Just today, I finished a deep-learning example in C# with Accord .NET. It solves a simple XOR function, as well as classifying ascii digits.

If you're looking for a basic example in .NET, I think the XOR one is as simple as it gets.

Deep-Learning XOR: https://github.com/primaryobjects/deep-learning/blob/XOR/Dee...

Deep-Learning Digits: https://github.com/primaryobjects/deep-learning/blob/master/...