Solving the 3D Wave Equation Driven by PINN: MATLAB Code

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Abstract: MATLAB code for solving the 3D wave equation driven by PINN

Physics-Informed Neural Networks (PINNs) are a neural network method that combines deep learning with physical laws, aimed at solving problems involving partial differential equations (PDEs). Below is a detailed explanation of this problem:

1. Definition and Core Idea of Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) are a method that embeds physical laws into neural networks, guiding the training process of the neural network by using physical equations, boundary conditions, and initial conditions as constraints. This method can leverage known physical laws to enhance the model’s generalization ability, especially in cases of scarce or low-quality data.

2. Core Mechanism of PINNs

• Embedding of Physical Constraints: PINNs embed physical equations (such as partial differential equations) and boundary conditions into the loss function of the neural network, ensuring that the model automatically satisfies physical constraints during training. For example, the loss function typically includes terms for physical residuals and boundary condition residuals.
• Combination of Data-Driven and Physics-Driven Approaches: PINNs can utilize both data and physical knowledge for modeling, achieving good predictive results even with limited data.
• Flexibility and Scalability: PINNs can handle nonlinear problems, parameterized problems, and multiscale problems, and can address complex physical systems such as fluid dynamics, materials science, and geophysics.

3. Application Areas of PINNs

PINNs have been widely applied in various fields, including but not limited to:

• Fluid Mechanics and Fluid Dynamics: Used for simulating fluid flow, turbulence, wave propagation, etc.
• Materials Science and Engineering: Used for simulating material behavior, microstructural mechanics, heterogeneous materials, etc.
• Biomedical and Medical Imaging: Used for image reconstruction, medical imaging, biological signal processing, etc.
• Energy and Environmental Science: Used for simulating energy systems, climate modeling, environmental dynamics, etc.

4. Advantages and Challenges of PINNs

• Advantages:

• High Data Efficiency: In cases of scarce data, PINNs can improve the model’s generalization ability through physical constraints.
• Strong Interpretability: The introduction of physical constraints enhances the model’s interpretability, aiding in understanding the complex behavior of physical systems.
• High Computational Efficiency: Compared to traditional numerical methods (such as finite element methods), PINNs can have higher computational efficiency in certain cases.

• Challenges:

• Optimization Difficulty: The training process of PINNs can be complex, requiring careful tuning of hyperparameters (such as loss function weights, learning rates, etc.).
• Representation of Physical Equations: Accurately embedding complex physical equations and boundary conditions into the neural network remains a challenge.

5. Future Development Directions of PINNs

• Self-Adaptive Physics-Informed Neural Networks (SAPINNs): By introducing adaptive weights and dynamically adjusting physical constraints, further enhancing the model’s flexibility and adaptability.
• Multiscale and Multi-Physics Coupling: Developing PINNs models capable of handling multiscale and multi-physics coupling problems.
• Integration with Quantum Computing: Exploring the combination of PINNs with quantum computing to solve more complex physical problems.

Conclusion

Physics-Informed Neural Networks (PINNs) are an innovative method that combines deep learning with physical laws, effectively solving complex problems involving partial differential equations. By embedding physical constraints into the neural network, PINNs have significant advantages in cases of scarce data, low data quality, or complex physical systems. Despite challenges in optimization and representation of physical equations, PINNs show great application prospects in various fields.

2.1

This MATLAB code implements a 3D wave equation solver based on Physics-Informed Neural Networks (PINN). Below is a detailed analysis:

🧠 I. Main Functions

🔗 II. Logical Relationships

The code structure is clear and divided into five main parts:

1. Problem Definition and Data Generation: Define equation parameters, initial/boundary conditions, and generate training points.
2. Neural Network Construction: Construct a fully connected network with inputs as (x, y, z, t) and output as u.
3. Loss Function Definition: Includes three parts: PDE residual, initial condition, and boundary condition.
4. Training Loop: Train the network using the Adam optimizer.
5. Visualization: Plot loss curves, wave field slices, and generate propagation animations.

