Author: Zen and the Art of Computer Programming
1 Introduction
The average user who performs arithmetic on a computer does not understand the rules and working principles behind complex mathematical systems. For beginning learners, this “mathematical blind spot” is often unavoidable. However, by reading, watching or listening to other people's sharing, mastering some relevant basic knowledge or skills can help you understand and apply this knowledge faster, and even deepen your impression of what you have learned.
This time, we will look for some simple but widely influential systems from mathematics, and try to explain their basic logical structures in an easy-to-understand language, while giving some examples to prove their effectiveness and reliability. sex. This article will not describe mathematics professional vocabulary, but will only engage in basic introduction, definitions and conceptual descriptions.
1. System Overview
The word "system" is an important word in computer science and related fields because it defines how the various elements in a computer relate to each other and work together to produce different results. According to different classifications of systems, we can divide computer systems into the following categories:
1. Input/output device system (I/O system): refers to the external devices and interfaces of the computer system, including keyboard, mouse, screen, printer, etc. 2. Main memory system: refers to the storage of instructions and data in memory. 3. Processing unit system: refers to the computing core of the computer, that is, the part of the computer that performs various computing tasks. 4. Peripheral device system: refers to computer peripherals, such as disks, network cards, etc. 5. Bus system: refers to the data transmission between various components, connected by the bus.
In order to run various application software, in addition to the various subsystems listed above, the computer system also requires auxiliary support systems such as operating systems and file systems. The operating system is responsible for managing various resources and hardware devices, and the file system stores various files, such as documents, pictures, videos, etc.
To briefly summarize, a computer system is a high-performance device used to calculate, store and disseminate information. It includes input/output devices, main memory, processors, peripherals, buses and other subsystems. The core is the processor system. The processor system is mainly responsible for executing various algorithms and computing tasks, and communicating with other subsystems.
2. Physical system model
The physical system model is an intuitive and abstract model used to present various parameters of the system and their relationships. Physical system models usually use differential equations to represent the physical process of the system, and their form generally conforms to the Maxwell-Boltzmann distribution.
2.1 Newtonian mechanics model
The Newtonian mechanics model is the earliest physical system model proposed. It is based on Newton's laws and assumes that the three basic physical quantities of mass, velocity, and acceleration have calculus forms.
$$\frac{dx}{dt} = v \label{eq1}$$
$$\frac{dv}{dt} = \frac{-Gm_1m_2}{r^2}\theta \label{eq2}$$
$$\frac{d\theta}{dt} = \frac{\omega}{\sqrt{k}}\sin\left(\theta\right)\cos\left(f\right) + f\tan\left(\phi\right) \label{eq3}$$
Where $t$ represents time, $x, v$ represents position and velocity, $\theta,\omega$ represents angle and angular velocity, $m_i$ represents mass, $G$ represents the gravity constant, $r^2=x^2 +y^2$ represents the square of distance, $f,\phi$ represents angular velocity and steering angle.
This model assumes that mass, distance, and angle all have simple functional forms and therefore does not describe real-world physical systems well. However, due to its intuitiveness and ease of understanding, it has been popular for some time and has become a textbook for many physicists.
2.2 Boltzmann machine
The Boltzmann machine is an improved version of the physical system model. It uses a state space model to study the dynamic behavior of the system. It is composed of electronics, nuclear reactors, circuits, robotic arms, robots, etc.
Its basic state variables are position, velocity, charge, magnitude and direction.
$$x_1, x_2, x_3, v_1, v_2, v_3, n_{elec}, m_{\mathrm{core}}, l_{1} \cdots l_{n}$$
Among them, $x_1, x_2, x_3$ represents the position coordinates, $v_1, v_2, v_3$ represents the speed, $n_{elec}$ represents the number of electrons, $m_{\mathrm{core}}$ represents the reactor size, $l_1 \cdots l_n$ represents the circuit length.
