Integrating Out, Twice:The Open-System Case That Neural-Network Ensemble Theory Is Missing

Neural operators (NOs) are designed to learn maps between infinite-dimensional function spaces. We propose a novel reframing of their use. By introducing an auxiliary base-space, any finite-dimensional function can be viewed as an operator acting by composition on functions of the base-space. Through a range of benchmarks on analytic functions of increasing complexity and dimensionality, we demonstrate that NOs can match or outperform standard multilayer perceptrons and Kolmogorov--Arnold Networks in accuracy while requiring significantly fewer parameters and training time. As a real-world application, we apply a two-dimensional Tensorized Fourier Neural Operator (TFNO) to the nuclear chart, learning a correction to state-of-the-art nuclear mass models as a partially observed residual field. A TFNO ensemble reaches a held-out root-mean-square error of 198.2 keV, placing it among the best recent neural-network approaches while retaining high parameter efficiency and short training times. More broadly, these results introduce NOs as a scalable framework for finite-dimensional function interpolation, from analytic benchmarks to structured scientific data.

A persistent empirical observation is that trained neural networks outperform their neural tangent kernel (NTK) limit on tasks with compositional structure, yet a quantitative account of $\textbf{when}$ and $\textbf{by how much}$ has been lacking. Working on the unit circle, we give such an account through a dichotomy between two complexity measures of the target: its $\textbf{Fourier complexity}$, which controls NTK kernel regression, and its $\textbf{architectural complexity}$, which controls learning over depth-$L$, width-$w$ ReLU networks with the variation norm of the weights bounded by $R$. We first characterize the minimax rate of the architecture class $\mathcal{C}_{L,w,R}$, pinning it down up to a single factor of $L$: between $Ω(Lw^2R^2/n)$ and $\tilde{O}(L^2w^2R^2/n)$. We then show the NTK estimator sits $\textbf{exponentially}$ above this floor whenever the two complexities decouple: for the depth-$L$ iterated sawtooth, NTK regression needs $Ω(4^L)$ samples while the minimax floor is polynomial in $L$. Numerical experiments confirm the theoretical claims: on bandlimited smooth targets, the NTK is competitive or better, while on the hypercube sparse-parity model, a standard two-layer network beats the NTK by four to six orders of magnitude in test error. The gap is thus a function-space property, a mismatch between the kernel's smoothness bias and the target's compositional structure, rather than a generic kernel-versus-network phenomenon.

In training a neural network with gradient descent (GD), each iteration induces a linear operator that governs first-order updates to a model's internal state variables. We define this operator as the Global Empirical Neural Tangent Kernel (NTK). In finite-width networks, the NTK is typically intractable to form, leading prior work to focus on restrictive settings such as tracking outputs only or taking infinite-width limits. Here, we study the structure of the NTK for a range of models. Formulating the model state as the solution to a single global implicit constraint, we derive the NTK as a product of two operators: K, accounting for immediate parameter-to-state interactions, and P, describing internal state-to-state dependencies. For a broad class of weight-based models, including RNNs and transformers, we prove a universal Kronecker-core theorem showing that K admits an exact, computable form given by the Gram matrix of weight-site variables. This core structure reveals that the NTK is structurally bottlenecked, constraining its effective rank and giving rise to a self-referential bias whereby GD preferentially learns within dominant modes of joint hidden and input activity. For recurrent models, we examine the spectrum of the NTK and show when it is biased and low-rank in space or time under the proposed decomposition. We further demonstrate that model dynamics at initialization bias the NTK, restricting learning and preventing task components from being learned effectively. Finally, we show that the NTK associated with a self-attention transformer is likewise structurally constrained to be low-rank. Overall, we show that the NTK possesses tractable structure that explains GD bias toward task solutions and the emergence of low-rank representations. To enable use of the NTK as a practical metric, we build kpflow, a library relying on randomized matrix-free numerical linear algebra.