Geometry Induced Contraction Degradation and Stabilization of Learning Enabled Observers

What the math gives to ML

The paper identifies a concrete failure mode in learning-enabled nonlinear observers: the Jacobian of a learned measurement map multiplies the correction gain and can destroy a fixed-gain Euclidean contraction certificate. Its key transferable mechanism is local geometry normalization, choosing the correction scale inversely proportional to the norm of the product between the nominal gain and the learned measurement Jacobian. This can be transferred directly to recurrent neural state-space models and learned observers by normalizing measurement-driven latent-state updates at every time step, without retraining the measurement network. The theorem predicts a measurable contraction boundary: normalized updates should have a Jacobian norm bounded by the plant or latent transition Lipschitz constant plus a constant beta, with only an error-dependent residual term.