DeepXDE: A deep learning library for solving differential equations (1907.04502v2)

Abstract: Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation. The PINN algorithm is simple, and it can be applied to different types of PDEs, including integro-differential equations, fractional PDEs, and stochastic PDEs. Moreover, from the implementation point of view, PINNs solve inverse problems as easily as forward problems. We propose a new residual-based adaptive refinement (RAR) method to improve the training efficiency of PINNs. For pedagogical reasons, we compare the PINN algorithm to a standard finite element method. We also present a Python library for PINNs, DeepXDE, which is designed to serve both as an education tool to be used in the classroom as well as a research tool for solving problems in computational science and engineering. Specifically, DeepXDE can solve forward problems given initial and boundary conditions, as well as inverse problems given some extra measurements. DeepXDE supports complex-geometry domains based on the technique of constructive solid geometry, and enables the user code to be compact, resembling closely the mathematical formulation. We introduce the usage of DeepXDE and its customizability, and we also demonstrate the capability of PINNs and the user-friendliness of DeepXDE for five different examples. More broadly, DeepXDE contributes to the more rapid development of the emerging Scientific Machine Learning field.

• The paper introduces DeepXDE, a Python library that applies physics-informed neural networks to solve various differential equations.
• The paper demonstrates a residual-based adaptive refinement method that improves training efficiency and accuracy, particularly for steep solution gradients.
• The work compares PINNs with traditional FEM, highlighting PINNs' mesh-free flexibility and capability in handling complex inverse problems and geometries.

Overview of DeepXDE: A Deep Learning Library for Solving Differential Equations

The paper "DeepXDE: A deep learning library for solving differential equations" by Lu Lu, Xuhui Meng, Zhiping Mao, and George E. Karniadakis, introduces a significant advancement in the field of Scientific Machine Learning (SciML). The work primarily introduces DeepXDE, a Python library designed to facilitate the application of physics-informed neural networks (PINNs) for solving a wide range of differential equations, including partial differential equations (PDEs), integro-differential equations (IDEs), fractional differential equations (FDEs), and stochastic differential equations (SDEs).

Physics-Informed Neural Networks (PINNs)

PINNs employ deep neural networks to approximate the solution of differential equations by embedding the PDE constraints directly into the loss function using automatic differentiation. Unlike traditional numerical methods such as the finite element method (FEM) and finite difference method (FDM), PINNs are mesh-free and harness the power of neural networks for function approximation. This allows PINNs to potentially overcome the curse of dimensionality and handle higher-dimensional PDEs more effectively.

Key Contributions

1. DeepXDE Library: The paper presents DeepXDE, an open-source Python library that serves as both an educational tool and a research tool for computational science and engineering. The library supports solving both forward and inverse problems across various geometries and initial/boundary conditions.
2. Residual-Based Adaptive Refinement (RAR): To enhance the training efficiency, the authors propose a residual-based adaptive refinement (RAR) method. This technique adaptively improves the distribution of residual points based on the magnitude of the PDE residual, ensuring better accuracy, especially for solutions with steep gradients.
3. Comparison with FEM: The authors provide a detailed comparison between PINNs and FEM. While FEM uses piecewise polynomial functions and requires mesh generation, PINNs use neural networks for nonlinear approximation and are mesh-free. This comparison highlights the potential advantages of PINNs in terms of flexibility and capability to handle complex geometries.

Theoretical and Practical Implications

The theoretical foundations of PINNs are discussed, emphasizing the approximation capabilities of deep neural networks and their error components: approximation error, generalization error, and optimization error. This provides a comprehensive understanding of how PINNs compare to traditional methods and the challenges involved.

Practically, the DeepXDE library is demonstrated through several examples:

1. Poisson Equation: The method is applied to solve the Poisson equation over an L-shaped domain, demonstrating the accuracy of the PINN solution in comparison with the Spectral Element Method (SEM).
2. Burgers' Equation with RAR: The effectiveness of RAR is showcased by solving the 1D and 2D Burgers' equations, highlighting the improved accuracy and efficiency in capturing solutions with sharp fronts.
3. Inverse Problems: The paper presents examples of inverse problems, including the Lorenz system and a diffusion-reaction system. These examples demonstrate the capability of PINNs to identify unknown parameters from observed data.
4. Volterra Integro-Differential Equation: The method's flexibility is further exhibited by solving an IDE using Gaussian quadrature for integral approximation.

Speculations on Future Developments

The paper outlines several potential areas for future development:

1. Acceleration of Training: Techniques such as time-parallel methods and adaptive activation functions could further enhance the efficiency of PINNs.
2. Automated Neural Architecture Search: Emerging meta-learning techniques could automate the process of selecting effective neural network architectures.
3. Weak/Variational Formulations: Exploring alternative formulations might provide additional advantages, especially for complex problems.
4. Handling Mixed Data Types: Extending the framework to handle data from diverse sources, such as images and point measurements, could broaden the applicability of PINNs in multi-physics and multi-scale problems.

Conclusion

The introduction of DeepXDE marks a substantial step forward in the application of deep learning to solve differential equations. By combining the strengths of machine learning with the principles of numerical analysis, this library opens new avenues for researchers and educators in computational science. The proposed enhancements and future directions suggest a promising trajectory for further advancements in this intersection of fields.