מושגי ליבה
This paper introduces a novel structure-preserving physics-informed neural network (PINN) algorithm that embeds information about initial and boundary conditions directly into the neural network architecture, reducing the reliance on spectral bases and improving the training accuracy and stability for solving various types of time-dependent PDEs with periodic boundary conditions.
תקציר
The authors present a structure-preserving PINN algorithm that aims to address the challenges in training PINNs for solving time-dependent partial differential equations (PDEs) with periodic boundary conditions. The key insight is to incorporate the initial and boundary condition data directly into the neural network structure, rather than relying on spectral bases and collocation points during training.
The paper first introduces the general setup for a family of time-dependent PDEs with periodic boundary conditions. It then describes the proposed structure-preserving PINN approach, where the initial and boundary condition information is embedded into the neural network architecture through a transformation involving the functions ψ and φ. This reduces the reliance on the multi-objective training loss and simplifies the optimization problem, particularly for stiff PDEs.
The authors also discuss how the structure-preserving PINN can be augmented with other training enhancement techniques, such as mini-batching, self-adaptive weights, causal PINNs, and time-marching PINNs, to further improve the prediction accuracy.
The effectiveness of the proposed approach is demonstrated through numerical experiments on various time-dependent PDEs, including the Viscous Burgers' Equation, Allen-Cahn Equation, Cahn-Hilliard Equation, Kuramoto-Sivashinsky Equation, Gray-Scott Equation, Belousov-Zhabotinsky Equation, and Nonlinear Schrödinger Equation. The results show that the structure-preserving PINN outperforms the baseline PINN and the re-sampling technique proposed by Wight and Zhao (2020), particularly in handling stiff PDEs with sharp moving interfaces.
סטטיסטיקה
The authors provide numerical results that compare the relative L2, L1, and L∞ errors of the learned solutions using the baseline PINN, the re-sampling technique proposed by Wight and Zhao (2020), and the proposed structure-preserving PINN approach for the Allen-Cahn equation.
ציטוטים
"By integrating initial and boundary condition data into the neural network structure—thus preserving the underlying problem structure—we simplify the PINN training process, particularly for stiff time-dependent PDEs."
"Our key insight lies in recognizing that collocation-based machine learning solvers, utilized for training PINNs, can be viewed as a specialized form of regularized regression."