A solution to the inverse generator problem and related questions

What the math gives to ML

The paper constructs highly nonnormal finite-dimensional generators whose forward semigroups are uniformly bounded and exponentially decaying, while inverse dynamics exhibit large amplification. The transferable lesson is that eigenvalue-based stability of a recurrent or state-space block does not control transient growth, inverse sensitivity, or the stability of time discretization. A practical neural-network adaptation is to measure and regularize finite-horizon operator norms of continuous-time state matrices and their discretizations, rather than constraining only their eigenvalues. The theorem supplies a sharp stress-test family in which stable forward dynamics coexist with amplification of order n^alpha under inverse evolution.