Evolutional deep neural network.

Area of Science:

• Computational Mathematics
• Artificial Intelligence
• Fluid Dynamics

Background:

• Solving complex partial differential equations (PDEs) often requires computationally intensive methods.
• Traditional neural network approaches struggle with long-term predictions and enforcing constraints.

Purpose of the Study:

• Introduce and validate an Evolutional Deep Neural Network (EDNN) for solving PDEs.
• Demonstrate the EDNN's ability to predict indefinite state-space trajectories and satisfy boundary conditions.
• Showcase the EDNN's versatility across various fluid dynamics and heat transfer equations.

Main Methods:

• EDNNs are trained on initial conditions only, with parameters dynamically updated using governing equations.
• Network parameters are treated as functions and marched in parameter space for temporal evolution.
• Boundary conditions are embedded as hard constraints within the neural network architecture.
• Divergence-free constraints for Navier-Stokes equations are implicitly handled by network design.

Main Results:

• EDNN accurately predicts system evolution without further training.
• The method successfully solves diverse PDEs including heat, advection, Burgers, Kuramoto-Sivashinsky, and Navier-Stokes equations.
• Incompressible Navier-Stokes solutions demonstrate implicit satisfaction of the divergence-free constraint.
• Numerical results show high accuracy compared to analytical and benchmark solutions for transient and statistical dynamics.

Conclusions:

• EDNN offers a novel and accurate framework for solving PDEs with long-term prediction capabilities.
• The approach effectively integrates boundary and physical constraints into the network.
• EDNN presents a versatile and efficient alternative to existing numerical methods for dynamic systems.