MSA-PINN: A Structure-Adaptive Physics-Informed Neural Network for Maxwell's Equations

This article introduces MSA-PINN, a structure-adaptive physics-informed neural network specifically designed to solve Maxwell's equations. To address the slow convergence and limited accuracy of standard PINNs under strong field coupling, wave propagation, and complex boundary conditions, MSA-PINN provides an efficient solution for electromagnetic field simulation through innovations such as structure-adaptive mechanisms, multi-scale feature extraction, and dynamic loss weight scheduling. It is expected to be applied in fields like antenna design and microwave device analysis.

Numerical solution of Maxwell's equations is a core challenge in computational physics. Traditional methods like Finite Element Method (FEM) and Finite-Difference Time-Domain (FDTD) face issues such as difficult mesh generation and high resource consumption. PINNs achieve mesh-free solution by embedding physical laws, but standard PINNs perform poorly in scenarios like strong field coupling regions (gradient interference), wave propagation (difficulty capturing high-frequency oscillations), and complex boundary conditions (hard to satisfy constraints), leading to slow convergence and limited accuracy.

MSA-PINN proposes three innovations to address the limitations of standard PINNs: 1. Structure-adaptive network architecture: dynamically adjusts the number of neurons and connection patterns based on physical properties, allocating more resources to regions with intense field changes; 2. Multi-scale feature extraction: parallel sub-networks process different frequency components to capture fast oscillations and slow changes; 3. Adaptive loss weight scheduling: prioritizes satisfying boundary conditions in the early training stage, enhances PDE residual weight in the later stage, and automatically adjusts weights in strong coupling regions to accelerate convergence.

MSA-PINN is implemented based on PyTorch, with core components including: 1. Network architecture module: supports residual connections and attention mechanisms, takes spatiotemporal coordinates as input and outputs electromagnetic field components; 2. Physical constraint layer: calculates the residual of Maxwell's equations via automatic differentiation (time domain/frequency domain optional); 3. Adaptive training engine: includes modules for learning rate scheduling, loss weight optimization, and network structure adjustment; 4. Evaluation and visualization tools: post-processing functions like field distribution visualization and S-parameter extraction.

MSA-PINN has wide applications in the field of electromagnetic engineering: 1. Antenna design optimization: obtains high-precision results under coarse-grained sampling, shortening the design cycle; 2. Microwave device analysis: efficiently handles multi-physics field coupling in complex structures like filters and couplers; 3. Electromagnetic compatibility evaluation: quickly predicts electromagnetic interference of PCBs/packaging, assisting layout optimization; 4. Inverse problem solving: material parameter inversion, source reconstruction, etc., providing tools for non-destructive testing and medical imaging.

MSA-PINN has limitations: 1. Training time: adaptive mechanisms increase complexity, and training time for large-scale 3D problems may exceed traditional methods; 2. Hyperparameter sensitivity: needs tuning for different problems; 3. Generalization ability: currently focused on Maxwell's equations, and extension to other physical equations needs verification. Future directions: combining neural operators to achieve cross-geometry generalization, using transfer learning to accelerate solving, and developing industry-specific versions.