Thermodynamics-informed Graph Neural Networks

The paper states the GENERIC/metriplectic evolution ż = L∇E + M∇S with L skew-symmetric, M symmetric positive semidefinite, and the degeneracy conditions L∇S = 0 and M∇E = 0, and then asserts the consequences dE/dt = {E,E} = 0 and dS/dt = [S,S] ≥ 0 (its Eq. (5)) as the first and second laws of thermodynamics in this framework . The candidate solution gives the standard chain-rule derivation using exactly these hypotheses (skew-symmetry, symmetry/PSD, and degeneracy) and rewrites the results in bracket form, which is the same reasoning the paper relies on (though the paper presents it succinctly). Hence both are correct and essentially the same proof idea.