Thermodynamics-informed neural networks for physically realistic mixed reality

The paper states the GENERIC dynamical form ż = L∇E + M∇S and explicitly claims that, under the degeneracy conditions L∇S = 0 and M∇E = 0 together with L skew-symmetric and M symmetric positive semidefinite, the flow conserves energy and satisfies the entropy inequality; this matches the standard chain-rule argument the model provides . The candidate solution supplies the detailed dot-product/chain-rule derivation implied—but not written out—in the paper, using x·Lx = 0 for skew L, symmetry/PSD of M, and the degeneracy conditions to obtain dE/dt = 0 and dS/dt ≥ 0. Hence both are correct, and the proof ideas are essentially the same.