Physics-Informed Neural Networks (PINN): A New Scientific Computing Method Integrating Physical Laws and Deep Learning

Physics-Informed Neural Networks (PINN) are an emerging scientific computing method that combines physical laws and deep learning, aiming to address the challenges of traditional numerical methods (such as finite element and finite difference methods) in high-dimensional problems, inverse problems, and real-time simulations. Its core is embedding physical laws as soft constraints into neural network training, creating a new paradigm of "small data + strong prior knowledge". It can obtain physically consistent prediction results without a large amount of labeled data, providing an innovative solution for solving partial differential equations and scientific computing.

Traditional scientific computing relies on numerical methods like finite element and finite difference, but they are costly or difficult to implement in high-dimensional problems, inverse problem solving, and real-time simulation scenarios. Traditional deep learning models are purely data-driven and require a large number of labeled samples, while high-quality data is scarce in the scientific field. The core idea of PINN is to explicitly embed physical laws—a reliable source of knowledge—into the learning process, changing the paradigm of AI for Science by integrating physical constraints.

PINN uses a fully connected deep neural network as a function approximator, taking spatiotemporal coordinates (x,y,z,t) as input and outputting the physical quantities to be solved (e.g., velocity field, pressure field). Its uniqueness lies in the loss function, which includes three types of terms:

1. Data fitting term: Measures the consistency between predictions and observed data;
2. Physical residual term: Penalizes the residual of physical equations (e.g., heat conduction equation);
3. Boundary/initial condition term: Ensures compliance with the problem's boundary and initial conditions. By optimizing these three types of losses through multi-task learning, the network is made to both fit the data and satisfy physical laws.

PINN has significant advantages over traditional methods:

• Mesh independence: No need for fine mesh division, suitable for high-dimensional problems;
• Inverse problem solving capability: Can perform parameter identification and field quantity prediction simultaneously. Application scenarios include fluid mechanics (cardiovascular blood flow field reconstruction), heat conduction (inference of internal material coefficients), electromagnetism, geological science, financial modeling, etc., showing broad potential.

PINN faces challenges such as training stability (difficult gradient propagation and slow convergence caused by high-order derivatives) and multi-scale problems (a single network is hard to capture features of different scales). Research frontiers include:

• Architecture innovation: Fourier feature embedding, attention mechanism, multi-scale structure;
• Training strategies: Adaptive loss weighting, curriculum learning, transfer learning;
• Theoretical research: Convergence theory and generalization error analysis framework.

For beginners, it is recommended to start with classic forward problems (solving field distribution with known parameters) and gradually transition to inverse problems and data-driven discovery. Open-source frameworks like DeepXDE, NeuroDiffEq, and SimNet provide high-level APIs that facilitate the definition of geometry, PDEs, and boundary conditions. Learning requires combining deep learning knowledge with an understanding of physical problems, and successful application relies on interdisciplinary collaboration (physicists, computer scientists, domain experts).

PINN is an important direction in AI for Science. With the improvement of computing power and the maturity of algorithms, it is expected to play a greater role in real-time digital twins, accelerated scientific discovery, complex system modeling, and other fields. Its intelligent paradigm of integrating "knowledge + data" is an important path to more reliable and interpretable AI systems.