Hi Raymond,
I understand you have two vectors "x" and "y" of sizes 3 and 4 with a non-linear relationship between them. It is clear that "y" cannot be obtained analytically from "x", and you want to know if there is a way to train a neural network or other similar models to learn the mapping between the two.
Given the nonlinear relationships between the vectors "x" and "y" that you've described, it is indeed possible to use machine learning models, such as neural networks, to approximate the inverse mapping from "x" to "y". Since you have explicit formulas for "x" in terms of "y", you can generate a synthetic dataset where you sample values of "y" and calculate the corresponding "x" values. With this dataset, you can train a model to learn the mapping from "x" back to "y".
Here's an example of how you might implement this in MATLAB:
% Generate a dataset of x and y pairs
N = 10000; % Number of data points
Y = rand(N, 4); % Randomly generate Y vectors
X = [Y(:,1) + Y(:,3) - Y(:,3).^2 ./ (Y(:,2) + Y(:,3)), ...
Y(:,3).^2 ./ (Y(:,2) + Y(:,3)), ...
Y(:,3) .* Y(:,4) ./ (Y(:,2) + Y(:,3)).^2]; % Calculate corresponding X vectors
% Now we need to choose a machine learning model. A feedforward neural network with one or more hidden layers could work well for this task.
% Create a feedforward neural network
layers = [
featureInputLayer(3, 'Normalization', 'zscore')
fullyConnectedLayer(64)
reluLayer
fullyConnectedLayer(64)
reluLayer
fullyConnectedLayer(4) % Output layer size is 4, since Y has 4 elements
regressionLayer];
% Training the model
% Set training options; you can experiment with these according to your needs
options = trainingOptions('adam', ...
'MaxEpochs',1000, ...
'MiniBatchSize', 64, ...
'InitialLearnRate', 1e-3, ...
'LearnRateSchedule', 'piecewise', ...
'LearnRateDropPeriod', 25, ...
'LearnRateDropFactor', 0.8, ...
'Shuffle', 'every-epoch', ...
'ValidationData', {X, Y}, ...
'ValidationFrequency', 30, ...
'Verbose', false, ...
'Plots', 'training-progress');
% Train the neural network
net = trainNetwork(X, Y, layers, options);
% Predict Y from new X using our trained model
new_X = [0.5, 0.2, 0.3]; % Example new X vector
predicted_Y = predict(net, new_X);
% Display the predicted Y
disp(predicted_Y);
I hope this helps!
