DeepXDE: A Deep Learning Library for Solving Differential Equations

By Lu Lu, Xuhui Meng, Zhiping Mao, G. Em Karniadakis. 2019

2,276 citations recorded by the source corpus.

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.

Subfields: Electronic, Optical and Magnetic Materials, Statistical and Nonlinear Physics, Statistics and Probability

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Electronic, Optical and Magnetic Materials

1. G. Em Karniadakis
2. Lu Lu
3. Jian-Xun Wang
4. Han Gao
5. Zhiping Mao
6. Pengzhan Jin
7. Shuai Meng
8. Xuhui Meng

In Statistical and Nonlinear Physics

1. G. Em Karniadakis
2. Lu Lu
3. Jian-Xun Wang
4. Han Gao
5. Zhiping Mao
6. Shengze Cai
7. Anima Anandkumar
8. Xuhui Meng

In Statistics and Probability

1. Jian-Xun Wang
2. Han Gao
3. G. Em Karniadakis
4. Luning Sun
5. Lu Lu
6. Sankaran Mahadevan
7. Xuhui Meng
8. Zhen Hu

Citations in the corpus

• PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain (2020, methodology): “The effectiveness of the data-free FC-NN based PDE solution algorithm has been demonstrated on a number of canonical PDEs [41, 42] and stochastic PDEs [1, 43, 44].”
• Characterizing possible failure modes in physics-informed neural networks (2021, background): “A number of tools and methodologies now exist to solve scientific problems by combining ML and domain insights [14, 20, 27, 28, 38].”
• A comprehensive review of digital twin — part 1: modeling and twinning enabling technologies (2022, methodology): “• Approach 1: Physics-Informed Loss Function This approach modifies the loss function of a data-driven ML model by adding a physicsinformed loss term that penalizes model predictions not compliant with first principles (or physics), thus constraining the training of the ML model toward solutions tha”
• Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks (2020, background): “In the past few years, there has been intensive research on understanding how deep neural networks can be adopted to solve and discover differential equations [1–3,7,8,15, 17, 20–22, 24, 28].”
• MIONet: Learning multiple-input operators via tensor product (2022, methodology): “The code in this study is implemented by using the library DeepXDE [29], and is publicly available from the GitHub repository https://github.”
• Physics-informed neural networks for inverse problems in nano-optics and metamaterials. (2019, methodology): “We use DeepXDE [10], a user-friendly Python library, for the code implementation in this paper.”
• Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport (2019, result): “Similar convergence results for solving partial differential equations with the PINN method were also observed in [22, 28, 25, 23, 29].”
• PDEBENCH: An Extensive Benchmark for Scientific Machine Learning (2022, methodology): “The DeepXDE project [34] is available under an open license and provides an impressive range of capabilities, but is largely restricted to PINN and DeepONet methods [47].”
• Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks (2021, background): “[13] present an overview of PINN for the solution”
• Stiff-PINN: Physics-Informed Neural Network for Stiff Chemical Kinetics (2020, methodology): “[15] L.”
• Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next (2022, methodology): “DeepXDE is used by the authors for: for inverse scattering (Chen et al, 2020b), or deriving mechanical characteristics from materials with MFNNs (Lu et al, 2020).”
• Transformer for Partial Differential Equations' Operator Learning (2022, background): “The pioneering work DeepONet (Lu et al., 2021a) introduces a new data-driven modeling paradigm that aims to learn the solution operator of PDEs and provides a practical realization of the general universal nonlinear operator approximation theorem (Chen & Chen, 1995).”
• Physics-Informed Neural Operator for Learning Partial Differential Equations (2021, positive): “The baseline method PINN, LAAF-PINN, and SA-PINN are implemented us-ing library DeepXDE [22] with TensorFlow as the backend.”