Matlab 深度学习工具箱 案例学习与测试————求二阶微分方程

clc
clear% 定义输入变量
x = linspace(0,2,10000)';% 定义网络的层参数
inputSize = 1;
layers = [featureInputLayer(inputSize,Normalization="none")fullyConnectedLayer(10)sigmoidLayerfullyConnectedLayer(1)sigmoidLayer];
% 创建网络
net = dlnetwork(layers);% 训练轮数
numEpochs = 15;
% 每个Batch的数据个数
miniBatchSize = 100;

% SGDM优化方法设置的参数
initialLearnRate = 0.5;
learnRateDropFactor = 0.5;
learnRateDropPeriod = 5;
momentum = 0.9;
velocity = [];

% 损失函数里面考虑初始条件的系数
icCoeff = 7;% ArrayDatastore
ads = arrayDatastore(x,IterationDimension=1);
% 创建一个用于处理管理深度学习数据的对象
mbq = minibatchqueue(ads, ...MiniBatchSize=miniBatchSize, ...PartialMiniBatch="discard", ...MiniBatchFormat="BC");% 用于迭代过程监控
numObservationsTrain = numel(x);
numIterationsPerEpoch = floor(numObservationsTrain / miniBatchSize);
numIterations = numEpochs * numIterationsPerEpoch;% 创建监控对象 
% 由于计时器在您创建监控器对象时启动，因此请确保在靠近训练循环的位置创建对象。
monitor = trainingProgressMonitor( ...Metrics="LogLoss", ...Info=["Epoch" "LearnRate"], ...XLabel="Iteration");% Train the network using a custom training loop
epoch = 0;
iteration = 0;
learnRate = initialLearnRate;
start = tic;% Loop over epochs.
while epoch < numEpochs  && ~monitor.Stopepoch = epoch + 1;% Shuffle data，打乱数据.mbq.shuffle% Loop over mini-batches.while hasdata(mbq) && ~monitor.Stopiteration = iteration + 1;% Read mini-batch of data.X = next(mbq);% Evaluate the model gradients and loss using dlfeval and the modelLoss function.[loss,gradients] = dlfeval(@modelLoss, net, X, icCoeff);% Update network parameters using the SGDM optimizer.[net,velocity] = sgdmupdate(net,gradients,velocity,learnRate,momentum);% Update the training progress monitor.recordMetrics(monitor,iteration,LogLoss=log(loss));updateInfo(monitor,Epoch=epoch,LearnRate=learnRate);monitor.Progress = 100 * iteration/numIterations;end% Reduce the learning rate.if mod(epoch,learnRateDropPeriod)==0learnRate = learnRate*learnRateDropFactor;end
endxTest = linspace(0,4,1000)';yModel = minibatchpredict(net,xTest);yAnalytic = exp(-xTest.^2);figure;
plot(xTest,yAnalytic,"-")
hold on
plot(xTest,yModel,"--")
legend("Analytic","Model")

在深度学习中，被求导的对象（样本/输入）一般是多元的（向量x），绝大多数情况是标量y对向量x进行求导，很少向量y对向量x进行求导，否则就会得到复杂的微分矩阵。所以经常把一个样本看做一个整体，它包含多个变量（属性），对其所有属性求导后再加和，就得到了这个样本的偏导数之和。

% 损失函数
function [loss,gradients] = modelLoss(net, X, icCoeff)% 前向传播计算y = forward(net,X);% Evaluate the gradient of y with respect to x. % Since another derivative will be taken, set EnableHigherDerivatives to true.dy = dlgradient(sum(y,"all"),X,EnableHigherDerivatives=true);% Define ODE loss.eq = dy + 2*y.*X;% Define initial condition loss.ic = forward(net,dlarray(0,"CB")) - 1;% Specify the loss as a weighted sum of the ODE loss and the initial condition loss.loss = mean(eq.^2,"all") + icCoeff * ic.^2;% Evaluate model gradients.gradients = dlgradient(loss, net.Learnables);end