A hybrid approach for solving deterministic and stochastic partial differential equations : application to smart energy management systems
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Keywords
Partial Differential Equations (PDEs) ; Parabolic PDEs ; Allen–Cahn equation ; Cahn–Hilliard equation ; Physics-Informed Neural Networks (PINNs) ; Neural Network Solver for PDEs ; Hybrid neural network design ; Quasi-Monte Carlo sampling ; Adaptive sampling strategy ; Fourier methods ; Semi-Implicit schemes ; Gradient clipping ; AdamW optimizer ; Sobol sequences ; Smart energy management systems ; Energy functional ; Loss function evolution ; Coupled Multi-PDE system ; Performance evaluation ; Multi-scale modelingAbstract
This thesis presents a hybrid methodology for solving coupled deterministic and stochastic Partial Differential Equations (PDEs), with an application to smart energy management systems. The Allen–Cahn and Cahn–Hilliard equations are employed to model energy redistribution and variability, respectively. The combination of these equations enables the simultaneous representation of predictable dynamics and uncertain fluctuations within energy networks. To efficiently solve these equations, a Physics-Informed Neural Network (PINN) framework is implemented, which incorporates physical laws into the training process of neural networks, thus reducing the reliance on large datasets. Furthermore, an advanced system of five coupled PDEs is developed to model complex energy flows, storage behavior, and stochastic demand. The study leverages techniques such as Quasi-Monte Carlo sampling, Sobol sequences, and optimized training algorithms. Results demonstrate the proposed model's ability to deliver accurate, efficient, and scalable solutions for real-world energy management scenarios.
Postgraduate Studies Programme
Προηγμένα Συστήματα ΠληροφορικήςDepartment
Σχολή Τεχνολογιών Πληροφορικής και Επικοινωνιών. Τμήμα Ψηφιακών ΣυστημάτωνNumber of pages
74Language
EnglishCollections
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Αναφορά Δημιουργού-Μη Εμπορική Χρήση-Όχι Παράγωγα Έργα 3.0 Ελλάδα
Αναφορά Δημιουργού-Μη Εμπορική Χρήση-Όχι Παράγωγα Έργα 3.0 Ελλάδα