Data-driven Derivation of Partial Differential Equations using Neural Network Model

2021 ◽  
Author(s):  
K. Koyamada ◽  
Y. Long ◽  
T. Kawamura ◽  
K. Konishi
Keyword(s):  
Neural Network ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Network Model ◽  
Neural Network Model ◽  
Data Driven ◽  
Partial Differential

Applying neural network for identification of land surface model parameters

2021 ◽  
Author(s):  
Ruslan Chernyshev ◽  
Mikhail Krinitskiy ◽  
Viktor Stepanenko
Keyword(s):  
Neural Network ◽  
Machine Learning ◽  
Neural Networks ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Thermodynamic Model ◽  
Land Surface ◽  
Land Surface Model ◽  
Surface Model ◽  
Partial Differential

<p>This work is devoted to development of neural networks for identification of partial differential equations (PDE) solved in the land surface scheme of INM RAS Earth System model (ESM). Atmospheric and climate models are in the top of the most demanding for supercomputing resources among research applications. Spatial resolution and a multitude of physical parameterizations used in ESMs continuously increase. Most of parameters are still poorly constrained, many of them cannot be measured directly. To optimize model calibration time, using neural networks looks a promising approach. Neural networks are already in wide use in satellite imaginary (Su Jeong Lee, et al, 2015; Krinitskiy M. et al, 2018) and for calibrating parameters of land surface models (Yohei Sawada el al, 2019). Neural networks have demonstrated high efficiency in solving conventional problems of mathematical physics (Lucie P. Aarts el al, 2001; Raissi M. et al, 2020). </p><p>We develop a neural networks for optimizing parameters of nonlinear soil heat and moisture transport equation set. For developing we used Python3 based programming tools implemented on GPUs and Ascend platform, provided by Huawei. Because of using hybrid approach combining neural network and classical thermodynamic equations, the major purpose was finding the way to correctly calculate backpropagation gradient of error function, because model trains and is being validated on the same temperature data, while model output is heat equation parameter, which is typically not known. Neural network model has been runtime trained using reference thermodynamic model calculation with prescribed parameters, every next thermodynamic model step has been used for fitting the neural network until it reaches the loss function tolerance.</p><p>Literature:</p><p>1.     Aarts, L.P., van der Veer, P. “Neural Network Method for Solving Partial Differential Equations”. Neural Processing Letters 14, 261–271 (2001). https://doi.org/10.1023/A:1012784129883</p><p>2.     Raissi, M., P. Perdikaris and G. Karniadakis. “Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations.” ArXiv abs/1711.10561 (2017): n. pag.</p><p>3.     Lee, S.J., Ahn, MH. & Lee, Y. Application of an artificial neural network for a direct estimation of atmospheric instability from a next-generation imager. Adv. Atmos. Sci. 33, 221–232 (2016). https://doi.org/10.1007/s00376-015-5084-9</p><p>4.     Krinitskiy M, Verezemskaya P, Grashchenkov K, Tilinina N, Gulev S, Lazzara M. Deep Convolutional Neural Networks Capabilities for Binary Classification of Polar Mesocyclones in Satellite Mosaics. Atmosphere. 2018; 9(11):426.</p><p>5.     Sawada, Y.. “Machine learning accelerates parameter optimization and uncertainty assessment of a land surface model.” ArXiv abs/1909.04196 (2019): n. pag.</p><p>6.     Shufen Pan et al. Evaluation of global terrestrial evapotranspiration using state-of-the-art approaches in remote sensing, machine learning and land surface modeling. Hydrol. Earth Syst. Sci., 24, 1485–1509 (2020)</p><p>7.     Chaney, Nathaniel & Herman, Jonathan & Ek, M. & Wood, