The paper introduces SW-KAN, a Kolmogorov‑Arnold Network that replaces traditional B‑spline activations with Stieltjes‑Wigert q‑orthogonal polynomials defined on the semi‑infinite domain (0, ∞). It addresses the domain mismatch between unbounded inputs and bounded polynomial bases by applying a smooth exponential‑of‑tanh mapping, and uses a numerically stable three‑term recurrence to evaluate polynomial expansions efficiently. Experiments on image classification and continuous function approximation show that SW‑KAN achieves better accuracy‑efficiency trade‑offs than existing polynomial KANs, especially in resource‑constrained scenarios with limited data or feature dimensionality.
By Amirhosein Azarpour, Seyyed Moein Kazemi
Polynomial-Augmented Neural Networks (PANNs) merge deep neural networks with polynomial expansions to leverage the flexibility of DNNs and the rapid convergence of polynomials. The architecture introduces orthogonality constraints, basis pruning, and polynomial preconditioning to stabilize training and improve accuracy across diverse problems. Experiments show that PANNs outperform both pure DNNs and polynomial methods in approximating smooth and limited‑smoothness functions, as well as in solving partial differential equations.
By Madison Cooley, Shandian Zhe, Robert M. Kirby, Varun Shankar
The paper discusses how backpropagation enables deep learning but does not inherently organize parameters for reusable functional components, leading to weight entanglement where overlapping parameter sets hinder independent modification. It introduces weight operators—parameterized modules that can be composed at inference—to address this, proposing a two-stage learning process that first infers operator composition and then updates only the selected operators. Vector Networks (VNs) are presented as an implementation that couples operator selection to local error-driven updates, demonstrating that learned operators can be recombined in unseen ways while keeping updates confined to the relevant parameter sets.
By Giuseppe Chindemi, Benjamin F. Grewe
arXiv:2609.35938v1 Announce Type: new
Abstract: This paper proposes an interpretable neural operator framework, the Kernel Operator Network (KernelOnet), which incorporates kernel functions explicitl...
By Yuan Guo, Hanshu Chen, Qiang Xi, Timon Rabczuk, Zhuojia Fu
NeuralCert presents a framework that learns high‑dimensional variational trial functions in a compact separable form, then spectrally diagnoses, prunes, and exactly certifies them via multimodular evaluation. The method is fully explicit and independently verifiable, and can run on a standard personal computer. Applied to three extremal problems, it demonstrates that neural optimization can discover better constructions, reveal empirical invariants useful for proofs, and expose optimization barriers that inspire new analytic or numerical approaches.
By Mark Patrick Roeling
arXiv:2606. 26705v1 Announce Type: cross Abstract: Feedforward neural network (NN) expressivity is typically studied by emulating optimal basis-expansion schemes.
By Anastasis Kratsios, Simone Brugiapaglia, Bum Jun Kim, Gregory Cousins, Haitz S\'aez de Oc\'ariz Borde
arXiv:2310.16295v2 Announce Type: replace-cross
Abstract: Neural network have achieved remarkable successes in many scientific fields. However, the interpretability of the neural network model is sti...
By Zhimin Li, Shusen Liu, Kailkhura Bhavya, Peer-Timo Bremer, Valerio Pascucci
arXiv:2503. 10496v2 Announce Type: replace-cross Abstract: Modeling natural phenomena with artificial neural networks (ANNs) often provides highly accurate predictions.
By Eirik H{\o}yheim, Lars Skaaret-Lund, Solve S{\ae}b{\o}, Aliaksandr Hubin
arXiv:2606. 06861v1 Announce Type: cross Abstract: Understanding nonlinear feature interactions is crucial in science and engineering, yet standard multilayer perceptrons (MLPs) often capture such interactions only implicitly, leading to entangled representations that can impair robustness and interpretability.
By Ziyuan Li, Uwe Jaekel, Babette Dellen
arXiv:2606. 28065v1 Announce Type: cross Abstract: Understanding model predictions is essential for physical applications, where outputs often inform safety-critical decisions, such as structural load assessment, weather warnings, and clinical diagnosis.
By Joshua Stiller, Santo M. A. R. Thies, Felix Czaja, Eyke H\"ullermeier
arXiv:2607. 02203v1 Announce Type: new Abstract: Operator learning has emerged as a powerful tool for modeling complex physical systems in functional spaces.
By Mojgan Alishiri, Amirhossein Arzani
arXiv:2608. 13572v1 Announce Type: cross Abstract: We present The Architect, a system that turns Microsoft Excel into an interactive view of deep learning mathematics.
By Mohammad Imrul Jubair, Tom Yeh