Parallel Solution of Nonlinear Projection Equations in a Multitask Learning Framework.

Area of Science:

• Computational Mathematics
• Machine Learning
• Optimization

Background:

• Nonlinear projection equations (NPEs) are crucial for constrained nonlinear optimization and engineering.
• Traditional numerical methods struggle with solving multiple NPEs due to independent, iterative processing.
• There is a need for more efficient methods to handle systems of NPEs.

Purpose of the Study:

• To develop a novel, efficient approach for solving multiple nonlinear projection equations (NPEs) simultaneously.
• To leverage multitask learning (MTL) and physics-informed neural networks (PINNs) for parallel NPE solutions.
• To demonstrate superior computational performance over existing methods.

Main Methods:

• Modeled each NPE as a system of ordinary differential equations (ODEs) via neurodynamic optimization.
• Employed physics-informed neural networks (PINNs) to solve individual ODE systems.
• Implemented a multibranch multitask learning (MTL) framework, with each branch representing a PINN.

Main Results:

• The proposed MTL framework enables parallel solving of multiple NPEs.
• Experimental results confirm superior computational performance of the MTL approach.
• Efficiency gains are particularly pronounced when solving a large number of NPEs.

Conclusions:

• The MTL-based approach offers a significant advancement in solving multiple NPEs.
• This method provides a more efficient and scalable solution for complex optimization and engineering problems.
• The framework demonstrates the potential of integrating MTL and PINNs for advanced scientific computing.