Different Singular Limits in a Gene Regulatory Network with Multiple Small Parameters

What the math gives to ML

The paper shows that a system with two independently small steepness parameters cannot generally be reduced by setting both parameters to zero simultaneously: the limiting phase portrait depends on the relative asymptotic scaling of the parameters. Its transferable mechanism is a blow-up of parameter space, replacing the singular corner $(\varepsilon_1,\varepsilon_2)=(0,0)$ by directional charts such as $\varepsilon_2=r\varepsilon_1$ or $\varepsilon_1=r\varepsilon_2$, followed by separate bifurcation analysis in each chart. In neural networks, this suggests treating sharpness or time-scale parameters of different nonlinear modules as independently scheduled variables rather than using one global annealing parameter. A practical implementation can monitor Jacobian stability and chart transitions while sharpening activations, predicting that training dynamics will change qualitatively when the ratio of the two sharpness scales crosses a measurable boundary.