Operator networks are used as deep learning tools for approximating solutions of partial differential equations (PDEs). Robust methods for solving PDEs have already been developed, such as the finite element method (FEM), but they have a high computational cost. The architecture of a variationally mimetic operator network (VarMiON) mimics the form of the numerical solution obtained from an approximate variational or weak formulation of a PDE making it possible to obtain an efficient method that can solve multiple instances of a problem with different input functions. The literature on operator networks has already considered the steadystate heat equation. This thesis explores the problem of solving the timedependent heat equation with a VarMiON.
Operator networks are used as deep learning tools for approximating solutions of partial differential equations (PDEs). Robust methods for solving PDEs have already been developed, such as the finite element method (FEM), but they have a high computational cost. The architecture of a variationally mimetic operator network (VarMiON) mimics the form of the numerical solution obtained from an approximate variational or weak formulation of a PDE making it possible to obtain an efficient method that can solve multiple instances of a problem with different input functions. The literature on operator networks has already considered the steadystate heat equation. This thesis explores the problem of solving the timedependent heat equation with a VarMiON.
Variationally mimetic operator neural networks for the timedependent heat equation
DELL'ORTO, MARCO
2023/2024
Abstract
Operator networks are used as deep learning tools for approximating solutions of partial differential equations (PDEs). Robust methods for solving PDEs have already been developed, such as the finite element method (FEM), but they have a high computational cost. The architecture of a variationally mimetic operator network (VarMiON) mimics the form of the numerical solution obtained from an approximate variational or weak formulation of a PDE making it possible to obtain an efficient method that can solve multiple instances of a problem with different input functions. The literature on operator networks has already considered the steadystate heat equation. This thesis explores the problem of solving the timedependent heat equation with a VarMiON.File  Dimensione  Formato  

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https://hdl.handle.net/20.500.12608/64777