arXiv:2602. 05977v2 Announce Type: replace Abstract: We introduce Clifford Kolmogorov-Arnold Network (ClKAN), a flexible and efficient architecture for function approximation in arbitrary Clifford Algebra spaces.
arXiv:2604. 21174v3 Announce Type: replace-cross Abstract: Kolmogorov-Arnold Networks (KANs) replace fixed activations with learnable univariate edge functions whose behavior depends strongly on the chosen basis.
arXiv:2606. 02936v1 Announce Type: new Abstract: In this manuscript, we propose and analyze hierarchical Kolmogorov--Arnold neural network architectures employing radial basis functions as activation functions for approximating deterministic functions and random field models.
arXiv:2606. 22984v2 Announce Type: replace-cross Abstract: Efficient sampling of the Boltzmann distribution in frustrated spin glasses is central to statistical mechanics and combinatorial optimization.
The paper introduces the Physics-Informed Stochastic Configuration Machine (PI‑SCM), a backpropagation‑free neural network designed for solving nonlinear differential equations. By analytically evaluating local Jacobians, PI‑SCM linearizes the physical loss, enabling optimal weight determination through generalized linear least squares and avoiding iterative nonlinear optimization. The authors present a progressive algorithmic suite—PI‑SC‑I, PI‑SC‑II, and PI‑SC‑III—prove their universal approximation properties, and show through experiments that PI‑SCM achieves high‑fidelity predictions and parameter identification while accelerating training by orders of magnitude compared to standard PINNs.
arXiv:2209. 01432v4 Announce Type: replace-cross Abstract: In this paper we study probabilistic and neural network approximations for solutions to Poisson equation subject to Holder data in general bounded domains of $\mathbb{R}^d$.
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.
AlphaClifford is a model‑based reinforcement learning framework that uses Monte Carlo Tree Search to synthesize Clifford circuits from the H, S, and CNOT gate set. By modeling the state space with the algebraic properties of the symplectic group, it consistently reduces total and two‑qubit gate counts compared to existing heuristics. The approach also extends to hardware‑constrained transpilation and serves as a post‑synthesis optimizer in a full Clifford+T pipeline.
arXiv:2604. 19341v2 Announce Type: replace-cross Abstract: Scientific discovery often requires many cycles of proposing, testing, and refining candidate solutions.
arXiv:2606. 09434v1 Announce Type: new Abstract: The Fokker-Planck equation (FPE) plays a pivotal role in describing the time evolution of probability density functions (PDFs) for systems governed by stochastic dynamics.
arXiv:2508. 11522v4 Announce Type: replace Abstract: Neural tangent kernels (NTKs) are a powerful tool for analyzing deep, non-linear neural networks.
arXiv:2501. 17296v4 Announce Type: replace-cross Abstract: Multiphysics simulations play an essential role in accurately modeling complex interactions across diverse scientific and engineering domains Although neural operators especially the Fourier Neural Operator FNO have significantly improved computational efficiency they often fail to effectively capture intricate correlations inherent in coupled physical processes To address this limitation we introduce COMPOL a novel coupled multiphysics operator learning framework COMPOL extends conventional operator architectures by incorporating sophisticated recurrent and attentionbased aggregation mechanisms effectively modeling interdependencies among interacting physical processes within latent feature spaces Our approach is architectureagnostic and seamlessly integrates into various neural operator frameworks that involve latent space transformations Extensive experiments on diverse benchmarksincluding biological reactiondiffusion systems patternforming chemical reactions multiphase geological flows and thermohydromechanical processes demonstrate that COMPOL consistently achieves superior predictive accuracy compared to stateoftheart methods.
These lecture notes accompany the PhD course "Physics Informed Neural Network" offered at the University of Bozen/Bolzano in 2025/2026. They introduce Physics Informed Deep Neural Networks (PINNs) and Neural Operators (NOs), covering implementation from scratch in PyTorch and with libraries such as NVIDIA PhysicsNeMo. The notes also discuss advanced topics like Mixture-of-Models, Fourier Neural Operators, and Physics‑Informed Kolmogorov‑Arnold Networks (PIKANs).