用深度学习拟合多项式系数

实现来自https://blog.csdn.net/cuixing001/article/details/82317630示例。

首先给出真值多项式参考方程：

其对应参数形式为：

程序实现思路为 ：根据方程1，给定一些样本数据点坐标（x，y）点集，及最小均方差为目标，求解最佳参数（w1,w2,w3,b）。

下图为动态显示在深度学习训练过程中多项式逐渐逼近真值曲线！

step1: Data preprocess

xsample = linspace(-1,1,30);
w = [0.5,3,2.4];
b = 0.9;
ysample = b+w(1).*xsample+w(2).*xsample.^2+w(3).*xsample.^3;

% input network data
x = 2*rand(10000,1)-1; % [-1,1] range
Xdata = [x,x.^2,x.^3];
Ydata = b+w(1).*x+w(2).*x.^2+w(3).*x.^3;

step2: define model and train

numFeatures = 3;
layers = [featureInputLayer(numFeatures,'Name','input')
    fullyConnectedLayer(1, 'Name','fc')];
net = dlnetwork(layers);

arrdsX = arrayDatastore(Xdata);
arrdsY = arrayDatastore(Ydata);
arrds = combine(arrdsX,arrdsY);
mbq = minibatchqueue(arrds,2,MiniBatchFormat={'BC','BC'});

% Initialize the training progress plot.
figure;
tiledlayout(2,1)
nexttile;
hold on;grid on;
gtLine = plot(xsample,ysample,LineWidth=2,DisplayName="ground truth");
predictLine = plot(xsample,nan(1,length(xsample)),LineWidth=2,DisplayName="predict");
legend

nexttile;
C = colororder;
lineLossTrain = animatedline(Color=C(2,:),LineWidth=2);
ylim([0 inf])
xlabel("Iteration")
ylabel("Loss")
grid on

%
iteration = 0;
numEpochs = 5;
velocity = [];
start = tic;

% Loop over epochs.
for epoch = 1:numEpochs

    % Shuffle data.
    shuffle(mbq);
    
    % Loop over mini-batches.
    while hasdata(mbq)
        iteration = iteration + 1;
        
        % Read mini-batch of data.
        [X,T] = next(mbq);
        
        % Evaluate the model gradients, state, and loss using dlfeval and the
        % modelLoss function and update the network state.
        [loss,gradients,state] = dlfeval(@modelLoss,net,X,T);
        net.State = state;
        
        % Update the network parameters using the SGDM optimizer.
        [net,velocity] = sgdmupdate(net,gradients,velocity);

        % plot fit line
        ww = extractdata(net.Learnables.Value{1});
        bb= extractdata(net.Learnables.Value{2});
        predictPt = bb+ww(1).*xsample+ww(2).*xsample.^2+ww(3).*xsample.^3;
        predictLine.YData = predictPt;
        
        % Display the training progress.
        D = duration(0,0,toc(start),Format="hh:mm:ss");
        loss = double(loss);
        addpoints(lineLossTrain,iteration,loss)
        title("Epoch: " + epoch + ", Elapsed: " + string(D))

       exportgraphics(gcf,"polyfit.gif","Append",true)
        drawnow
    end
end

function [f,g,state] = modelLoss(net,X,T)
% Calculate objective using supported functions for dlarray
    [Y,state] = forward(net,X);
    f = mse(Y,T);%sum((Y-T).^2);
    g = dlgradient(f,net.Learnables); % Automatic gradient
end