simulating neural networks with mathematica download usc

Moreover, we prove that only O ( | log ( ϵ ) | ) layers are needed for an ϵ-accurate solution to the inverse problem.ĭeep learning (DL) has demonstrated unparalleled accomplishments in fields ranging from image classification and computer vision ( 1– 3), to voice recognition and automated diagnosis in medicine ( 4– 6), to inverse problems and image reconstruction ( 7– 12). We introduce fast iterative restarted networks (FIRENETs), which we both prove and numerically verify are stable. We begin this theory by establishing sufficient conditions for the existence of algorithms that compute stable NNs in inverse problems. These results imply a classification theory describing conditions under which (stable) NNs with a given accuracy can be computed by an algorithm. For any positive integers K > 2 and L, there are cases where simultaneously 1) no randomized training algorithm can compute a NN correct to K digits with probability greater than 1/2 2) there exists a deterministic training algorithm that computes a NN with K –1 correct digits, but any such (even randomized) algorithm needs arbitrarily many training data and 3) there exists a deterministic training algorithm that computes a NN with K –2 correct digits using no more than L training samples.

We address this paradox by demonstrating basic well-conditioned problems in scientific computing where one can prove the existence of NNs with great approximation qualities however, there does not exist any algorithm, even randomized, that can train (or compute) such a NN. However, current DL methods typically suffer from instability, even when universal approximation properties guarantee the existence of stable neural networks (NNs). Deep learning (DL) has had unprecedented success and is now entering scientific computing with full force.

Simulating neural networks with mathematica download usc