Neural Integral Equations

The paper clearly states the NIE/ANIE integral-equation form y(t)=f(t)+∫_{α(t)}^{β(t)}G(y,t,s)ds and its specialization y(t)=f(t)+∫ K(t,s)F(y(s))ds, and notes that choosing α,β recovers the Fredholm (α=a,β=b) and Volterra (α=a,β=t) cases. It also claims iterative solution and points to standard existence/uniqueness results via fixed-point theorems, and interprets self-attention as an integral/quadrature approximation. However, it does not supply a complete, self-contained proof of a contraction regime, mapping properties on L^∞, or a rigorous mask-to-integral-limits convergence result; these are only referenced or discussed at a high level. By contrast, the candidate solution provides a correct Banach fixed-point argument on X=L^∞ with explicit hypotheses (L_f, L_F, M) yielding a contraction when L_f+ML_F<1, and a clean Riemann/consistent quadrature demonstration of how masked attention sums converge to Fredholm/Volterra integrals as the grid densifies. Hence, for the posed tasks, the paper is incomplete while the model’s solution is correct. See the paper’s formulation of the operator and Fredholm/Volterra limits, and its reliance on external theorems and the attention–Nyström connection for integration via self-attention (e.g., definitions and α,β choices; existence/uniqueness via Schauder/Tychonoff; solver iteration; and self-attention ≈ Nyström/Kernel quadrature) .