Certified Rational Neural Layers: Stability and Prescribed Gain for Sequence Modeling

Long-range sequence models require expressive memory together with efficient parallel computation. Rational transfer-function layers represent long-memory linear dynamics using finitely many coefficients and enable a state-free, full-sequence implementation through parallel FFT-based convolution. However, unconstrained rational filters may become unstable during training, while the Montel-certified RTF parameterization is conservative and neither formulation directly controls input–output amplification. We introduce Certified Rational Neural Layers (CERNEL), based on two complementary constructions. First, an unconstrained reflection-coefficient parameterization represents every real monic Schur-stable polynomial of a prescribed degree. It guarantees stability throughout training without computationally expensive and conservative projections. Second, a finite Schur-recursive construction enforces a user-specified induced ${\ell }_2$-gain bound on the rational operator. Both constructions admit parallel implementation over the filter order and FFT-based full-sequence evaluation. We extensively evaluate CERNEL across several experiments and benchmark tasks spanning long-range sequence modeling, nonlinear system identification, and visuomotor control. Across these settings, CERNEL remains competitive with unconstrained rational models while providing stability or gain guarantees by construction, and improves over conservative certified parameterizations when their representational restrictions become limiting.