Automatic Differentiation to Simultaneously Identify Nonlinear Dynamics and Extract Noise Probability Distributions from Data

The paper asserts that the global optimum of the modified SINDy objective L(Ξ, N̂)=es+ed must satisfy N̂=N and f(x)=Θ(x)Ξ, but gives no formal assumptions or proof, and the objective as written admits degenerate zero-valued solutions without a fidelity/regularization term on N̂. The candidate solution supplies idealized assumptions and a structured proof, but it incorrectly concludes X̂=X and N̂=N: because N̂ is unconstrained, any self-consistent trajectory X̂ for some Ξ yields L=0 by setting N̂=Y−X̂, so recovery is not unique. Hence both arguments are incomplete.