Studio delle reti neurali informate dalla fisica per la dinamica delle onde.

Abstract

Detecting structural defects is a critical issue in engineering. Researchers often use the acoustic wave equation (AWE), which is vital for various applications such as seismic imaging inversion and non-destructive testing to study wave propagation in fluids and solids. However, traditional numerical solvers have limitations, as they require discrete model representations with many restrictions placed on the shape and spacing of grid el- ements. The acoustic wave equation is a second-order linear partial differential equation, and finding its analytical solution for complex media is challenging and sometimes even impossible. A new class of methods has emerged that combines Partial Differential Equa- tions (PDEs) with a Neural Network (NN). These methods are commonly referred to as Physics-Informed Neural Networks (PINNs), and they minimize a loss function which in- corporates the given PDE by using automatic differentiation. The objective of this thesis is to gain a clear understanding of these methods for wave dynamics. Il rilevamento dei difetti strutturali è un problema critico in ingegneria. I ricercatori utilizzano spesso l'equazione delle onde acustiche (AWE), che è vitale per varie applicazioni come l'inversione dell'imaging sismico e i test non distruttivi per studiare la propagazione delle onde nei fluidi e nei solidi. Tuttavia, i tradizionali solutori numerici presentano dei limiti, poiché richiedono rappresentazioni di modelli discreti con molte restrizioni poste sulla forma e sulla spaziatura degli elementi della griglia. L’equazione dell’onda acustica è un’equazione alle derivate parziali lineari del secondo ordine e trovare la sua soluzione analitica per mezzi complessi è impegnativo e talvolta addirittura impossibile. È emersa una nuova classe di metodi che combina le equazioni differenziali parziali (PDE) con una rete neurale (NN). Questi metodi sono comunemente indicati come reti neurali informate dalla fisica (PINN) e minimizzano una funzione di p

Detecting structural defects is a critical issue in engineering. Researchers often use the acoustic wave equation (AWE), which is vital for various applications such as seismic imaging inversion and non-destructive testing to study wave propagation in fluids and solids. However, traditional numerical solvers have limitations, as they require discrete model representations with many restrictions placed on the shape and spacing of grid el- ements. The acoustic wave equation is a second-order linear partial differential equation, and finding its analytical solution for complex media is challenging and sometimes even impossible. A new class of methods has emerged that combines Partial Differential Equa- tions (PDEs) with a Neural Network (NN). These methods are commonly referred to as Physics-Informed Neural Networks (PINNs), and they minimize a loss function which in- corporates the given PDE by using automatic differentiation. The objective of this thesis is to gain a clear understanding of these methods for wave dynamics. The study explores the performance of PINNs in various scenarios, identifying and ad- dressing their limitations. It addresses the challenges of PINN scaling to high frequency and proposes transfer learning to overcome them. Lastly, we study the 2D acoustic wave equation for homogenous and heterogenous velocity models, highlighting the need for hard constraints on initial conditions. Through this work, we contribute insights into the prac- tical application of PINNs in wave dynamics, highlighting potential advancements and avenues for future research.