Connections Between Numerical Algorithms for PDEs and Neural Networks

Area of Science:

• Computational Mathematics
• Machine Learning
• Neural Network Architectures

Background:

• Partial differential equations (PDEs) are fundamental in science and engineering.
• Neural networks have shown great success in various applications.
• Bridging these fields can lead to more robust and efficient models.

Purpose of the Study:

• To explore structural connections between numerical PDE algorithms and neural network architectures.
• To transfer mathematical foundations from PDEs to neural networks.
• To develop novel, mathematically-grounded neural building blocks.

Main Methods:

• Investigated numerical schemes for PDEs (explicit, implicit, multigrid) and their connection to neural networks (ResNets, RNNs, U-Nets).
• Developed a symmetric residual network with stability guarantees.
• Implemented multigrid techniques within U-Net architectures for PDE solutions.

Main Results:

• Inspired a novel symmetric residual network design with provable stability.
• Justified the effectiveness of skip connections from a numerical standpoint.
• Demonstrated U-Net architectures implementing multigrid for efficient PDE learning.
• Achieved up to 50% reduction in trainable parameters with improved performance.

Conclusions:

• The study provides a mathematical foundation for popular neural architectures.
• The findings offer a blueprint for designing new, mathematically sound neural building blocks.
• This interdisciplinary approach enhances neural network efficiency and performance in scientific applications.