I need to try a Neural Network on an interesting research problem I'm faced with.
Any language is fine, although I suspect the best NN's are probably in C.
I would like to be able to integrate with Java or PHP, but it's not 100% necessary.
What is the best choice from the dozens of open-source NN frameworks out there?
If you are just exploring your options, you should go with ease of use + correctness, and worry about performance once you get promising results. For that, the neural network libraries for R are probably the most comprehensive. See, for example, here.
If you have access to Matlab, its neural networks and biostatistics toolboxes have a lot of algorithms in them; again, these are easy to use and correct if not the fastest possible.
Once you have promising results, then you can refine your question to something like, "What is the fastest implementation of a random forest algorithm?"
I am currently writing a Neural Network Admin written purely in PHP (and optimized to oblivion for respectable speed) with a helpful GUI and all.
As mentioned: http://neuralmesh.com
Research, eh?
MATLAB's Neural Network Toolbox is the way to go then, if you can get your hands on it.
Fast Artificial Neural Network Library (FANN) - its my recommendation: its fastest lib I found, nothing difficult in study.
About:
Fast Artificial Neural Network Library is a free open source neural network library, which implements multilayer artificial neural networks in C with support for both fully connected and sparsely connected networks. Cross-platform execution in both fixed and floating point are supported. It includes a framework for easy handling of training data sets. It is easy to use, versatile, well documented, and fast. PHP, C++, .NET, Ada, Python, Delphi, Octave, Ruby, Prolog Pure Data and Mathematica bindings are available. A reference manual accompanies the library with examples and recommendations on how to use the library. A graphical user interface is also available for the library.