solving the fokker-planck equation with neural networks: a performance improvement approach

The fokker-planck equation models the evolution of probability densities in fields such as physics, biology, and finance. traditional numerical methods for solving this equation can be computationally expensive and struggle with complex, high-dimensional problems. in this work, we propose a physics-informed neural networks (pinns) approach to efficiently approximate solutions of the fokker-planck equation. our method employs a fully connected feedforward neural network using two activation functions--tanh and silu--with fixed learning rates (0.001 for tanh and 0.01 for silu) and varied spatial discretization. the loss function is designed to enforce the governing differential equation as well as the initial and boundary conditions. experimental results, evaluated using standard error metrics (rms, relative $l_2$-norm error, and mae), demonstrate that our pinn approach achieves competitive accuracy with improved convergence and lower computational costs compared to traditional methods. this study underscores the potential of neural network-based solvers for complex differential equations and sets the stage for future optimization.