The Viscosity Budget: Why Softmax Is Not Just a Knob
TL;DR for operators A new paper by Jose Marie Antonio Miñoza, Erika Fille T. Legara, and Christopher P. Monterola argues that a log-sum-exp neural layer is not merely analogous to a viscous Hamilton-Jacobi equation. Under the paper’s parameterisation, it is exactly the Hopf-Cole solution of one, evaluated at the input point.1 The operational point is not “neural networks are physics now”, although someone will certainly try to put that on a slide. The point is cleaner: one parameter, $\varepsilon$, simultaneously controls softmax temperature, PDE viscosity, and entropy-regularised convex optimisation. That makes smoothness, expressiveness, robustness, attribution sharpness, and scaling behaviour mathematically coupled. ...