cd2d_inverse_domain

Loss Function Implementation for Convection-Diffusion 2D Inverse Problems.

This module implements the loss function for solving inverse problems in 2D convection-diffusion equations using neural networks. It focuses on computing residuals in the weak form of the PDE for parameter identification.

Key functions

• pde_loss_cd2d_inverse_domain: Computes domain-based PDE loss

Note

The implementation is based on the FastVPINNs methodology [1] for efficient computation of Variational residuals of PDEs.

References

[1] FastVPINNs: Tensor-Driven Acceleration of VPINNs for Complex Geometries DOI: https://arxiv.org/abs/2404.12063

pde_loss_cd2d_inverse_domain(test_shape_val_mat, test_grad_x_mat, test_grad_y_mat, pred_nn, pred_grad_x_nn, pred_grad_y_nn, forcing_function, bilinear_params, inverse_params_list)

Computes domain-based loss for 2D convection-diffusion inverse problem.

Implements the weak form residual calculation for parameter identification in 2D convection-diffusion equations. The loss includes diffusion, convection, and reaction terms.

Parameters:

Name	Type	Description	Default
test_shape_val_mat	Tensor	Test function values at quadrature points Shape: (n_elements, n_test_functions, n_quad_points)	required
test_grad_x_mat	Tensor	Test function x-derivatives at quadrature points Shape: (n_elements, n_test_functions, n_quad_points)	required
test_grad_y_mat	Tensor	Test function y-derivatives at quadrature points Shape: (n_elements, n_test_functions, n_quad_points)	required
pred_nn	Tensor	Neural network solution at quadrature points Shape: (n_elements, n_quad_points)	required
pred_grad_x_nn	Tensor	x-derivative of NN solution at quadrature points Shape: (n_elements, n_quad_points)	required
pred_grad_y_nn	Tensor	y-derivative of NN solution at quadrature points Shape: (n_elements, n_quad_points)	required
forcing_function	callable	Right-hand side forcing term	required
bilinear_params	dict	Dictionary containing: - b_x: x-direction convection coefficient - b_y: y-direction convection coefficient - c: reaction coefficient	required
inverse_params_list	list	List containing: - diffusion coefficient neural network	required

Returns:

Type	Description
Tensor	Cell-wise residuals averaged over test functions Shape: (n_cells,)

Notes

The weak form includes: - Diffusion term: ∫ε∇u·∇v dΩ - Convection term: ∫(b·∇u)v dΩ - Reaction term: ∫cuv dΩ where ε is the diffusion coefficient to be identified.

Source code in scirex/core/sciml/fastvpinns/physics/cd2d_inverse_domain.py

def pde_loss_cd2d_inverse_domain(
    test_shape_val_mat: tf.Tensor,
    test_grad_x_mat: tf.Tensor,
    test_grad_y_mat: tf.Tensor,
    pred_nn: tf.Tensor,
    pred_grad_x_nn: tf.Tensor,
    pred_grad_y_nn: tf.Tensor,
    forcing_function: callable,
    bilinear_params: dict,
    inverse_params_list: list,
) -> tf.Tensor:
    """Computes domain-based loss for 2D convection-diffusion inverse problem.

    Implements the weak form residual calculation for parameter identification
    in 2D convection-diffusion equations. The loss includes diffusion,
    convection, and reaction terms.

    Args:
        test_shape_val_mat: Test function values at quadrature points
            Shape: (n_elements, n_test_functions, n_quad_points)
        test_grad_x_mat: Test function x-derivatives at quadrature points
            Shape: (n_elements, n_test_functions, n_quad_points)
        test_grad_y_mat: Test function y-derivatives at quadrature points
            Shape: (n_elements, n_test_functions, n_quad_points)
        pred_nn: Neural network solution at quadrature points
            Shape: (n_elements, n_quad_points)
        pred_grad_x_nn: x-derivative of NN solution at quadrature points
            Shape: (n_elements, n_quad_points)
        pred_grad_y_nn: y-derivative of NN solution at quadrature points
            Shape: (n_elements, n_quad_points)
        forcing_function: Right-hand side forcing term
        bilinear_params: Dictionary containing:
            - b_x: x-direction convection coefficient
            - b_y: y-direction convection coefficient
            - c: reaction coefficient
        inverse_params_list: List containing:
            - diffusion coefficient neural network

    Returns:
        Cell-wise residuals averaged over test functions
            Shape: (n_cells,)

    Notes:
        The weak form includes:
        - Diffusion term: ∫ε∇u·∇v dΩ
        - Convection term: ∫(b·∇u)v dΩ
        - Reaction term: ∫cuv dΩ
        where ε is the diffusion coefficient to be identified.
    """

    # The first values in the inverse_params_list is the number of inverse problems
    diffusion_coeff_NN = inverse_params_list[0]

    # ∫ε.du/dx. dv/dx dΩ
    pde_diffusion_x = tf.transpose(
        tf.linalg.matvec(test_grad_x_mat, pred_grad_x_nn * diffusion_coeff_NN)
    )

    # ∫ε.du/dy. dv/dy dΩ
    pde_diffusion_y = tf.transpose(
        tf.linalg.matvec(test_grad_y_mat, pred_grad_y_nn * diffusion_coeff_NN)
    )

    # eps * ∫ (du/dx. dv/dx + du/dy. dv/dy) dΩ
    # Here our eps is a variable which is to be learned, Which is already premultiplied with the predicted gradient of the neural network
    pde_diffusion = pde_diffusion_x + pde_diffusion_y

    # ∫du/dx. v dΩ
    conv_x = tf.transpose(tf.linalg.matvec(test_shape_val_mat, pred_grad_x_nn))

    # # ∫du/dy. v dΩ
    conv_y = tf.transpose(tf.linalg.matvec(test_shape_val_mat, pred_grad_y_nn))

    # # b(x) * ∫du/dx. v dΩ + b(y) * ∫du/dy. v dΩ
    conv = bilinear_params["b_x"] * conv_x + bilinear_params["b_y"] * conv_y

    # reaction term
    # ∫c.u.v dΩ
    reaction = bilinear_params["c"] * tf.transpose(
        tf.linalg.matvec(test_shape_val_mat, pred_nn)
    )

    residual_matrix = (pde_diffusion + conv + reaction) - forcing_function

    # Perform Reduce mean along the axis 0
    residual_cells = tf.reduce_mean(tf.square(residual_matrix), axis=0)

    return residual_cells