The best replacement for neural network is svm.

The best alternatives to neural network are rated as following:
★★★★★  svm
★★★★★  decision tree
★★★★  regression
★★★★  genetic algorithm
★★★  expert system
★★  hmm
★★  knn
★  random forest
★  fuzzy logic
★  hidden markov model
★  bayesian network
★  artificial neural network
  support vector machine
  kalman filter
  gaussian process
  linear regression
  artificial intelligence

Your alternative for neural network is not listed? Suggest!
Speaking about neural network, one probably shouldn't forget about genetic algorithm.

Other search terms related to neural network and their ratings are:
★★★★★ genetic algorithm
★★★ matlab tutorial
★★ feedback
★★ weka
★ memory
★ backpropagation
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neural network
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