📐 III. Algorithm Steps

1. Data Sampling:

• Internal Points: Randomly sample (x, y, z, t)
• Initial Condition Points:t=0 sampled at (x, y, z)
• Boundary Condition Points: Sample (x, y, z, t) on six boundary surfaces.

2. Network Forward Propagation:

• Input:[x, y, z, t] (converted to dlarray format, supporting automatic differentiation)
• Output:u(x, y, z, t)

3. Loss Calculation:

• PDE Residual Loss: Use automatic differentiation to compute second-order derivatives and construct the wave equation residual.
• Initial Condition Loss: Ensure u(x,y,z,0) and u_t(x,y,z,0) meet the initial conditions.
• Boundary Condition Loss: Ensure that on the boundary u=0.

4. Backpropagation and Optimization:

• Use dlgradient to compute gradients, and adamupdate to update parameters.

5. Visualization:

• Plot loss curves (logarithmic scale)
• Plot the wave field distribution on the z=0 plane at different times
• Generate wave field propagation animations

🧪 IV. Technical Route

• PINN Framework: Embed physical equations into the loss function to guide the network in learning physical laws.
• Automatic Differentiation (AD): Use dlgradient to compute higher-order derivatives, avoiding numerical differentiation errors.
• Deep Learning Toolbox: Use dlnetwork to construct the network, supporting custom training loops.
• Multi-Task Loss: Jointly optimize PDE residuals, initial conditions, and boundary conditions.

📘 V. Formula Principles

⚙️ VI. Parameter Settings

Parameter	Value	Description
`<span>c</span>`	1.0	Wave speed
`<span>x_min, x_max</span>`	-1, 1	x range
`<span>y_min, y_max</span>`	-1, 1	y range
`<span>z_min, z_max</span>`	-1, 1	z range
`<span>t_min, t_max</span>`	0, 1	Time range
`<span>N_r</span>`	5000	Number of internal residual points
`<span>N_ic</span>`	1000	Number of initial condition points
`<span>N_bc</span>`	1000	Number of boundary condition points
`<span>numLayers</span>`	5	Number of hidden layers
`<span>numNeurons</span>`	100	Number of neurons per layer
`<span>numEpochs</span>`	1000	Number of training epochs
`<span>learningRate</span>`	1e-3	Learning rate

💻 VII. Operating Environment

• Software: MATLAB (recommended R2024 or higher)

✅ Summary

This code implements a complete process for solving the 3D wave equation using PINN, combining deep learning with physical modeling, suitable for wave problems with no analytical solution or complex boundary conditions. By using automatic differentiation to compute higher-order derivatives, it avoids the discretization errors of traditional numerical methods, demonstrating strong generality and scalability.

2.2

Running Effects

Complete code link:（It is recommended to copy the code link to a computer browser, use a certain F treasure to place an order, and click on the paid content to directly download the code）：https://mbd.pub/o/bread/mbd-YZWXmJlrZw==

Also can scan the QR code:

Partial Source Code

30px; } .lineNode {font-size: 10pt; font-family: Menlo, Monaco, Consolas, "Courier New", monospace; font-style: normal; font-weight: normal; }% Create a deeper neural network.rtcContent { padding: 30px; } .lineNode {font-size: 10pt; font-family: Menlo, Monaco, Consolas, "Courier New", monospace; font-style: normal; font-weight: normal; }%% Define neural network architecture (input: x,y,z,t; output: u(x,y,z,t))% -------------------------------------------------------------------------inputSize = 4;    % Input dimension: [x,y,z,t]numLayers = 5;    % Number of hidden layersnumNeurons = 100; % Number of neurons per hidden layeroutputSize = 1;   % Output dimension: wave field u% Construct network layerslayers = featureInputLayer(inputSize, 'Normalization', 'none', 'Name', 'input');for i = 1:numLayers-1    layers = [layers              fullyConnectedLayer(numNeurons, 'Name', ['fc', num2str(i)])              tanhLayer('Name', ['tanh', num2str(i)])]; % tanh activation functionendlayers = [layers          fullyConnectedLayer(outputSize, 'Name', 'output')];% Convert to dlnetwork (supports automatic differentiation)net = dlnetwork(layers);analyzeNetwork(net); % View network structure

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