Physics-informed neural networks based on adaptive weighted loss functions for Hamilton-Jacobi equations

Area of Science:

• Computational Mathematics
• Machine Learning
• Applied Physics

Background:

• Physics-informed neural networks (PINN) are increasingly used for forward and inverse problems.
• Solving Hamilton-Jacobi equations presents challenges in computational mathematics.

Purpose of the Study:

• To explore the generality of PINN for Hamilton-Jacobi equations.
• To propose an adaptive weighted physics-informed neural network (AW-PINN) for improved unsupervised learning with fewer data.

Main Methods:

• Developed AW-PINN with adaptive loss function weights using a logarithmic mean to balance constraints automatically.
• Implemented periodicity requirements for boundary conditions and gradients.
• Utilized fully connected feedforward neural networks with Adam and L-BFGS-B optimizers.

Main Results:

• AW-PINN demonstrated significant improvements in predictive accuracy and convergence rate compared to standard PINN.
• The algorithm successfully approximated solutions even for nonconvex Hamiltonians.
• Achieved more accurate solutions with fewer iterations.

Conclusions:

• AW-PINN offers a more efficient and accurate approach for solving Hamilton-Jacobi equations.
• The adaptive weighting mechanism effectively handles multiple constraints without additional hyperparameters.
• This method shows promise for unsupervised learning tasks in computational mathematics.