From the mathematical perspective, most engineering problems boil down to the problem of finding a mapping, $f$$f$ff$f$, from the input, $x$$x$xx$x$, to the output, $y$$y$yy$y$ :

Traditional physics-based engineering methods and data-driven methods like numerical AI (machine learning) are just two different ways of finding such a mapping. Coupled physics-based analysis represented by multiphysics, multiscale, and multi-fidelity and AI analysis represented by deep and reinforcement learning reflect the trends of collaboration and automation in modern industries, respectively, and both address the needs for more complicated and accurate analysis. Thus, both types of methods represent the future directions of modern industries and deserve close attention regardless of the rises and falls of AI compared with traditional engineering analysis. Traditional physics-based methods use mathematical equations derived on the basis of physics rules to construct models. These models are employed to explain or predict phenomena, e.g. states and processes.

By contrast, machine learning aims to find a model that approximates the solution to a real-world problem by analyzing the data. Such models need to be constructed with tools that computers can conveniently process and human beings can easily understand. Thus, mathematical models like basic mathematical functions (like polynomials) or linear algebra equations (formulated using arrays) are intuitive choices. In fact, mathematical models are what have been predominantly adopted. These mathematical functions are also referred to as "mathematical models" or just models. As a result, machine learning models are usually viewed as mathematical equations/functions that represent or model real-world problems/scenarios. Terms such as "mathematical form", "mathematical being", and "machine" are also used in the replace of
"models" in some literature. In some cases, machine learning models are also called function approximations. This is because it is usually difficult to find exact functions to represent real-world problems.

Therefore, data-driven AI, like machine learning, also uses models. However, distinct from physics-based methods, machine learning derives the model from data during the analysis instead of constructing the model based on physics before the analysis.

As illustrated in Fig. 2.3, in traditional engineering analysis, we first build the model, e.g., a mathematical model that derives the analytical solution and a numerical model like a finite element analysis model, for a problem based on its underlying physics. Then this model, together with some data that informs the initial/boundary conditions and material properties, will be fed into the model to obtain an answer, which can be predictions of some state variables or others.

In machine learning analysis, the construction of the model occurs during the analysis instead of before the analysis. Also, the construction of the model is a significant or even a major part of machine learning analysis in many cases. The word "training" is used to refer to this process of generating the model. Due to this reason, this word is used everywhere in data-driven methods. Notwithstanding, it may be totally new to people who are not familiar with machine learning. To generate a model in machine learning, we need to get the basic construction materials:

In particular, algorithms dictate how data can be used to generate models. Therefore, algorithms form a major body of the knowledge for machine learning. That is, machine learning is usually introduced in terms of different types of algorithms. By contrast, "models" represent the knowledge extracted from the data based on the employed algorithms. In a simple way, we can understand data as cooking materials, algorithms as recipes, and models as cooked food or dishes. Thus, a typical machine learning cooking book like this book does not talk about "models" specifically in detail, but instead, it focuses more on common recipes, typical ingredients, and their treatments, as well as tricks for cooking and evaluating dishes.

In machine learning, as illustrated in Fig. 2.3, after a model is obtained or "trained", this trained model will be used to obtain answers to new problems based on new data for these problems. This "testing" process is similar to the traditional engineering analysis.

Figure 2.3: Workflows of physics-based methods (left) and data-driven methods (right)

A simple engineering problem is employed here to show how data-driven methods and traditional physics-based engineering methods can be used to solve the sample problem. The problem to be considered is to analyze and predict the trajectory of a ball. As shown in Fig. 2.4, the ball moves off the edge of an object with a horizontal velocity of $v_0=5$$v_0=5$v_(0)=5v_{0}=5$v_0=5$ $\mathrm{m}/\mathrm{s}$$\mathrm{m}/\mathrm{s}$m//s\mathrm{m} / \mathrm{s}$\mathrm{m}/\mathrm{s}$. The goal is to find out the trajectory of the moving ball from 0 s to 10 s in terms of a function $y=f(x)$$y=f(x)$y=f(x)y=f(x)$y=f(x)$ with a gravity constant of $g=9.81\mathrm{m}/{\mathrm{s}}^2$$g=9.81\mathrm{m}/{\mathrm{s}}^2$g=9.81m//s^(2)g=9.81 \mathrm{~m} / \mathrm{s}^{2}$g=9.81\mathrm{m}/{\mathrm{s}}^2$.

