Generalized Quadratic-Embeddings for Nonlinear Dynamics using Deep Learning

The paper formulates a data-driven lifting framework (Problem 3.1) in which a learned encoder Ψ produces lifted states z that are required, via the loss L = λ1||x − CΨ(x)|| + λ2||ẋ − C(Az + H(z ⊗ z) + B)|| + λ3||∇xΨ(x)ẋ − (Az + H(z ⊗ z) + B)||, to obey quadratic dynamics and a linear projection back to x . It also illustrates exact quadratic embeddings for analytic systems (including the rational example ẋ = −x/(1+x) with a linear projection C) . The candidate model constructs, for any finite ODE-consistent dataset, a smooth lift with added bump-function coordinates and linear lifted dynamics (H = 0) that drives all three loss terms to zero; this is a valid finite-sample feasibility result under the paper’s constraints and hyperparameters. Hence, it does not contradict the paper’s aims (which emphasize small lifted dimension and generalization), but it exposes that L admits trivial zero-loss interpolants if the lifted dimension is allowed to scale with the number of distinct samples. The candidate’s verification for the rational example matches the paper’s construction in spirit and corrects what appears to be a sign typo in the printed ẏ3 equation (the derivation yields ẏ3 = −y2y3 + 2y3^2, not y3(y2 − 2y3)) . Overall: the paper’s methodology is sound for its stated scope and demonstrations, and the model’s construction is a correct existence proof for zero training loss under Problem 3.1’s constraints.