Last update: 2020/03/24 (I will update this answer from time to time...)
Simple Implementations of Neural Networks
- FANN is a very popular implementation in C/C++ and has bindings for many other languages.
- In scikit-learn (Python) 0.18 (current developement version) there will be an implementation of feed-forward neural networks (API documentation).
- And I must mention my own project, which is called OpenANN (Documentation). It is written in C++ and has Python bindings.
Deep Learning
Because neural networks are quite popular in research and industry at the moment ("deep learning") there are many research libraries available. Most of them are kind of easy to set up, integrate, and use. Although not as easy as the libraries mentioned above. They provide leading edge functionality and high performance (with GPUs etc.). Most of these libraries also have automatic differentiation. You can easily specify new architectures, loss functions etc. and don't have to specify the backpropagation manually.
- TensorFlow from Google (C++/Python)
- PyTorch from Facebook, in Python, can be extended with C/C++
- mxnet (C++, Python, R, Scala, Julia, Matlab, Javascript)
- Deeplearning4j (Java)
- CNTK from Microsoft (training in Python / evaluation in C++/C#/Java/Python)
- Chainer (Python)
- PaddlePaddle from Baidu in CUDA/C++ with Python bindings
- NNabla from Sony in Cuda/C++11 with Python bindings
A performance comparison for GPU-accelerated libraries can be found here (a bit outdated unfortunately). A comparison of GPUs and library versions can be found here.
Inactive:
- Keras: It could use Tensorflow, Theano, and CNTK as a backend. (Now part of tensorflow as its high-level interface.)
- Caffe from Berkeley Vision and Learning Center in C++ with Python bindings
- Darknet: CNNs in C, known for the implementations of the YOLO object detector.
- Neon from Intel Nervana provides very efficient implementations (Python)
- MatConvNet (Matlab)
- Theano (Python) and its high-level APIs:
- Pylearn 2
- Theanets
- scikit-neuralnetwork
- Lasagne
- Blocks based on Theano (Python)
- cuda-convnet2 in CUDA/C++ with Python bindings
- Hebel (Python)
- Caffe2 from Facebook in C++ with Python bindings; has been joined with PyTorch
- Neural Networks for Torch 7 (Lua, Torch 7 is a "Matlab-like environment", overview of machine learning algorithms in Torch)
- PyBrain (Python) contains different types of neural networks and training methods.
- Encog (Java and C#)