Kyriakos Georgiou

Kyriakos Georgiou, Gianluca Fabiani, Constantinos Siettos et al.

We deal with the solution of the forward problem for high-dimensional parabolic PDEs with random feature (projection) neural networks (RFNNs). We first prove that there exists a single-hidden layer neural network with randomized heat-kernels arising from the fundamental solution (Green's functions) of the heat operator, that we call HEATNET, that provides an unbiased universal approximator to the solution of parabolic PDEs in arbitrary (high) dimensions, with the rate of convergence being analogous to the ${O}(N^{-1/2})$, where $N$ is the size of HEATNET. Thus, HEATNETs are explainable schemes, based on the analytical framework of parabolic PDEs, exploiting insights from physics-informed neural networks aided by numerical and functional analysis, and the structure of the corresponding solution operators. Importantly, we show how HEATNETs can be scaled up for the efficient numerical solution of arbitrary high-dimensional parabolic PDEs using suitable transformations and importance Monte Carlo sampling of the integral representation of the solution, in order to deal with the singularities of the heat kernel around the collocation points. We evaluate the performance of HEATNETs through benchmark linear parabolic problems up to 2,000 dimensions. We show that HEATNETs result in remarkable accuracy with the order of the approximation error ranging from $1.0E-05$ to $1.0E-07$ for problems up to 500 dimensions, and of the order of $1.0E-04$ to $1.0E-03$ for 1,000 to 2,000 dimensions, with a relatively low number (up to 15,000) of features.

Kyriakos Georgiou, Guillaume Cluzel, Paul Butcher et al.

This white paper demonstrates how the assurance, reliability, and security of an existing professional-grade, open-source embedded TCP/IP stack implementation written in the C programming language is significantly enhanced by adopting the SPARK technology. A multifaceted approach achieves this. Firstly, the TCP layer's C code is being replaced with formally verified SPARK, a subset of the Ada programming language supported by formal verification tools. Then the lower layers, still written in C and on which the TCP layer depends, are modeled using SPARK contracts and validated using symbolic execution with KLEE. Finally, formal contracts for the upper layers are defined to call the TCP layer. The work allowed the detection and correction of two bugs in the TCP layer. In an increasingly connected world, where Cyber Security is of paramount importance, the powerful approach detailed in this work can be applied to any existing critical C library to harden their reliability and security significantly.

Kyriakos Georgiou, Constantinos Siettos, Athanasios N. Yannacopoulos

Building on our previous work introducing Fredholm Neural Networks (Fredholm NNs/ FNNs) for solving integral equations, we extend the framework to tackle forward and inverse problems for linear and semi-linear elliptic partial differential equations. The proposed scheme consists of a deep neural network (DNN) which is designed to represent the iterative process of fixed-point iterations for the solution of elliptic PDEs using the boundary integral method within the framework of potential theory. The number of layers, weights, biases and hyperparameters are computed in an explainable manner based on the iterative scheme, and we therefore refer to this as the Potential Fredholm Neural Network (PFNN). We show that this approach ensures both accuracy and explainability, achieving small errors in the interior of the domain, and near machine-precision on the boundary. We provide a constructive proof for the consistency of the scheme and provide explicit error bounds for both the interior and boundary of the domain, reflected in the layers of the PFNN. These error bounds depend on the approximation of the boundary function and the integral discretization scheme, both of which directly correspond to components of the Fredholm NN architecture. In this way, we provide an explainable scheme that explicitly respects the boundary conditions. We assess the performance of the proposed scheme for the solution of both the forward and inverse problem for linear and semi-linear elliptic PDEs in two dimensions.

Kyriakos Georgiou, Zbigniew Chamski, Kris Nikov et al.

Energy modeling can enable energy-aware software development and assist the developer in meeting an application's energy budget. Although many energy models for embedded processors exist, most do not account for processor-specific configurations, neither are they suitable for static energy consumption estimation. This paper introduces a comprehensive energy model for Arm's Cortex-M0 processor, ready to support energy-aware development of edge computing applications using either profiling- or static-analysis-based energy consumption estimation. The model accounts for the Frequency, PreFetch, and WaitState processor configurations which all have a significant impact on the execution time and energy consumption of edge computing applications. All models have a prediction error of less than 5%.

Kyriakos Georgiou, Samuel Xavier-de-Souza, Kerstin Eder

The Internet of Things (IoT) sparks a whole new world of embedded applications. Most of these applications are based on deeply embedded systems that have to operate on limited or unreliable sources of energy, such as batteries or energy harvesters. Meeting the energy requirements for such applications is a hard challenge, which threatens the future growth of the IoT. Software has the ultimate control over hardware. Therefore, its role is significant in optimizing the energy consumption of a system. Currently, programmers have no feedback on how their software affects the energy consumption of a system. Such feedback can be enabled by energy transparency, a concept that makes a program's energy consumption visible, from hardware to software. This paper discusses the need for energy transparency in software development and emphasizes on how such transparency can be realized to help tackling the IoT energy challenge.