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Assuming we have big $m \times n$ input dataset, with $m \times 1$ output vector. It's a classification problem with only two possible values: either $1$ or $0$.

Now, the problem is that almost all elements of the output vector are $0$s with a very few $1$s (i.e. it's a sparse vector), such that if the neural network would "learn" to give always 0 as output, this would produce high accuracy, while I'm also interested in learning when the 1s occurs.

I thought one possible approach could be to write a custom loss function giving more weight to the 1s, but I'm not completely sure if this would be a good solution.

What kind of strategy can be applied to detect such outliers?

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As described in this post, this problem is known as "unbalanced dataset" problem, which can have different solution approaches. If you use supervised learning, augmentation approaches could help. Otherwise, unsupervised approaches need some proper distance measure for outliers detection.

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