Learning Time Delay Systems with Neural Ordinary Differential Equations

The paper introduces a clear NODE-to-NDDE construction via history discretization and trainable delay taps, and supports it with empirical results on Mackey–Glass, including learned bifurcations and over-parameterized delay clustering, but provides no formal theorems or proofs (only methodology and experiments). Its key components—discretization X(t) with DM, interpolation matrix P, TDNN right-hand side, loss and delay-gradient—are explicit and internally consistent . The candidate model solution supplies plausible theoretical underpinnings: (i) a compact-set universal-approximation statement for the lifted discrete-delay vector field, and (ii) finite-horizon solution closeness plus bifurcation persistence under C^1-small perturbations, and (iii) identifiability-driven delay clustering when over-parameterizing. However, these arguments rely on substantial additional assumptions (e.g., C^1-uniform closeness of the learned nonlinearity over a parameter-state neighborhood for τ∈[0,2]; quantitative delay identifiability/persistent excitation) that are not derived from the paper’s setup or training protocol. Empirically, the paper’s bifurcation and clustering claims match the reported results (Hopf near 0.24; period doublings near 0.61 and 0.84; clustering of learned delays near 0 and 1) , but the model’s formal claims remain conditional. Therefore, both the paper and the model solution are incomplete: the paper lacks proofs; the model provides reasonable theoretical scaffolding but leaves key hypotheses unverified.