Mesh-Uniform Power Stability of Two-Relaxation-Time Vector Lattice Boltzmann Schemes with Reversible Boundaries

What the math gives to ML

The paper provides a transferable recipe for constructing recurrent linear operators whose powers remain uniformly bounded despite non-normal ordering of transport and relaxation. Its key ingredients are an equilibrium-metric unitary transport, velocity-reversal symmetry, and paired even/odd relaxation rates satisfying s_+ + s_- = 2 with both rates in (0,2). This suggests a recurrent or state-space neural block in which feature channels are split into reversible pairs, mixed by an orthogonal or metric-unitary operator, and damped by complementary relaxation. The numerical-range ellipse and Crouzeix–Palencia argument also provide a practical spectral diagnostic for detecting transient amplification before expensive training.