我要PINN的matlab程序

以下是使用 MATLAB 实现的 PINN（Physics-Informed Neural Networks）程序的示例代码：

% PINN for solving a 1D Poisson equation

% Define the domain
L = 1;
x = linspace(0, L, 100);

% Define the boundary conditions
u0 = 0;
uL = 1;

% Define the neural network architecture
layers = [
    featureInputLayer(1)
    fullyConnectedLayer(10)
    reluLayer
    fullyConnectedLayer(10)
    reluLayer
    fullyConnectedLayer(10)
    reluLayer
    fullyConnectedLayer(1)
    regressionLayer];

% Define the loss function
function [loss, grad] = pinnLoss(theta, x, u, f, model)
    % Unpack the parameters
    [params, weights] = unpack(theta, model);

    % Evaluate the neural network
    u_pred = predict(model, x, params, 'OutputAs', 'cell');

    % Compute the residual
    r = pdeResidual(u_pred{1}, x, f);

    % Compute the boundary conditions
    bc = boundaryConditions(u_pred{1}(1), u_pred{end}(end), u(1), u(end));

    % Compute the loss and gradient
    loss = bc.loss + mean(r.^2);
    grad = bc.gradient + jacobian(model, x, params, r);
end

% Define the residual function
function r = pdeResidual(u, x, f)
    % Compute the second derivative of u
    d2u_dx2 = diff(u, 2) ./ diff(x).^2;

    % Compute the residual
    r = d2u_dx2 - f;
end

% Define the boundary conditions function
function bc = boundaryConditions(u_left, u_right, u0, uL)
    % Compute the loss
    loss = (u_left - u0)^2 + (u_right - uL)^2;

    % Compute the gradient
    gradient = [
        2 * (u_left - u0)
        zeros(size(u_left))
        zeros(size(u_left))
        2 * (u_right - uL)
        zeros(size(u_left))
        zeros(size(u_left))
        zeros(size(u_left))
        zeros(size(u_left))];

    bc = struct('loss', loss, 'gradient', gradient);
end

% Define the PDE parameters
f = -1;
u_true = x.*(1-x);
u_init = zeros(size(x));

% Train the PINN
theta0 = randn(numel(layers) + 1, 1);
model = dlnetwork(layers);
options = optimoptions('adam', 'MaxIterations', 1000);
theta_star = adam(@(theta) pinnLoss(theta, x, u_true, f, model), theta0, options);

% Evaluate the trained PINN
u_pred = predict(model, x, theta_star, 'OutputAs', 'cell');

这个代码是一个简单的示例，可以根据实际情况进行修改和扩展。如果您需要更多帮助，可以查看 PINN 的相关文献或参考其他 MATLAB PINN 实现的代码。