Introduction to neural networks driven by physical information

Physical information-based neural network (PINN) is a method that combines physical models and neural networks. By integrating physical methods into neural networks, PINN can learn the dynamic behavior of nonlinear systems. Compared with traditional physical model-based methods, PINN has higher flexibility and scalability. It can adaptively learn complex nonlinear dynamic systems while meeting the requirements of physical specifications. This article will introduce the basic principles of PINN and provide some practical application examples.

The basic principle of PINN is to integrate physical methods into neural networks to learn the dynamic behavior of the system. Specifically, we can express the physical method in the following form:

F(u(x),\frac{\partial u}{\partial x},x,t) =0

Our goal is to understand the behavior of the system by learning the time evolution of the system state change u(x) and the boundary conditions around the system. To achieve this goal, we can use a neural network to simulate the development of the state change u(x) and use automatic differentiation techniques to calculate the gradient of the state change. At the same time, we can also use physical methods to constrain the relationship between the neural network and state changes. In this way, we can better understand the state evolution of the system and predict future changes.

Specifically, we can use the following loss function to train PINN:

L_{pinn}=L_{data} L_{ Physics}

where L_{data} is data loss, used to simulate the known state change value. Generally, we can use the mean square error to definitely define L_{data}:

L_{data}=\frac{1}{N}\sum_{i=1}^{ N}(u_i-u_{data,i})^2

where $N$ is the number of samples in the data set, u_i is the state change value predicted by the neural network, u_{data ,i} is the corresponding real state change value in the data set.

L_{physics} is the physical constraint loss, which is used to ensure that the neural network and state changes satisfy the physical method. In general, we can use the number of residuals to definitely define L_{physics}:

L_{physics}=\frac{1}{N}\sum_{i=1}^{ N}(F(u_i,\frac{\partial u_i}{\partial x},x_i,t_i))^2

where F is the physical method,\frac{\partial u_i}{\partial x} is the slope of the state change predicted by the neural network, x_i and t_i are the space and time coordinates similar to this i.

By minimizing L_{pinn}, we can simultaneously simulate data and satisfy physical methods, thereby learning the dynamic behavior of the system.

Now let’s look at some realistic PINN demonstrations. One typical example is learning the dynamic behavior of the Navier-Stokes method. The Navier-Stokes method describes the motion behavior of the fluid, which can be written in the following form:

\rho(\frac{\partial u}{\partial t} u\cdot\nabla u)=-\nabla p \mu\nabla^2u f

where \rho is the density of the fluid, u is the velocity of the fluid, p is the pressure of the fluid, \mu is the Density, f is the external force. Our goal is to learn the time evolution of the velocity and pressure of the fluid, as well as the boundary conditions at the fluid boundaries.

To achieve this goal, we can fill in the Navier-Stokes method into the neural network to facilitate learning the time evolution of speed and pressure. Specifically, we can use the following loss to train PINN:

L_{pinn}=L_{data} L_{physics}

The definitions of L_{data} and L_{physics} are the same as before. We can use a fluid dynamics model to generate a set of state variable data including velocity and pressure, and then use PINN to simulate state changes and satisfy the Navier-Stokes method. In this way, we can learn the dynamic behavior of flowing bodies, including phenomena such as wet flows, vortices, and boundary layers, without having to first determine a complex physical model or manually derive the analysis.

Another example is the kinematic behavior of learning nonlinear wave motion methods. The nonlinear wave motion method describes the propagation behavior of wave motion in the introduction, which can be written in the following form:

\frac{\partial^2u}{\partial t^2} -c^2\nabla^2u f(u)=0

where u is the amplitude of the wave speed, c is the wave speed, and f(u) is the item of nonlinear quality. Our goal is to learn the time evolution of the wave dynamics and boundary conditions at the introductory boundaries.

To achieve this goal, we can incorporate nonlinear wave processes into neural networks to facilitate learning of the epochal evolution of wave motion. Specifically, we can use the following damage numbers to train PINN:

L_{pinn}=L_{data} L_{physics}

The definitions of L_{data} and L_{physics} are the same as before. We can use numerical methods to generate a set of state change data containing amplitudes and steps, and then use PINN to simulate the state changes and satisfy the nonlinear wave method. In this way, we can study the time evolution of waves in a medium, including phenomena such as shape changes, refraction and reflection of wave packets, without first defining complex physical models or manually deriving the analysis.

In short, the neural network based on physical information is a method that combines physical models and neural networks, which can adapt to the earth's learning of complex non-linear dynamic systems while maintaining strict satisfaction of physical laws. PINN has been widely used in fluid mechanics, acoustics, structural mechanics and other fields, and has achieved some remarkable results. In the future, with the continuous development of neural networks and automated differential technology, PINN will hopefully become a larger, stronger and more versatile tool for solving various nonlinear dynamics problems.

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