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Data generation and network training workflow
Generation if training data for a neural network that represents linear elastic isotropic material behaviour. Two
neural networks are trained based on the generated data. One representing the normal stress response and the
second one representing the shear stress response.
Generate data
This part generates normal stress and trains the corresponding neural network
clear;close all;clc;
Set control flags
generate = 0; % data generation flag 0: load data 1: generate data training = 0; % neural network training flag 0: load neural network training info 1: t
Definition of input data
ndpoint = 200; epslim1 = -0.01; epslim2 = 0.01;
Material properties
E=68900; % Young's modulus nu=0.33; % Poisson's ratio lam = E/((1+nu)(1-2nu));
Allocate arrays for speed
dsize = ndpointndpointndpoint; ndata = zeros(6,dsize); sdata = zeros(2,ndpoint);
step = (epslim2-epslim1)/(ndpoint-1);
Generate data
if generate eps1=epslim1:step:epslim2; eps2=epslim1:step:epslim2; eps3=epslim1:step:epslim2; ndata(1:3,:) = combvec(eps1,eps2,eps3); sdata(1,:) = epslim1:step:epslim2; for i=1:size(ndata,2) ndata(4,i) = lam(1-nu)ndata(1,i) + lamnundata(2,i) + lamnundata(3,i); ndata(5,i) = lamnundata(1,i) + lam(1-nu)ndata(2,i) + lamnundata(3,i); ndata(6,i) = lamnundata(1,i) + lamnundata(2,i) + lam(1-nu)ndata(3,i); end disp('normal data generated');
for i=1:ndpoint sdata(2,i) = lam0.5(1-2nu)sdata(1,i); end disp('Shear data generated'); save('SEartificialData.mat','ndata','sdata');
else load('SEartificialData.mat'); end
Plot data
plot3(ndata(4,:),ndata(5,:),ndata(6,:),'o'); axis tight equal xlabel('Stress sigmazz');
figure plot(sdata(1,:),sdata(2,:)); title('Shear stress');
msize = size(ndata,2); k = randperm(msize); datared = ndata(:,k); input = datared(1:3,:); output = datared(4:6,:);
2.Normalise data
inputreg =(max(abs(input')))'; input = input./inputreg; outputreg = (max(abs(output')))'; output =output./outputreg;
3.Create network (2 hidden layers with 10 units each)
net1=feedforwardnet([10 10]);
4.Set training parameters
- trainFcn: optimisation algorithm to use (trainbr has demonstrated fastest convergence and highest accuracy for this dataset)
- epochs: number of iterations during training
- trainRatio: % of data to be used for training (default is 0.7)
- testRatio: % of data to be used for testing the network (default is 0.15)
- valRatio: % of data to be used for validating the network (default is 0.15)
- lr: Learning rate (default 0.001)
- processFcns: define what operations should be performed on the data before training, we alter the defaults as we do our own normalisation
This example uses Tansig activation functions for the hidden layers and a linear activation for the output
layer. This is a typical setting for regression problems. The actiivation functions can be changed by setting
net1.layers{i}.transferFcn to the required value. Please see Matlab documentation for details.
Note that the parameters discussed here are not complete. There are much more settings to control the
network.
% net.trainFcn = 'traingd'; % Gradient descent net1.trainFcn = 'trainbr'; % Bayesian regularisation (does not require validation d % net.trainFcn = 'trainlm'; % Levenberg-Marquardt optimization. % net.trainFcn = 'trainscg'; % Scaled Conjugate Gradient. % switch of pre and post processing net1.trainParam.epochs = 500; % net.divideParam.trainRatio=0.7; % net.divideParam.testRatio=0.15; % net.divideParam.valRatio=0.15; % net.trainParam.lr=0.001; % net.trainParam.min_grad=1e-20;
net1.inputs{1,1}.processFcns={'removeconstantrows'}; net1.outputs{1,3}.processFcns={'removeconstantrows'}; % net.trainParam;
5.Initialise training
% [net,tr] = train(net,input,output,'useParallel','yes','useGPU','yes'); [net1,tr1] = train(net1,input,output,'useParallel','yes');
save('trainedNN_normal.mat','input','inputreg','net1','output','outputreg','tr1'); else load('trainedNN_normal.mat'); end
Warning: Class 'network' is an unknown object class or does not have a valid 'loadobj' method. Object 'net1' of this class has been converted to a structure.
