A Scalable and Generalizable Pathloss Map Prediction

Large-scale channel prediction, i.e., estimation of the pathloss from geographical/morphological/building maps, is an essential component of wireless network planning. Ray tracing (RT)-based methods have been widely used for many years, but they require significant computational effort that may become prohibitive with the increased network densification and/or use of higher frequencies in B5G/6G systems. In this paper, we propose a data-driven, model-free pathloss map prediction (PMP) method, called PMNet. PMNet uses a supervised learning approach: it is trained on a limited amount of RT (or channel measurement) data and map data. Once trained, PMNet can predict pathloss over location with high accuracy (an RMSE level of $10^{-2}$) in a few milliseconds. We further extend PMNet by employing transfer learning (TL). TL allows PMNet to learn a new network scenario quickly (x5.6 faster training) and efficiently (using x4.5 less data) by transferring knowledge from a pre-trained model, while retaining accuracy. Our results demonstrate that PMNet is a scalable and generalizable ML-based PMP method, showing its potential to be used in several network optimization applications

Daugavet and diameter two properties in Orlicz-Lorentz spaces

In this article, we study the diameter two properties (D2Ps), the diametral diameter two properties (diametral D2Ps), and the Daugavet property in Orlicz-Lorentz spaces equipped with the Luxemburg norm. First, we characterize the Radon-Nikod\'ym property of Orlicz-Lorentz spaces in full generality by considering all finite real-valued Orlicz functions. To show this, the fundamental functions of their K\"othe dual spaces defined by extended real-valued Orlicz functions are computed. We also show that if an Orlicz function does not satisfy the appropriate $\Delta_2$-condition, the Orlicz-Lorentz space and its order-continuous subspace have the strong diameter two property. Consequently, given that an Orlicz function is an N-function at infinity, the same condition characterizes the diameter two properties of Orlicz-Lorentz spaces as well as the octahedralities of their K\"othe dual spaces. The Orlicz-Lorentz function spaces with the Daugavet property and the diametral D2Ps are isometrically isomorphic to $L_1$ when the weight function is regular. In the process, we observe that every locally uniformly nonsquare point is not a $\Delta$-point. This fact provides another class of real Banach spaces without $\Delta$-points. As another application, it is shown that for Orlicz-Lorentz spaces equipped with the Luxemburg norm defined by an N-function at infinity, their K\"othe dual spaces do not have the local diameter two property, and so as other (diametral) diameter two properties and the Daugavet property.Comment: 19 page

Diameter two properties and the Radon-Nikod\'ym property in Orlicz spaces

Some necessary and sufficient conditions are found for Banach function lattices to have the Radon-Nikod\'ym property. Consequently it is shown that an Orlicz space $L_\varphi$ over a non-atomic $\sigma$-finite measure space $(\Omega, \Sigma,\mu)$, not necessarily separable, has the Radon-Nikod\'ym property if and only if $\varphi$ is an $N$-function at infinity and satisfies the appropriate $\Delta_2$ condition. For an Orlicz sequence space $\ell_\varphi$, it has the Radon-Nikod\'ym property if and only if $\varphi$ satisfies condition $\Delta_2^0$. In the second part the relationships between uniformly $\ell_1^2$ points of the unit sphere of a Banach space and the diameter of the slices are studied. Using these results, a quick proof is given that an Orlicz space $L_\varphi$ has the Daugavet property only if $\varphi$ is linear, so when $L_\varphi$ is isometric to $L_1$. The other consequence is that the Orlicz spaces equipped with the Orlicz norm generated by $N$-functions never have local diameter two property, while it is well-known that when equipped with the Luxemburg norm, it may have that property. Finally, it is shown that the local diameter two property, the diameter two property, the strong diameter two property are equivalent in function and sequence Orlicz spaces with the Luxemburg norm under appropriate conditions on $\varphi$

Pseudo-Differential Neural Operator: Generalized Fourier Neural Operator for Learning Solution Operators of Partial Differential Equations

Learning the mapping between two function spaces has garnered considerable research attention. However, learning the solution operator of partial differential equations (PDEs) remains a challenge in scientific computing. Fourier neural operator (FNO) was recently proposed to learn solution operators, and it achieved an excellent performance. In this study, we propose a novel \textit{pseudo-differential integral operator} (PDIO) to analyze and generalize the Fourier integral operator in FNO. PDIO is inspired by a pseudo-differential operator, which is a generalized differential operator characterized by a certain symbol. We parameterize this symbol using a neural network and demonstrate that the neural network-based symbol is contained in a smooth symbol class. Subsequently, we verify that the PDIO is a bounded linear operator, and thus is continuous in the Sobolev space. We combine the PDIO with the neural operator to develop a \textit{pseudo-differential neural operator} (PDNO) and learn the nonlinear solution operator of PDEs. We experimentally validate the effectiveness of the proposed model by utilizing Darcy flow and the Navier-Stokes equation. The obtained results indicate that the proposed PDNO outperforms the existing neural operator approaches in most experiments.Comment: 23 pages, 13 figure