Research on Solving Convection-Diffusion Equations and Stiff Phase-Field Equations Using Physics-Informed Neural Networks

This article focuses on the research of using Physics-Informed Neural Networks (PINN) to solve two types of challenging partial differential equations (convection-diffusion equations and stiff phase-field equations) in the field of computational fluid dynamics. PINN embeds physical laws into the neural network's loss function and does not require mesh discretization, providing a new solution to the bottlenecks of traditional numerical methods (finite difference, finite element) in high-dimensional problems, complex geometries, or inverse problems. The research covers the PINN framework, solution strategies for the two types of equations, experimental verification, and future directions.

In the fields of computational fluid dynamics and materials science, numerical solution of partial differential equations is a core challenge. Traditional methods (finite difference, finite element) have problems such as high computational cost and difficult mesh generation in high-dimensional problems, complex geometries, or inverse problems. As an emerging paradigm, PINN's core is to embed physical laws (in the form of partial differential equations) into the neural network's loss function. During training, it both fits observed data and satisfies physical constraints, and has mesh independence, making it suitable for handling high-dimensional and complex boundary conditions.

This research targets two important equations in engineering and physical applications:

1. Convection-diffusion equations: Widely used in fluid dynamics, heat transfer, and environmental science, they describe the transport of substances/heat in flowing media. When convection is dominant, numerical oscillations and spurious diffusion are prone to occur;
2. Stiff phase-field equations: Used for simulating interface evolution such as material phase transitions and crystal growth. Due to large differences in time scales, explicit integration requires extremely small time steps, posing great stability challenges.

The core architecture of PINN includes:

1. Neural network approximator: Takes spatiotemporal coordinates as input and outputs the solution of the equation, with flexible structures (fully connected, residual network, etc.);
2. Physics-constrained loss: Constructs residuals by calculating derivatives via automatic differentiation, and the loss includes equation residuals and initial/boundary condition residuals;
3. Multi-task optimization: Minimizes both data fitting errors and violations of physical constraints. For convection-diffusion equations, PINN's advantages are: avoiding mesh discretization errors, adaptively focusing on boundary layers, and naturally supporting inverse problems (joint optimization of solutions and unknown parameters).

For the stiff characteristics of phase-field equations (time evolution stiffness caused by rapid changes in the interface region), the research may adopt:

1. Adaptive sampling: Increase sampling density near the interface;
2. Multi-scale network: Capture characteristic scales of different regions;
3. Time-parallel training: Avoid cumulative errors and efficiency bottlenecks of sequential integration;
4. Loss function improvement: Introduce energy functional constraints to ensure thermodynamic consistency.

Typical PINN verification experiments include:

1. Accuracy verification: Compare indicators such as L2 error with analytical solutions or high-precision reference solutions;
2. Parameter sensitivity analysis: Evaluate the impact of hyperparameters such as network structure, activation function, and loss weight;
3. Computational efficiency: Compare computational costs under the same accuracy with traditional methods;
4. Generalization ability: Test the prediction performance of unseen spatiotemporal points.

Current limitations of PINN:

1. Difficult training (complex loss landscape and local minima);
2. Insufficient capture of high-frequency modes (neural networks' preference for low frequencies);
3. Error accumulation in long-term evolution. Future directions: Develop efficient optimization algorithms and loss weighting strategies, partitioned PINN combined with traditional methods, application of uncertainty quantification, and expansion to multi-physics coupling problems.