Ten independently seeded 50-ReLU networks are each fitted to the Lennard-Jones pair…
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
Ten independently seeded 50-ReLU networks are each fitted to the Lennard-Jones pair potential U(r)=4ϵ[(σ/r)12−(σ/r)6] using training data only on r∈[0.95σ,2.5σ]. Inside that interval all ten agree with U(r) and with each other to within 0.05ϵ. Extrapolated to r=0.85σ, the ten predictions cluster around 6.9ϵ, with a standard deviation across the ensemble of only 0.25ϵ. Someone argues: "Universal approximation guarantees that a network with enough hidden units can approximate any continuous function to any accuracy, and the tight agreement of ten independently trained networks at r=0.85σ confirms the prediction there is reliable."
((a)) Compute the true value of U(0.85σ)/ϵ from the Lennard-Jones formula.
((b)) State what universal approximation actually guarantees, and explain why it says nothing about r=0.85σ.
((c)) Explain why the ensemble's tight agreement at r=0.85σ does not certify the prediction there, given what the ten members have in common.