Deep-learning of parametric partial differential equations from sparse and noisy data

This paper gives an acceleration scheme for deep backward stochastic differential equation (BSDE) solver, a deep learning method for solving BSDEs introduced in Weinan et al. [Weinan, E, J Han and A Jentzen (2017). Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Communications in Mathematics and Statistics, 5(4), 349–380]. The solutions of nonlinear partial differential equations are quickly estimated using technique of weak approximation even if the dimension is high. In particular, the loss function and the relative error for the target solution become sufficiently small through a smaller number of iteration steps in the new deep BSDE solver.

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Fitting parameters in partial differential equations from partially observed noisy data

Physica D Nonlinear Phenomena ◽

10.1016/s0167-2789(02)00546-8 ◽

2002 ◽

Vol 171 (1-2) ◽

pp. 1-7 ◽

Cited By ~ 27

Author(s):

T.G Müller ◽

J Timmer

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Noisy Data ◽

Partial Differential ◽

Partially Observed ◽

Fitting Parameters

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Numerical solution for high-dimensional partial differential equations based on deep learning with residual learning and data-driven learning

International Journal of Machine Learning and Cybernetics ◽

10.1007/s13042-021-01277-w ◽

2021 ◽

Author(s):

Zheng Wang ◽

Futian Weng ◽

Jialin Liu ◽

Kai Cao ◽

Muzhou Hou ◽

...

Keyword(s):

Deep Learning ◽

Partial Differential Equations ◽

Differential Equations ◽

Numerical Solution ◽

Data Driven ◽

High Dimensional ◽

Partial Differential ◽

Residual Learning

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Solving Partial Differential Equations Using Deep Learning and Physical Constraints

Applied Sciences ◽

10.3390/app10175917 ◽

2020 ◽

Vol 10 (17) ◽

pp. 5917

Author(s):

Yanan Guo ◽

Xiaoqun Cao ◽

Bainian Liu ◽

Mei Gao

Keyword(s):

Neural Networks ◽

Deep Learning ◽

Partial Differential Equations ◽

Numerical Methods ◽

Differential Equations ◽

Language Processing ◽

Kdv Equation ◽

Great Success ◽

Partial Differential ◽

Physical Constraints

The various studies of partial differential equations (PDEs) are hot topics of mathematical research. Among them, solving PDEs is a very important and difficult task. Since many partial differential equations do not have analytical solutions, numerical methods are widely used to solve PDEs. Although numerical methods have been widely used with good performance, researchers are still searching for new methods for solving partial differential equations. In recent years, deep learning has achieved great success in many fields, such as image classification and natural language processing. Studies have shown that deep neural networks have powerful function-fitting capabilities and have great potential in the study of partial differential equations. In this paper, we introduce an improved Physics Informed Neural Network (PINN) for solving partial differential equations. PINN takes the physical information that is contained in partial differential equations as a regularization term, which improves the performance of neural networks. In this study, we use the method to study the wave equation, the KdV–Burgers equation, and the KdV equation. The experimental results show that PINN is effective in solving partial differential equations and deserves further research.

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The New Simulation of Quasiperiodic Wave, Periodic Wave, and Soliton Solutions of the KdV-mKdV Equation via a Deep Learning Method

Computational Intelligence and Neuroscience ◽

10.1155/2021/8548482 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Yong Zhang ◽

Huanhe Dong ◽

Jiuyun Sun ◽

Zhen Wang ◽

Yong Fang ◽

...

Keyword(s):

Deep Learning ◽

Partial Differential Equations ◽

Initial Data ◽

Differential Equations ◽

Nonlinear Partial Differential Equations ◽

Soliton Solutions ◽

Periodic Wave ◽

Mkdv Equation ◽

Learning Method ◽

Partial Differential

How to solve the numerical solution of nonlinear partial differential equations efficiently and conveniently has always been a difficult and meaningful problem. In this paper, the data-driven quasiperiodic wave, periodic wave, and soliton solutions of the KdV-mKdV equation are simulated by the multilayer physics-informed neural networks (PINNs) and compared with the exact solution obtained by the generalized Jacobi elliptic function method. Firstly, the different types of solitary wave solutions are used as initial data to train the PINNs. At the same time, the different PINNs are applied to learn the same initial data by selecting the different numbers of initial points sampled, residual collocation points sampled, network layers, and neurons per hidden layer, respectively. The result shows that the PINNs well reconstruct the dynamical behaviors of the quasiperiodic wave, periodic wave, and soliton solutions for the KdV-mKdV equation, which gives a good way to simulate the solutions of nonlinear partial differential equations via one deep learning method.

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Solving high-dimensional partial differential equations using deep learning

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.1718942115 ◽

2018 ◽

Vol 115 (34) ◽

pp. 8505-8510 ◽

Cited By ~ 108

Author(s):

Jiequn Han ◽

Arnulf Jentzen ◽

Weinan E

Keyword(s):

Deep Learning ◽

Partial Differential Equations ◽

Differential Equations ◽

Ad Hoc ◽

Difficult Problem ◽

High Dimensional ◽

Parabolic Pdes ◽

Partial Differential ◽

Long Time ◽

Hamilton Jacobi Bellman

Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the “curse of dimensionality.” This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function. Numerical results on examples including the nonlinear Black–Scholes equation, the Hamilton–Jacobi–Bellman equation, and the Allen–Cahn equation suggest that the proposed algorithm is quite effective in high dimensions, in terms of both accuracy and cost. This opens up possibilities in economics, finance, operational research, and physics, by considering all participating agents, assets, resources, or particles together at the same time, instead of making ad hoc assumptions on their interrelationships.

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