dynamics on unstructured...

Area of Science:

• Computational physics
• Machine learning
• Scientific computing

Background:

• Physics-aware recurrent convolutional networks (PARC) excel at spatiotemporal dynamics but are limited to uniform Cartesian grids.
• Existing graph-based physics-aware deep learning (PADL) methods struggle with extreme nonlinear regimes and irregular geometries.
• Pixel-based convolutions are inefficient for evolving localized structures.

Purpose of the Study:

• To introduce Graph PARC (G-PARC), a novel physics-aware deep learning framework that integrates graph neural networks with differential operators for enhanced nonlinear dynamics prediction.
• To overcome the limitations of existing methods in handling irregular spatial discretizations and extreme nonlinearities.
• To enable accurate modeling of complex phenomena on unstructured and deforming computational domains.

Main Methods:

• G-PARC utilizes moving least squares (MLS) kernels for approximating spatial derivatives on unstructured graphs.
• Differential operators of governing partial differential equations are embedded directly into the neural network's computational graph.
• The traditional encoder-processor-decoder framework is replaced by analytically computed differential operators.

Main Results:

• G-PARC demonstrates superior accuracy with 2-3× fewer parameters compared to MeshGraphNet, MeshGraphKAN, and GraphSAGE.
• The method generalizes effectively across non-uniform spatial and temporal discretizations and handles moving meshes for structural deformation.
• G-PARC outperforms existing graph-based PADL methods on nonlinear benchmarks, including fluvial hydrology, planar shock waves, and elastoplastic dynamics.

Conclusions:

• G-PARC successfully integrates the flexibility of GNNs with explicit physical operators for accurate modeling of extreme nonlinear phenomena.
• This approach extends PADL beyond idealized Cartesian grids, enabling applications on complex, evolving computational domains.
• G-PARC represents a significant advancement in deep learning for scientific computing, offering improved efficiency and accuracy for challenging dynamic systems.