Finite

In this post, we will discuss various distributions. The role of these distributions is to model the probability distribution $p(x)$ of a random variable $x$ when a set of observed data is given, which is known as density estimation. Before addressing Gaussian distributions for continuous variables, we will first discuss distributions for discrete variables. Bernoulli Distribution To give an exa..

In the previous post, we covered basic and popular distribution methods, and in this post, we will study one of the most important probability distributions for continuous variables, the Gaussian distribution. The Gaussian distribution for a single real-valued variable $x$ can be defined as follows. The Gaussian distribution represents the probability density for $x$ with two parameters, referre..

Understanding distributions in deep learning is important in several ways. Firstly, since data itself follows various probability distributions, understanding these distributions helps in grasping the intrinsic characteristics of the data. For example, exponential or Laplace distributions reflect certain features that real-world data can possess. This understanding is essential for effectively p..

The content on machine learning is based on studies from various books and papers, primarly focusing on Chris Bishop's "Deep Learning - Foundations and Concepts" In machine learning, it is essential to deal with uncertainty. For example, this is because a system that classifies dogs and cats cannot actually achieve perfect accuracy. Uncertainty can be broadly divided into two types. The first is..

We derive the fundamental differential equation for one-dimensional heat flow due to thermal conduction, convection, and mass transfer in a fluid. The Galerkin's residual method is used for problems where variational methods are not applicable. We start with the equation for thermal energy conservation discussed in the previous post(42_One-Dimensional Heat Conduction). The mass transfer term(${q..

Today, based on what we studied in the previous post, I'll solve an example problem. In the figure above, we calculate the temperature distribution along the insulated external boundary. The temperature at the left side is ${40^ \circ }{\text{C}}$, the convection coefficient $h = 100{\text{W/}}{{\text{m}}^{\text{2}}}{ \times ^{\text{o}}}{\text{C}}$, the free-steam temperature ${T_\infty } = {10^..

Having recently addressed a 1-D problem, today I aim to solve a 2-D problem. The goal of this post is to study the necessary formulas before solving an example. First, the basic element is a triangle with three nodes, and the temperature function is given as follows. Where, ${t_i}$, ${t_j}$ and ${t_m}$ are the nodal temperatures, and the shape functions are as follows. The same applies to ${N_j}..

Similar to the problem addressed in the previous post, in this post, I'll solve an example that calculates the temperature distribution and rate of heat in a rod fixed to a wall, without any insulated parts. In the figure above, the temperature at the left end is fixed at ${100^{\text{o}}}{\text{C}}$, ${K_{xx}} = 60{\text{W/(m}}{{\text{ - }}^{\text{o}}}{\text{C)}}$, radius is $50mm$, $h = 800{\t..

Based on what we studied in the previous post, I'll solve an example problem. In the figure above, we calculate the temperature distribution along the insulated external boundary. The temperature at the left end is fixed at ${40^ \circ }{\text{C}}$, and the free-stream temperature is ${-10^\circ}{\text{C}}$. The value of $h$ is ${\text{55W(}}{{\text{m}}^{\text{2}}}{{\text{ - }}^{\text{o}}}{\text..

Continuing from the previous post, I will write the remaining part. In the heat transfer problem, we derived the element thermal conductivity matrix. From the thermal conductivity matrix, the following equation can be derived. The above equation represents the heat conduction and convection equation. Therefore, it can be summarized as follows. Using the first integral term, the thermal conductio..