# Solving Partial Differential Equations Using Deep Learning and Physical Constraints
## 3 Experiments and Results
### KdV-Burgers Equation  
We have studied the KdV-Burgers equation to further analyze the ability of physics-informed neural networks to solve complex partial differential equations. The KdV-Burgers equation is a nonlinear partial differential equation containing higher-order derivatives that has been of interest to many researchers [56; 57]. Today, the KdV-Burgers equation is widely studied and applied in many fields, such as the study of the flow of liquids containing bubbles, the flow of liquids in elastic tubes, and other problems [58; 59]. In mathematical form, the KdV-Burgers equation is defined, as follows:  
\[u_{t}+\alpha uu_{x}+\beta u_{xx}+\gamma u_{xxx}=0 \tag{15}\]  
where \(\alpha\), \(\beta\) and \(\gamma\) are all non-zero real constants, i.e., \(\alpha\beta\gamma\
eq 0\). Equation (15) can be reduced to Burgers equation [60] or Korteweg-de Vries (KdV) equation [61] in special cases. Specifically, when \(\gamma=0\), the Equation (15) is simplified to Burgers equation. \[u_{t}+\alpha uu_{x}+\beta u_{xx}=0 \tag{16}\]  
The Burgers equation is a second-order nonlinear partial differential equation, which is used to simulate the propagation and reflection of shock waves. This equation is used in various fields of research, such as fluid dynamics and nonlinear acoustics (NLA) [60; 62]. Besides, Equation (15) becomes the Korteweg-de Vries (KdV) equation when \(\beta\) is zero. \[u_{t}+\alpha uu_{x}+\gamma u_{xxx}=0 \tag{17}\]  
Figure 4: Comparison of the prediction given by physics-informed neural networks with the exact solution. Figure 3: Solution of the wave equation given by physics-informed neural networks. The Korteweg-de Vries (KdV) equation was first introduced in 1985 by [PERSON] and [PERSON]. It is a very important equation, both mathematically and practically, for the description of small amplitude shallow-water waves, ion-phonon waves, and fluctuation phenomena in biological and physical systems [63; 64]. This equation, which differs from the Burgers equation in that it does not introduce dissipation and it can explain the existence of solitary waves, is of great interest to physicists and mathematicians. Therefore, the study of this equation is of great scientific significance and research value. The KdV-Burgers equation can be viewed as a combination of the Korteweg-de Vries equation and the Burgers equation, containing the nonlinearity \(uu_{xx}\), the dispersion \(u_{xxx}\), and the dissipation \(u_{xx}\), with high complexity. The equation has been applied in many fields and it has received a great deal of attention from many researchers. In this section, we use physics-informed neural networks to develop new methods for solving the KdV-Burgers equation. In this experiment, an important task is to construct a high-quality training data set based on partial differential equations. For Equation (15), the values of \(\alpha,\,\beta,\,\gamma\) are given as \(1\),\(-0.075,\pi/1000\), respectively, and deterministic initial conditions are given. In mathematical form, the nonlinear KdV-Burgers equation with periodic boundary conditions studied in this section is defined, as follows  
\[ \begin{array}{l}u_{t}+uu_{x}-0.075u_{xx}+\pi/1000u_{xxx}=0,\quad x\in[-1,1], \quad t\in[0,1]\\ u(0,x)=e^{(0.005*cos(\pi*x))}sin(\pi*x)\\ u(t,-1)=u(t,1)\\ u_{x}(t,-1)=u_{x}(t,1)\end{array}  \tag{18}\]  
For Equation (18), we simulate it using conventional spectral methods and use the Chebfun package [65] in the programming implementation. Specifically, we integrate Equation (18) from the initial moment \(t=0\) to the final time \(t=1.0\) using a time step \(t=10^{-6}\), depending on the initial and periodic boundary conditions. Besides, we use a fourth-order explicit Runge-Kutta temporal integrator and a spectral Fourier discretization with 512 modes to ensure the accuracy of the integration. After obtaining the high-resolution dataset, we next constructed a neural network to approximate the solution of Equation (18), which is denoted as \(\hat{u}(t,x)\).