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I calculated the distance correlation among the independent variables and the dependent variables to verify the nonlinear relationship among the variables and the values I am getting for each independent and dependent variable pair is not more than 0.1, which means that the relationship among the variables is not strong.

So is there any way of, still, applying the neural network model to get the better the results.

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Yes you can use neural network for any non linear function. Maybe your variables don't have a strong linear correlation but you can train it using neural network to find the best pattern possible . Although it is tough to say whether it will generalize or not

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