A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations

By Mohammad Amin Nabian, Hadi Meidani. 2018

109 citations recorded by the source corpus.

Abstract Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution approach for these problems based on deep learning. This approach is intrusive, entirely unsupervised, and mesh-free. Specifically, the random PDE is approximated by a feed-forward fully-connected deep residual network, with either strong or weak enforcement of initial and boundary constraints. Parameters of the approximating deep neural network are determined iteratively using variants of the Stochastic Gradient Descent (SGD) algorithm. The satisfactory accuracy of the proposed approach is numerically demonstrated on diffusion and heat conduction problems, in comparison with the converged Monte Carlo-based finite element results.

Subfields: Computational Mechanics, 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 Computational Mechanics

1. Jian-Xun Wang
2. Han Gao
3. Xiaoying Zhuang
4. G. Em Karniadakis
5. Luning Sun
6. Lu Lu
7. Zhiping Mao
8. Xuhui Meng

In Statistical and Nonlinear Physics

1. Jian-Xun Wang
2. Han Gao
3. G. Em Karniadakis
4. Luning Sun
5. Lu Lu
6. Xuhui Meng
7. Zhiping Mao
8. Timon Rabczuk

In Statistics and Probability

1. Jian-Xun Wang
2. Han Gao
3. Luning Sun
4. Lu Lu
5. G. Em Karniadakis
6. Xuhui Meng
7. Ilias Bilionis
8. Rohit Tripathy

Citations in the corpus

• Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks (2019, background): “For example, in [57] the authors solved high-dimensional deterministic PDEs from variational principles using DNN approximators, and in [58] they solved SPDEs with input random fields up to 100 dimensions following a solver free approach.”
• DeepXDE: A Deep Learning Library for Solving Differential Equations (2019, background): “PINNs can also be easily extended to solve FDEs [25] and SDEs [38, 36, 24, 37], but we do not discuss here such cases due to the page limit.”
• A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate (2021, background): “Nabian and Meidani studied the presentation of high-dimensional random partial differential equations with a feed-forward fully-connected deep neural networks [41,42].”
• Weak Adversarial Networks for High-dimensional Partial Differential Equations (2019, methodology): “In [10], deep learning technique is employed to solve low-dimensional random PDEs based on both strong form and variational form.”
• PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain (2020, background): “Nabian and Meidani [54] and Karumuri et al.”
• Recent advances and applications of surrogate models for finite element method computations: a review (2022, methodology): “…Papadopoulos et al. (2018) × N/A Carbon nanotubes Chu et al. (2019) × N/A Analysis of dental implants Ghorbel et al. (2019) × Adaptive run parameterization Fluid dynamics Nabian and Meidani (2019) × Mesh-free framework for solving PDEs Heat conduction and diffusion problems Schulz et al.…”