A New Low-Rank Cholesky-Factor ADI Algorithm Allowing Shifts Anywhere in the Complex Plane with Applications to Data-Driven Model Reduction

What the math gives to ML

The paper contributes a generalized low-rank Cholesky-factor ADI method for Lyapunov, frequency-limited Lyapunov, and Riccati equations whose shifts may lie anywhere in the complex plane, including the imaginary axis. Its most transferable mechanism is frequency-targeted Gramian computation: imaginary-axis shifts allow controllability and observability information to be extracted from experimentally accessible transfer samples rather than unstable right-half-plane samples. A practical neural-network transfer is to compress stable linear state-space or recurrent layers using frequency-limited balanced truncation, preserving input-output behavior in frequencies relevant to the task. The method predicts a measurable tradeoff between retained Hankel singular values and frequency-domain error, while the ADI residual provides an implementation-time certificate.