Sparse Identification of Time Delay Systems via Pseudospectral Collocation

Claims (1), (3), and (4) in the candidate solution match the paper exactly: the paper defines the E‑SINDy and P‑SINDy objectives ϵ(τ; w) and ϵM(τ̄; w) and explicitly states that external optimization over unknown delays is k‑dimensional for E‑SINDy but univariate (only the maximum delay) for P‑SINDy; it also shows the pseudospectral reduction yielding a block ODE where only the first block depends on F, with the rest being known differentiation relations (their system (6) and discussion in §3.2) . The empirical complexity reduction from product‑grid N^k down to N for delays is borne out by their Tables and discussion (e.g., 10,000 calls for E‑BF with two delays versus 1,000 for P‑BF with only the maximum delay) . However, the paper does not provide a formal identifiability/consistency theorem of the type asserted in candidate claim (2). The candidate’s correctness argument for exact representation is mathematically reasonable but relies on strong assumptions (noise‑free data, exact simulation of the identified model, full‑rank active library) that are not spelled out or proved in the paper; moreover, P‑SINDy’s external search can include other unknown parameters (e.g., the Mackey–Glass exponent α), so the optimization may exceed one dimension in general, even though the delays reduce to a single scalar τ̄ (see §4.2 and Table 3) . Net: the paper is correct on the method and dimensionality/complexity and the collocation reduction, but theoretically incomplete; the model’s argument fills that gap at a high level yet also omits practical algorithmic subtleties (e.g., STLSQ thresholding) and assumes exact simulation.