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Loss Function Implementation for 2D Convection-Diffusion Problem

Overview

This module implements the loss function for solving 2D convection-diffusion equations using Physics-Informed Neural Networks (PINNs). It focuses on computing residuals of the Partial Differential Equation (PDE) with known coefficients.

Key Functions

  • pde_loss_cd2d: Computes the PDE loss for 2D convection-diffusion equations.

Function: pde_loss_cd2d

Description

Calculates residuals for the 2D convection-diffusion problem using the PINNs methodology.
The loss function includes: - Diffusion term: -ε∇²(u) - Convection term: b·∇u - Reaction term: cu
where ε, b, and c are known coefficients.

Arguments

  • pred_nn (tf.Tensor): Neural network solution at quadrature points.
    Shape: (N_points, 1)
  • pred_grad_x_nn (tf.Tensor): x-derivative of the neural network solution at quadrature points.
    Shape: (N_points, 1)
  • pred_grad_y_nn (tf.Tensor): y-derivative of the neural network solution at quadrature points.
    Shape: (N_points, 1)
  • pred_grad_xx_nn (tf.Tensor): Second-order x-derivative of the neural network solution at quadrature points.
    Shape: (N_points, 1)
  • pred_grad_yy_nn (tf.Tensor): Second-order y-derivative of the neural network solution at quadrature points.
    Shape: (N_points, 1)
  • forcing_function (callable): Right-hand side forcing term.
  • bilinear_params (dict): A dictionary containing:
  • eps: Diffusion coefficient.
  • b_x: x-direction convection coefficient.
  • b_y: y-direction convection coefficient.
  • c: Reaction coefficient.

Returns

  • tf.Tensor: Cell-wise residuals averaged over test functions.
    Shape: (1,)

Notes

  • The methodology combines the effects of diffusion, convection, and reaction in a unified residual formulation.