The circuit is the core component of the Boltzmann machine. It consists of a series of binary gates called CNOT or NOT gates. The input voltage signal is multiplied by a matrix to activate or inhibit certain circuit paths.
$$U(x_1, x_2, x_3, v_1, v_2, v_3) = e^{-i \sum_{j=1}^nl_j \sigma(x_j)} \cdot U_\mathrm{circuit} \cdot e^{i \sum_{j=1}^nl_j \sigma(x_j)}$$
Where $\sigma(x)$ is the Hamiltonian operator, which is a unitary matrix evolved through time evolution.
Boltzmann machines can simulate many real-world physical systems, such as charged particles, semiconductors, high-temperature materials, circuits, robotic arms, etc.
2.3 Tensor network model
The tensor network model is another complex physical system model that uses tensor networks to describe the state of complex systems. Tensor networks are composed of nodes and edges. The nodes have different attribute values and the edges represent tensors.
$$\ket{\psi} = e^{-\beta H} \cdot \ket{\psi_0} \label{eq4}$$
Where $\beta$ is a bit string driving parameter, $H$ is an entropy term, and $\ket{\Psi_0}$ represents the initial state.
Tensor network models are able to capture the nonlinear and interactive characteristics of the system. However, it still has many limitations, such as the inability to simulate complex systems involving multiple modes.
3. Control system model
The control system model depicts the relationship between the input and output of the system and is used to describe the control strategy of the system and the response of the system.
3.1 Kalman filter
Kalman filter is one of the most common control system models. Its basic idea is to use the difference between measured values and predicted values to estimate the state of the system.
$$x_{k|k-1} = F_k x_{k-1} + B_k u_{k-1} + L_k w_{k-1} \label{eq5}$$
$$P_{k|k-1} = F_k P_{k-1} F^\top_k + Q_k \label{eq6}$$
Where $x_{k|k-1}$ is the current state estimate of the system, $F_k$, $B_k$, $L_k$ are the state transition matrix, control input and noise model of the system, $Q_k$ is the system noise covariance matrix.
According to the measured value $z_k$, the state estimate $x_{k|k}$ and the error covariance matrix $P_{k|k}$ can be updated.
$$K_k = P_{k|k-1} H_k (H_k^\top P_{k|k-1} H_k + R_k)^{-1} \label{eq7}$$
$$x_{k|k} = x_{k|k-1} + K_k (z_k - H_k x_{k|k-1}) \label{eq8}$$
$$P_{k|k} = (I - K_k H_k) P_{k|k-1} \label{eq9}$$
Where $H_k$ is the measurement model, $R_k$ is the measurement noise covariance matrix.
3.2 Linear Microcontroller
A linear microcontroller is a simple control system model that represents the system state as a linear function and derives an output based on the input.
$$y_k = C_k x_k + D_k u_k \label{eq10}$$
Where $y_k$ is the output of the system, $u_k$ is the control input of the system, $x_k$ is the state of the system, $C_k$ and $D_k$ are the linear transformation matrix and control gain of the system.
Linear microcontrollers can simulate many real-world systems such as DC motors, elevator propellers, booms, inertial steering wheels, controlled AC motors, etc.
3.3 Recurrent neural network controller
Recurrent neural network (RNN) is a recursive model capable of capturing and modeling time series data. Here, we can train an RNN for control. The parameters of the RNN are updated through backpropagation, and the network structure is optimized based on the system's state estimation, control input and output.
The RNN controller consists of input layer, hidden layer and output layer. The input layer receives system status, and the output layer outputs system control instructions.
For example, an RNN controller can be implemented using an LSTM unit, which is capable of capturing long-term dependencies and long-term memory.
4. Reinforcement Learning Model
Reinforcement learning models are an attempt to have a machine automatically choose an action to maximize reward. Its basic idea is to let machines learn through trial and exploration like humans do, rather than relying on fixed goals or rules.
