Corrector Operator to Enhance Accuracy and Reliability of Neural Operator Surrogates of Nonlinear Variational Boundary-Value Problems

This work focuses on developing methods for approximating the solution operators of a class of parametric partial differential equations via neural operators. Neural operators have several challenges, including the issue of generating appropriate training data, cost-accuracy trade-offs, and nontrivial hyperparameter tuning. The unpredictability of the accuracy of neural operators impacts their applications in downstream problems of inference, optimization, and control. A framework is proposed based on the linear variational problem that gives the correction to the prediction furnished by neural operators. The operator associated with the corrector problem is referred to as the corrector operator. Numerical results involving a nonlinear diffusion model in two dimensions with PCANet-type neural operators show almost two orders of increase in the accuracy of approximations when neural operators are corrected using the proposed scheme. Further, topology optimization involving a nonlinear diffusion model is considered to highlight the limitations of neural operators and the efficacy of the correction scheme. Optimizers with neural operator surrogates are seen to make significant errors (as high as 80 percent). However, the errors are much lower (below 7 percent) when neural operators are corrected following the proposed method.Comment: 34 pages, 14 figure

Interference modelling and management for cognitive radio networks

Radio spectrum is becoming increasingly scarce as more and more devices go wireless. Meanwhile, studies indicate that the assigned spectrum is not fully utilised. Cognitive radio (CR) technology is envisioned to be a promising solution to address the imbalance between spectrum scarcity and spectrum underutilisation. It improves the spectrum utilisation by reusing the unused or underutilised spectrum owned by incumbent systems (primary systems). With the introduction of CR networks, two types of interference originating from CR networks are introduced. They are the interference from CR to primary networks (CR-primary interference) and the interference among spectrum-sharing CR nodes (CR-CR interference). The interference should be well controlled and managed in order not to jeopardise the operation of the primary network and to improve the performance of CR systems. This thesis investigates the interference in CR networks by modelling and mitigating the CR-primary interference and analysing the CR-CR interference channels. Firstly, the CR-primary interference is modelled for multiple CR nodes sharing the spectrum with the primary system. The probability density functions of CR-primary interference are derived for CR networks adopting different interference management schemes. The relationship between CR operating parameters and the resulting CRprimary interference is investigated. It sheds light on the deployment of CR networks to better protect the primary system. Secondly, various interference mitigation techniques that are applicable to CR networks are reviewed. Two novel precoding schemes for CR multiple-input multipleoutput (MIMO) systems are proposed to mitigate the CR-primary interference and maximise the CR throughput. To further reduce the CR-primary interference, we also approach interference mitigation from a cross-layer perspective by jointly considering channel allocation in the media access control layer and precoding in the physical layer of CR MIMO systems. Finally, we analyse the underlying interference channels among spectrum-sharing CR users when they interfere with each other. The Pareto rate region for multi-user MIMO interference systems is characterised. Various rate region convexification schemes are examined to convexify the rate region. Then, game theory is applied to the interference system to coordinate the operation of each CR user. Nash bargaining over MIMO interference systems is characterised as well. The research presented in this thesis reveals the impact of CR operation on the resulting CR-primary network, how to mitigate the CR-primary interference and how to coordinate the spectrum-sharing CR users. It forms the fundamental basis for interference management in CR systems and consequently gives insights into the design and deployment of CR networks