A Comparative Study of Loss-Level Conservation Constraints in PINNs and Scheme-Level Structure-Preserving Approaches for Hamiltonian Systems

Abstract

Physics-Informed Neural Networks (PINNs) provide a neural network-based, mesh-free computational method for solving partial differential equations (PDEs) by incorporating PDE residuals, initial conditions, and boundary conditions into the loss function. However, standard PINNs may still suffer from issues such as conservation drift, insufficient training stability, and difficulty in preserving physical structure when dealing with long-term predictions and complex nonlinear problems. Centered on the core concept of “structure preservation” in scientific computing, this paper systematically compares methods for enhancing physical conservation constraints (Conservation-PINN) in PDE solving with methods for preserving geometric structure (symplex integrators) in the evolution of Hamiltonian systems. This paper selects the heat conduction equation, the linear advection equation, the Burgers equation, and the double-pendulum Hamiltonian system as test cases to conduct a comparative analysis of the basic PINN, Res-PINN, Conservation-PINN, as well as the RK4 method and various symplectic integration methods (including Implicit Midpoint and Gauss-Legendre IRK4).The results show that introducing a conservation loss term can reduce mass drift to some extent during the solution of partial differential equations, and residual connection can improve the training stability of deep PINNs; in Hamiltonian systems, high-order non-symplectic methods (RK4) exhibit excellent local accuracy, while symplectic integrators, by preserving the system’s symplectic structure, can effectively suppress cumulative energy divergence during long-term evolution. This paper provides a reference for understanding the connections and differences between loss function constraints in scientific machine learning and classical geometric numerical integration.