A one-hidden-layer network computes from the spins of a 10-site Ising chain using ,

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · High-performance and data-driven computational physics · High-performance and data-driven computational physics

Scientific Computing for Physicists, Ch Neural networks on physical data: backpropagation, convolution and symmetry

Problem

A one-hidden-layer network computes ∣m∣|m| from the spins of a 10-site Ising chain using h1=ReLU(m)h_1=\mathrm{ReLU}(m), h2=ReLU(−m)h_2=\mathrm{ReLU}(-m), with m=110∑isim=\tfrac{1}{10}\sum_i s_i, followed by a sigmoid output unit y=σ(a(h1+h2−c))y=\sigma\big(a(h_1+h_2-c)\big) with a=15a=15 and c=0.15c=0.15. Consider the configuration with 7 spins up (si=+1s_i=+1) and 3 spins down (si=−1s_i=-1). ((a)) Compute mm. ((b)) Compute h1h_1, h2h_2 and h1+h2h_1+h_2, and check this equals ∣m∣|m|. ((c)) Compute the output yy to three significant figures.

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