Learning differential equations from data

The paper introduces a forward-Euler residual loss MSE(θ)=∑|−û_{i+1}+û_i+Δt·N[û;θ]|^2 (its Eq. (6)) and reports that integrating the learned N with an ODE solver reproduces trajectories on clean data but fails under 5% noise . However, it provides no formal identifiability analysis or noise-induced label perturbation bounds. The model’s solution correctly establishes: (i) the characterization of zero-loss minimizers on the training grid for noise-free data (conditional on realizability by the network class), (ii) the uniqueness argument that exact agreement of N with the true RHS along the trajectory implies exact reproduction by integration, and (iii) the noise-difference label perturbation δ_i=(ε_{i+1}−ε_i)/Δt with magnitude ≤2ε/Δt, explaining the empirical 5% noise failure reported in the paper . The only caveat is that the model’s claim MSE_min=0 tacitly assumes sufficient expressivity/realizability; this assumption is unstated in the paper and should be made explicit.