Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the...

Area of Science:

• Computational physics
• Machine learning for scientific computing

Background:

• Physics-informed neural networks (PINNs) excel at modeling physical processes governed by PDEs.
• However, current PINN architectures struggle with non-linear PDEs, particularly hyperbolic conservation laws with shock waves, often requiring artificial dissipation.

Purpose of the Study:

• To investigate optimal neural network architectures for learning complex non-linear PDEs.
• To address the limitations of existing PINNs in modeling hyperbolic systems and shock phenomena.

Main Methods:

• Introduced physics-informed attention-based neural networks (PIANNs), combining recurrent neural networks and attention mechanisms.
• Focused on architectural improvements rather than residual regularization for enhanced PDE solving.

Main Results:

• PIANNs effectively capture shock fronts in hyperbolic model problems.
• The attention mechanism allows networks to adapt to non-linear solution features, overcoming PINN limitations.

Conclusions:

• PIANNs represent a significant advancement in neural network architectures for solving challenging non-linear PDEs.
• This methodology provides high-quality solutions within the training data's convex hull, particularly for hyperbolic systems.