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I really liked the question. Yes, we sum over derivatives. First of all think what backpropagation is trying to do: finding the affect of each parameter on the loss. So as you said: the same filter is used multiple times on the input while convolving meaning that each kernel affects the final loss in several ways, so those affects should be summed together,...

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The claim that Neural Network with a single hidden layer can model any functions is proven in Cybenko's Approximation by superpositions of a sigmoidal function. https://link.springer.com/article/10.1007/BF02551274 check also: https://en.wikipedia.org/wiki/Universal_approximation_theorem The thing is that the neural network using sigmoidal functions, which ...

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Don't know if you have this doubt anymore, but this would be helpful for those who are facing similar problems- You will need to find the correct weights with which you add these two loses by hyperparameter search. That is, find the best $\lambda$ for the loss- $$L = Loss_1 + \lambda(Loss_2)$$ Here $Loss_1$ and $Loss_2$ can be any losses. Here, we take ...

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