6.Plot traing performance
figure plot(tr1.perf,'x');hold on; plot(tr1.tperf,'--','linewidth',2); plot(tr1.vperf); legend('training','testing','validation'); set(gca,'YScale','log'); xlabel('epochs'); ylabel('lowest error (mean squared)'); title('Training performance for normal stress');
weights{1}(:,2:size(Iweights{1,1},2)+1) = Iweights{1,1}; % extract weights of all subsequent layers for i=2:size(layers,1) weights{i}(:,1) = bias{i,1}; ns = size(Lweights{i,i-1},2); weights{i}(:,2:ns+1) = Lweights{i,i-1}; end % output weights in Fortran format fid=fopen('weightsNormal.txt','w'); fprintf(fid,'c data normalisation\n'); for i=1:length(inputreg) fprintf(fid,[' Ninputreg(',num2str(i),') = %25.20f\n'],inputreg(i)); end for i=1:length(outputreg) fprintf(fid,[' Noutputreg(',num2str(i),') = %25.20f\n'],outputreg(i)); end fprintf(fid,'c weights of Neural Network\n'); for i=1:size(weights,2) for k=1:size(weights{1,i},1) for m=1:size(weights{1,i},2) fprintf(fid,[' Nweights(',num2str(i),',',num2str(k),',',num2str(m),') end end end fclose(fid);
clear weights Lweights bias Iweights datared;
Train Neural network for shear stress
This section of the script can be slow when training is performed on large data sets (but much faster than the
one for normal stress). Firstly, Matlab live scripts run much slower than standard m scripts. Consider to convert
the live script into an m file and then run it for large training sets. In addition, running it on a machine with
multiple CPU's also speeds up the process. The Matlab neural network training tool supports parallel processing
(this feature is also enables in this example).
Please note that a separate user interface will open during the network training. Training progress can
be monitored through that UI. Training can be stopped when the performance plot (open by clicking on
Performance button) does not show further improvement.
if training
Prepare input data
This network uses one input feature (one shear strain, for isotropic materials, the shear stiffness is the same for
all shear components, hence training only one is sufficient) and the corresponding shear stresse as label.
1.Randomise order
ssize = size(sdata,1); msize = size(sdata,2); k = randperm(msize);
datared = sdata(:,k); input2 = datared(1,:); output2 = datared(2,:);
2.Normalise data
inputreg2 =(max(abs(input2')))'; input2 = input2./inputreg2;
outputreg2 = (max(abs(output2')))'; output2 =output2./outputreg2;
3.Create network (2 hidden layers with 10 units each)
net2=feedforwardnet([10 10]);
4.Set training parameters
- trainFcn: optimisation algorithm to use (trainbr has demonstrated fastest convergence and highest accuracy for this dataset)
- epochs: number of iterations during training
- trainRatio: % of data to be used for training (default is 0.7)
- testRatio: % of data to be used for testing the network (default is 0.15)
- valRatio: % of data to be used for validating the network (default is 0.15)
- lr: Learning rate (default 0.001)
- processFcns: define what operations should be performed on the data before training, we alter the defaults as we do our own normalisation
This example uses Tansig activation functions for the hidden layers and a linear activation for the output
layer. This is a typical setting for regression problems. The actiivation functions can be changed by setting
net1.layers{i}.transferFcn to the required value. Please see Matlab documentation for details.
Note that the parameters discussed here are not complete. There are much more settings to control the
network.
% net.trainFcn = 'traingd'; % Gradient descent net2.trainFcn = 'trainbr'; % Bayesian regularisation % net.trainFcn = 'trainlm'; % Levenberg-Marquardt optimization. % net.trainFcn = 'trainscg'; % Scaled Conjugate Gradient. % switch of pre and post processing net2.trainParam.epochs = 500; % net.divideParam.trainRatio=0.7; % net.divideParam.testRatio=0.15; % net.divideParam.valRatio=0.15; % net.trainParam.lr=0.001; % net.trainParam.min_grad=1e-20; % net.trainParam.goal=1e-30;
Plot error distribution
% evaluate performance outerr2 = net2(input2); % predict outputs from trained network
Array indices must be positive integers or logical values.
lasterr2 = output2-outerr2; % calculate difference of labels to predictions figure nob = 40; % number of bins in histogram ploterrhist(lasterr2,'bins',nob); % set(gca,'YScale','log'); % set(gca,'XTickLabelRotation',90) % xlabel('error (label - prediction)'); title('Error distribution (label - predictions)');
Extraction of trained weights and export to text file for integration in a fortran based constitutive model
% get layers from network layers=net2.layers; % extract weights Iweights = net2.IW; Lweights = net2.LW;
% extract bias bias = net2.b; % get weights from first hidden layer weights{1}(:,1) = bias{1,1}; weights{1}(:,2:size(Iweights{1,1},2)+1) = Iweights{1,1}; % get weights of all layers that follow for i=2:size(layers,1) weights{i}(:,1) = bias{i,1}; ns = size(Lweights{i,i-1},2); weights{i}(:,2:ns+1) = Lweights{i,i-1}; end % output weights in Fortran format fid=fopen('weightsShear.txt','w'); fprintf(fid,'c data normalisation\n'); for i=1:length(inputreg2) fprintf(fid,[' Sinputreg(',num2str(i),') = %25.20f\n'],inputreg2(i)); end for i=1:length(outputreg2) fprintf(fid,[' Soutputreg(',num2str(i),') = %25.20f\n'],outputreg2(i)); end fprintf(fid,'c weights of Neural Network\n'); for i=1:size(weights,2) for k=1:size(weights{1,i},1) for m=1:size(weights{1,i},2) fprintf(fid,[' Sweights(',num2str(i),',',num2str(k),',',num2str(m),') end end end fclose(fid); clear datared;
