PINN-Obs: Physics-Informed Neural Network-Based Observer for Nonlinear Dynamical Systems

The paper’s Theorem 1 claims: (i) for any gain L(t) the observer ODE has a unique solution, and (ii) empirical minimizers hN converge in C0(Ω) to that solution. However, Step 1 conflates detectability with existence and proves only “there exists L” while the statement asserts “for any L,” and it omits standard growth/boundedness hypotheses needed to preclude finite-time blow-up on Ω; moreover, g is defined using x(t) at first and later replaced by y(t), creating a consistency gap around (5) and its use in the proof . Step 2 jumps from vanishing empirical loss to vanishing sup norms and uniform convergence without a uniform law of large numbers, uniform Hölder bounds, or compactness of the hypothesis class; it also assumes exact zero empirical loss is attainable for each N and cites a PDE PINN bound to justify ODE generalization (Lemma 2), which is not enough to conclude sup-norm control or argmin consistency in the stated setting . The candidate model solution fixes Step 1 rigorously (via Picard–Lindelöf + linear growth on bounded Ω and bounded Hölder L), and sketches a correct ERM/argmin route for Step 2 if one adds compact-parameter, equicontinuity, and a uniform SLLN. But it still presumes the population minimizer achieves zero risk inside the fixed architecture, which requires an attainability assumption not established. Hence both are incomplete.