Per your interest on predicting probabilities, you can compute the probabilities for classification, by using the "net3" and the "trainedNet3" which use the SoftmaxLayer and the output as a string (e.g., "1", not 1.0.

{finalNet3[{9987, 13}, "Probabilities"] 
 finalNet3[{5062, 4938}, "Probabilities"] 
 finalNet3[{0, 10000}, "Probabilities"] }

Note that the following three samples are classified correctly. For example, using the 1st, 25th, and 50th samples, the 3 samples are correctly classified with the highest probabilities.

• {9987,13} -> "1": {9987,13} show the highest probability to be classified as "1"
• {5062,4938}->"25": {5062,4938} show the highest probability to be classified as "25"
• {0,10000}-> "50": {0,10000} show the highest probability to be classified as "50"

I am not sure what you mean by: "the output in training has inverted commas " ", that is {{9988, 12} -> "1"} instead of {{9988, 12} -> 1}." I think you mean string not a number.

You can compute the probabilities for each class by using the SoftmaxLayer but the SoftmaxLayer requires the output to be a string, not a number, which I called classification.

By the way, you can check and plot the sensitivities of each input by changing the 1st input while holding the 2nd input constant and vice versa.

Table[{x1, finalNet3[{x1, 5000}]}, {x1, 4000, 6000, 100}]

Hope this helps.

Please see the attachment. I provided 3 examples.

You are predicting the numerical values, e.g., {{9997, 3} -> 1 but the SoftmaxLayer[] is to classify categorical values. In other words, your output data is numerical (e.g., 1) but your network uses the error function for categorical output. The SoftmaxLayer automatically invoke the CrossEntropyLossLayer as a default which is to compute the measurements for the purpose of classification (e.g., accuracy, confusion matrix), rather than prediction (e.g., r-square or mean square)

"For nets that contain CrossEntropyLossLayer, the following built-in measurements are available"

What I learned are:

• pay attention to the type of the output, whether for prediction (numerical values) or classification (categorical value). This will determine the type of loss layer (MeanSquaredLossLayer, CrossEntropyLossLayer)
• pay attention to the type of the output value (real value, vector, or a list}, this determines the last layer and the second last layer (SoftmaxLayer[] or, LinearLayer[], SoftmaxLayer[]) as well as the number of neurons (e.g., LinearLayer[1] or LinearLayer[50]).

• Use the "TrainingNet" option to check the type of loss layer (not "TrainedNet")

  trainedNet3["TrainingNet"]

  finalNet3 = trainedNet3["TrainedNet"]

• Use the "All" option in the NetTrain[net3, trainingData3, All] to check all properties

I noticed you applied PINN and it would be nice if you can share it (I am also learning PINN).

Attachments:

wolfram communit...nb

The syntax looks correct although the training and testing data sets are usually 70% and 30% split, that is, ValidationSet->Scaled[.3]. You can split the input data into a training dataset and a validation dataset also, e.g., ValidationSet->myValidationSet. By doing this way, you can apply the NetMeasurements function to compute various measurements for your validation set.

I noticed that you are changing the one-hot coding as a number, not a "string" and think about whether the "string" would make more sense or not. If string is used, then your problem becomes classification and thus your net has to be modified, especially the last layer.

• Number? e.g., {1,0,0,0,0.......} to 1, {0,1,0,0,0.......} to 2,
• String? e.g., {1,0,0,0,0.......} to "1", {0,1,0,0,0.......} to "2",

Adding more hidden layers does not necessarily increase prediction accuracy as demonstrated by the Stephen Wolframs' blog. https://writings.stephenwolfram.com/2024/03/can-ai-solve-science/.

The followings are mere suggestions.

• The inputs are a 100 x 2 matrix (i.e., only 2 input parameters), but the outputs are a 100 x 100 matrix (a vector of 100 values due to one-hot vectors) which is huge compared to the 2 predictors. It might be helpful to change the one-hot coding as a number or string. e.g., {1,0,0,0,0.......} to 1, {0,1,0,0,0.......} to 2, etc.

  trainingData2 = Thread[trainingData[[All, 1]] -> Range[100]]
  net = NetChain[{LinearLayer[50], ElementwiseLayer["ReLU"], 
      LinearLayer[50], ElementwiseLayer["ReLU"], LinearLayer[1]}];  
  

• The "net" has only very few hidden layers and it might be helpful to increase the number of hidden layers. Adding more layers does not necessarily increase the accuracy thus apply other layers such as batch normalization layer (adding which layer may not be straightforward)

• The last 4 samples of the "trainingData" have the same input values but the output values are different, which does not make sense.