Why neural networks apply to scientific computing?

In recent years, neural networks have become an increasingly powerful tool in scientific computing. The universal approximation theorem asserts that a neural network may be constructed to approximate any given continuous function at desired accuracy. The backpropagation algorithm further allows efficient optimization of the parameters in training a neural network. Powered by GPU's, effective computations for scientific and engineering problems are thereby enabled. In addition, we show that finite element shape functions may also be approximated by neural networks.