No Gelation and Global Existence for a Boltzmann Equation with Regularly Varying Mass-Exchange Rates

What the math gives to ML

The paper constructs a locally finite convex superlinear weight from dyadic hinge functions, \(\Phi(m)=m+\sum_k q_k(m-L_k)_+\), whose tail growth can be made arbitrarily strong while remaining finite under only physical-moment assumptions. Its transferable asset is a multiscale tail penalty: instead of choosing one fragile quadratic threshold, the model monitors excess mass above many scales and accumulates penalties only when those scales are crossed. A practical neural-network adaptation is a dyadic overload regularizer for MoE expert loads, with thresholds and coefficients calibrated from the expected load distribution; this should reduce rare catastrophic expert overload while imposing less pressure on normally loaded experts than a global squared penalty.