GAUSSIAN PROCESSES SIMPLIFY DIFFERENTIAL EQUATIONS

The paper formulates the Cole–Hopf learning problem as an RKHS-norm minimization with constraints (their equation (2.4)), notes affine-type gauge freedoms, and moves to a generalized representer formulation via linear functionals (equations (2.6)–(2.10)), ultimately deriving the ODE 1/2 D'' + ν D' = 0 by taking h → 0 (equations (2.11)–(2.13)). It also states a heuristic norm-boundedness/blow-up criterion as collocation points densify, but does not prove it. These elements are correctly presented at a high level, but the gauge discussion mixes two transformations (aD+b and D+cu) while imposing only two anchors, and the boundedness claim is heuristic rather than proved . The model solution supplies a rigorous Hilbert-space projection argument for convergence and norm blow-up under standard RKHS assumptions, which substantively strengthens the paper’s heuristic. However, it (i) treats the representer structure as point-evaluation only (span{K(·, s)}), whereas the paper’s final constraints are linear functionals that include derivatives (hence require generalized representers), and (ii) mischaracterizes a three-parameter gauge fully fixed by two anchors; algebraically, two anchors cannot fix three independent parameters. Thus, both are incomplete: the paper is conceptually sound but lacks some rigor and precise gauge accounting, while the model gives the missing convergence proof but oversimplifies the constraint/representer structure and overstates the gauge invariance .