4.1 Monte Carlo tree search
Monte Carlo Tree Search (MCTS) is a reinforcement learning method that uses the Monte Carlo method to search decision trees and build a systematic decision sequence.
$$s_0 \in S$$
$$a_0 = argmax_a Q(s_0, a; \theta), a \in A(s_0)$$
$$T(s_0, a_0) = s_1 \in S$$
$$r_0 = R(s_0, a_0, s_1)$$
$$Q(s_0, a_0;\theta) \gets r_0 + \gamma max_a' Q(s_1, a'; \theta)$$
This algorithm first selects the root node, then selects the optimal action to enter the next node; if the wrong action is selected, it will fall back to the previous node; finally, when the termination state is reached, it will return to the starting node. And perform simulations to get rewards.
4.2 Policy Gradient Method
Policy Gradient is another method of reinforcement learning that uses partial derivatives of policy parameters to optimize the action value function.
$$J(\theta) = E[R_t \delta]$$
$$\nabla_{\theta} J(\theta) = \frac{\partial}{\partial \theta}E[\sum_t \delta_t R_t]\bigg|{\theta=\theta^{\star}} = E[A_t G_t], A_t = (\pi_\theta(s_t)|s_t), G_t = \nabla{\theta} \log\pi_\theta(a_t|s_t)$$
The policy gradient method draws on the policy iteration method and uses stochastic gradient descent to solve the optimal update of policy parameters.
5. Bioinformatics model
Bioinformatics models are used to study the composition, function, and molecular mechanisms of living organisms.
5.1 Markov model
The Maslov model is one of the oldest models in bioinformatics. It describes the life cycle of a spider and a ball and assumes that their behavior has a Mahalanobis constant, the rate of fat absorption per unit time.
5.2 Spider reproductive system model
The spider reproductive system model (Spider System Model) is one of the newer models in bioinformatics. It simulates the architecture and molecular regulation process of spiders.
The model consists of five parts: the nucleus, the cyst secretion duct, the cyst secretion cyst, the ovary and the egg cleft. The cell nucleus contains growth factors, and cell cycle produces hormones that are responsible for regulating the distribution, permeability and metabolism of molecules.
The vesicle is a cell in the spider's body. It has different types of neurons and is responsible for regulating the absorption of various signal molecules and forming signal transmission between tissues.
The ovaries and egg clefts are structures within the spider's body that help complete the formation of the fetal hole and provide new seeds for reproduction.
The spider reproductive system model can predict the distribution, metabolism and interaction of molecules in the molecular regulation process of spiders, as well as the mechanisms of molecular allocation and resource allocation.
6. Engineering system model
Engineering system models focus on how to establish or maintain the stability and reliability of computer systems, computer networks, communication systems, and transportation systems to meet specific needs or characteristics.
6.1 Data Center Model
Data Center Model is a type of engineering system model that studies the architecture, layout, control and operation of data centers.
A data center consists of servers, storage devices, networks, power systems, security equipment, etc., and its scale can be expanded from a few servers to millions of devices. The data center model includes the overall design, layout and scaling plan of the data center, and also focuses on evaluating the efficiency, resource utilization and effectiveness of each system component.
6.2 Transportation system model
The Transportation System Model focuses on the system's management and optimization of passenger flow and traffic congestion.
The transportation system model includes transportation network model, traffic dispatch model, parking service model, driving behavior model and comprehensive evaluation index.
The transportation network model depicts the static network structure of the transportation system and analyzes factors such as network capacity, connection methods, and impedance matching.
The traffic dispatch model takes into account vehicle congestion and assumes differences in capabilities among different drivers in order to give optimal routes.
The parking service model studies the layout, supply and demand relationship, management mechanism, charging standards and other factors of the parking lot.
The driving behavior model studies factors such as the vehicle's driving habits, traffic conditions and environmental influences.
The comprehensive evaluation index combines all the above models and is used to determine whether the transportation system is reasonable, safe and cost-effective.