Hands-On Physics-Informed Neural Networks: Solving Multi-Dimensional Coupled Spring-Mass Systems with PyTorch

This article introduces the pytorch-pinn-coupled-spring-mass project, demonstrating how to use Physics-Informed Neural Networks (PINN) to solve N-dimensional coupled spring-mass systems without real data. The project integrates physical constraints with deep learning to achieve efficient numerical simulation, providing a solution for data-scarce scenarios in scientific computing.

Coupled spring-mass systems are important models in classical mechanics, applied in fields like molecular dynamics and structural engineering. The motion of an N-dimensional system is described by second-order ordinary differential equations, involving mass, stiffness, and damping matrices. Traditional numerical methods (e.g., finite element method) require fine meshes, and their complexity grows exponentially with dimension; PINN directly approximates the solution function and incorporates physical equations as part of the loss function, avoiding the limitations of mesh discretization.

The core innovation of the project lies in the design of the loss function: the total loss includes initial condition loss (to satisfy initial states), boundary condition loss (to constrain boundary behavior), and physical equation residual loss (to satisfy differential equations). Multi-objective optimization allows the network to automatically learn solutions that conform to physical laws without supervised data. PyTorch's automatic differentiation function efficiently computes high-order derivatives, which is a key support.

The project uses a fully connected neural network to approximate the solution, with inputs as time and spatial coordinates and outputs as particle displacements. The depth and width of the network can be flexibly adjusted. Training uses the Adam optimizer with learning rate decay, and adaptive weights are introduced to balance loss terms for stiff problems. The code structure separates modules like problem definition and network construction, making it easy to extend to other physical systems.

PINN has unique value in multiple fields: parameter inversion can learn both parameters and solutions simultaneously; real-time simulation inference is faster than traditional solvers; high-dimensional problems avoid the curse of dimensionality. This project provides an extensible benchmark implementation for engineers and researchers.

The current implementation is limited to simple spring-mass systems; training stability and accuracy for highly nonlinear or stiff problems need improvement. Future directions can explore Fourier feature networks, adaptive activation functions, hybrid strategies with traditional numerical methods; uncertainty quantification and multi-fidelity fusion are also important directions.