Let us first take a look at two common traditional engineering methods: analytical solution and numerical analysis. Both methods are performed based on the physics underlying the process and a mathematical formulation of the process. The analytical method involves framing the problem in a well-understood form and calculating the exact solution. Numerical analysis is based on a numerical procedure that approaches the solution to the problem in a continuous world using a numeric approximation.

To solve the above problem using the analytical method, we first need to recall the major physics: Newton's second law $(f=m\cdot a)$$(f=m\cdot a)$(f=m*a)(f=m \cdot a)$(f=m\cdot a)$. Along the $x$$x$xx$x$ (horizontal) direction, the initial velocity will remain as a constant, because no force along the horizontal direction implies no acceleration. Thus, the traveling distance is $x=v_0\cdot t$$x=v_0\cdot t$x=v_(0)*tx=v_{0} \cdot t$x=v_0\cdot t$, in which $v_0$$v_0$v_(0)v_{0}$v_0$ is the initial velocity

Figure 2.4: Example of problem solving using different types of methods
of the ball along the horizontal direction and $t$$t$tt$t$ is time. Along the $y$$y$yy$y$ (vertical) direction, $m\cdot \mathrm{g}=m\cdot a$$m\cdot \mathrm{g}=m\cdot a$m*g=m*am \cdot \mathrm{~g}=m \cdot a$m\cdot \mathrm{g}=m\cdot a$, so the acceleration along the vertical direction is g . Accordingly, the vertical velocity is $\mathrm{g}\cdot t$$\mathrm{g}\cdot t$g*t\mathrm{g} \cdot t$\mathrm{g}\cdot t$ and the traveling distance is $y=\frac{1}{2}\mathrm{g}{t}^2$$y=\frac{1}{2}\mathrm{g}{t}^2$y=(1)/(2)gt^(2)y=\frac{1}{2} \mathrm{~g} t^{2}$y=\frac{1}{2}\mathrm{g}{t}^2$. Based on the above formulations, we can easily derive the function $y=f(x)$$y=f(x)$y=f(x)y=f(x)$y=f(x)$ as follows:

Based on the above deduction, the analytical solution is obtained by substituting the known constants into the equation: $y=0.01962x^2$$y=0.01962x^2$y=0.01962x^(2)y=0.01962 x^{2}$y=0.01962x^2$.

Numerical analysis is usually preferred for complicated problems, especially those that cannot be easily addressed with an analytical solution, for example, problems with high nonlinearity and high dimensionality. Thus, it is usually not adopted for the above simple problem. Here, we just use it to show the idea of numerical analysis. For this purpose, we discretize the time, e.g., using a timestep of 0.1 s for the 10 s . Then, the numerical solution to the above problem can be obtained via the following iterative process.

Initialize distance and velocity: $x_i=0,v_{xi}=v_0,y_i=0,v_{yi}=0,i=0$$x_i=0,v_{xi}=v_0,y_i=0,v_{yi}=0,i=0$x_(i)=0,v_(xi)=v_(0),y_(i)=0,v_(yi)=0,i=0x_{i}=0, v_{x i}=v_{0}, y_{i}=0, v_{y i}=0, i=0$x_i=0,v_{xi}=v_0,y_i=0,v_{yi}=0,i=0$, and ${\mathrm{\Delta }}_t=0.1$${\mathrm{\Delta }}_t=0.1$Delta_(t)=0.1\Delta_{t}=0.1${\mathrm{\Delta }}_t=0.1$
Repeat until $i=100$$i=100$i=100i=100$i=100$ (i.e., 10/0.1):

Data-driven methods seek the solution from data. Thus, distinct from the above physics-based methods, data-driven methods need to obtain some data first. Such data can be obtained from experiments or computer simulations. To illustrate this process, let us generate some data from the analytical solution and add random noise to represent "experimental data" with different sources of errors in real-world systems and measurements. Then, if we use a second-order polynomial function to fit the "experimental data", we can also obtain a model that describes the data. Such a model can also be used for predicting future behavior, for example, the trajectory beyond 10 s .

The results of the three methods and the "experimental data" are shown in Fig. 2.5. In this simple example, we can see that these three methods can help us achieve the same goal with comparable performance. In real-world engineering practices, the selection of methods usually relies on the characteristics of the problem, such as the nature and complexity of the problem, the knowledge about the physics and material properties, the available computational power, and the expectation for computing time and accuracy. Such a selection can be complicated and may require a certain level of expertise.