Partial Differential Equations Meet Deep Neural Networks: A Survey.

Area of Science:

• Computational science and engineering
• Applied mathematics
• Scientific computing

Background:

• Partial differential equations (PDEs) model critical scientific and engineering problems.
• Traditional numerical methods for PDEs face computational inefficiency challenges.
• Deep learning (DL) presents a promising alternative for solving PDEs.

Purpose of the Study:

• To systematically review and categorize deep neural network (NN) approaches for solving PDEs.
• To provide a comprehensive overview of the current progress in DL for PDEs.
• To bridge the gap in existing literature by offering a broader taxonomy beyond specific methods like physics-informed neural networks (PINNs).

Main Methods:

• Categorization and review of diverse deep neural network (NN) architectures applied to PDEs.
• Analysis of applications across scientific, engineering, and medical domains.
• Historical overview of DL for PDEs, including key challenges and future trends.

Main Results:

• Deep neural networks (DNNs) demonstrate significant potential as efficient solvers for PDEs.
• A wide range of NN methodologies are being developed and applied to PDE problems.
• The survey provides a structured understanding of the landscape of DL in solving PDEs.

Conclusions:

• Deep learning offers a transformative approach to solving partial differential equations (PDEs).
• This survey provides a valuable resource for researchers and practitioners exploring DL for PDE applications.
• Future trends indicate continued advancements and broader adoption of NN-based methods in scientific and engineering fields.