Koopman Theory-Inspired Method for Learning Time Advancement Operators in Unstable Flame Front Evolution

The paper introduces kFNO/kCNN, defines single- and multi-step operators G and Ḡ, specifies the Sivashinsky → MS/KS setups with periodic domains, and provides strong empirical evidence that kFNO/kCNN outperform FNO/CNN in 1D and 2D, including a two-times reduction in 2D training/validation errors at β=15 (Table I) and improved autocorrelation statistics, but offers no rigorous theoretical proof of a uniform error bound or strict optimality over baseline hypothesis classes . The candidate solution sketches a semigroup/Duhamel-based argument and training-error decomposition but depends on unproven assumptions (e.g., exact realization of e^{ΔtL} in-latent across the retained spectrum, uniform Lipschitz bounds for the quadratic nonlinearity on the data support, and capacity/optimization comparisons that yield strict dominance). As such, neither document contains a complete, rigorous proof of the stated superiority claims; the paper is empirically correct but theoretically incomplete, while the model’s argument remains heuristic and under-specified.