Surrogate-Based Physics-Informed Neural Networks for Elliptic Partial Differential Equations

global results on some nonlinear partial differential equations for direct and inverse problems

در این رساله به بررسی رفتار جواب های رده ای از معادلات دیفرانسیل با مشتقات جزیی در دامنه های کراندار می پردازیم . این معادلات به فرم نیم-خطی و غیر خطی برای مسایل مستقیم و معکوس مورد مطالعه قرار می گیرند . به ویژه، تاثیر شرایط مختلف فیزیکی را در مساله، نظیر وجود موانع و منابع، پراکندگی و چسبندگی در معادلات موج و گرما بررسی می کنیم و به دنبال شرایطی می گردیم که متضمن وجود سراسری یا عدم وجود سراسر...

Partial Differential Equations of Physics

The physical world is traditionally organized into various systems: electromagnetism, perfect fluids, Klein-Gordon fields, elastic media, gravitation, etc. Our descriptions of these individual systems have certain features in common: Use of fields on a fixed space-time manifold M , a geometrical interpretation of the fields in terms of M , partial differential equations on these fields, an init...

Artificial neural networks for solving ordinary and partial differential equations

We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial/boundary conditions and contains no adjustable parameters. The second part is constructed so as not to affect the initial/boundary conditions. This part involves a feedforward neura...

Deep Relaxation: partial differential equations for optimizing deep neural networks

We establish connections between non-convex optimization methods for training deep neural networks (DNNs) and the theory of partial differential equations (PDEs). In particular, we focus on relaxation techniques initially developed in statistical physics, which we show to be solutions of a nonlinear Hamilton-Jacobi-Bellman equation. We employ the underlying stochastic control problem to analyze...

Lecture Notes on Elliptic Partial Differential Equations