Structure-Preserving Physics-Informed Neural Networks for Solving Time-Dependent PDEs with Periodic Boundary Conditions

Kernekoncepter

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.

Resumé

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.

Statistik

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.

Citater

"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."

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Structure Preserving PINN for Solving Time Dependent PDEs with Periodic Boundary

by Baoli Hao,Ul... kl. arxiv.org 04-26-2024

https://arxiv.org/pdf/2404.16189.pdf

Dybere Forespørgsler

How can the structure-preserving PINN approach be extended to handle multi-dimensional PDEs with more complex boundary conditions

To extend the structure-preserving Physics-Informed Neural Network (PINN) approach to handle multi-dimensional Partial Differential Equations (PDEs) with more complex boundary conditions, several key considerations need to be addressed:

Dimensionality: For multi-dimensional PDEs, the neural network architecture needs to be adapted to handle higher-dimensional input data. This involves modifying the network structure to accommodate additional spatial dimensions and ensuring that the network can effectively capture the complex relationships in multi-dimensional data.

Boundary Conditions: Handling more complex boundary conditions in multi-dimensional PDEs requires a careful design of the network architecture. The incorporation of boundary conditions into the neural network structure, similar to the approach used for periodic boundary conditions in the 1D case, can be extended to higher dimensions. This may involve developing specialized neural network layers or modules to enforce boundary constraints in multi-dimensional space.

Adaptive Transformations: In multi-dimensional settings, the ψ and φ functions used for transforming the input data and incorporating initial and boundary conditions may need to be adapted to higher dimensions. Introducing adaptive or learnable transformations that can adjust to the complexity of the data and boundary conditions in multi-dimensional PDEs can enhance the flexibility and performance of the structure-preserving PINN approach.

Training Strategy: Training structure-preserving PINNs for multi-dimensional PDEs requires careful consideration of the training strategy, including the selection of collocation points, mini-batching techniques, and regularization methods tailored to higher-dimensional data. Efficient training algorithms that can handle the increased complexity of multi-dimensional problems are essential for the success of the approach.

By addressing these aspects and customizing the structure-preserving PINN approach to suit the requirements of multi-dimensional PDEs with complex boundary conditions, it is possible to extend the method effectively to higher-dimensional settings.

What are the theoretical guarantees or convergence properties of the structure-preserving PINN algorithm compared to the baseline PINN approach

The theoretical guarantees and convergence properties of the structure-preserving PINN algorithm compared to the baseline PINN approach can be analyzed based on several key factors:

Stability and Well-Conditioning: The structure-preserving PINN algorithm, by incorporating initial and boundary conditions into the neural network architecture, aims to reduce the ill-conditioning of the PDE solution maps. This can lead to improved stability and well-conditioning of the solutions, especially for stiff time-dependent PDEs where traditional PINNs may struggle with propagating information forward in time.

Convergence Rates: The convergence properties of the structure-preserving PINN algorithm can be analyzed in terms of the convergence rates of the neural network training process. By reducing the training difficulties associated with fitting initial and boundary conditions separately, the structure-preserving approach may exhibit faster convergence and improved accuracy in approximating solutions to time-dependent PDEs.

Generalization and Robustness: The structure-preserving PINN algorithm's ability to generalize well to unseen data and robustly handle variations in the input data and boundary conditions can also contribute to its theoretical guarantees. By preserving the underlying problem structure in the neural network architecture, the algorithm may offer enhanced generalization capabilities compared to traditional PINNs.

Comparison Studies: Conducting comparative studies between the structure-preserving PINN and baseline PINN approaches on a variety of PDEs with different boundary conditions can provide insights into the relative performance and convergence properties of the two methods. Empirical evaluations and numerical experiments can complement theoretical analyses to validate the effectiveness of the structure-preserving approach.

By investigating these aspects and conducting rigorous theoretical analyses and empirical studies, it is possible to establish the theoretical guarantees and convergence properties of the structure-preserving PINN algorithm in comparison to the baseline PINN approach.

Can the proposed method be further improved by incorporating adaptive or learnable transformations for the ψ and φ functions, rather than using predefined forms

The proposed structure-preserving PINN method can potentially be further improved by incorporating adaptive or learnable transformations for the ψ and φ functions, rather than using predefined forms. By introducing adaptive transformations that can adjust to the specific characteristics of the data and boundary conditions, the algorithm can enhance its flexibility, adaptability, and performance. Here are some ways in which adaptive transformations can enhance the structure-preserving PINN approach:

Learnable Parameters: Introducing learnable parameters in the ψ and φ functions allows the neural network to adapt and optimize the transformations based on the training data. By learning the optimal transformations from the data, the network can better capture the underlying structure of the PDEs and boundary conditions.

Dynamic Adjustment: Adaptive transformations can dynamically adjust during the training process to accommodate variations in the input data and boundary conditions. This flexibility can improve the network's ability to model complex relationships and handle diverse scenarios effectively.

Regularization and Constraints: Incorporating regularization techniques and constraints on the learnable parameters of the transformations can prevent overfitting and ensure that the adaptive transformations generalize well to unseen data. By imposing constraints based on the problem domain knowledge, the network can learn more robust and accurate representations.

Gradient-Based Optimization: Leveraging gradient-based optimization methods to update the learnable parameters of the transformations can facilitate efficient training and convergence. Adaptive transformations that are optimized through backpropagation can enhance the overall performance of the structure-preserving PINN algorithm.

By integrating adaptive or learnable transformations into the structure-preserving PINN approach, researchers can explore new avenues for improving the algorithm's effectiveness in solving complex time-dependent PDEs with varying boundary conditions. This adaptive framework can enhance the algorithm's versatility and robustness, leading to more accurate and reliable solutions for a wide range of PDE problems.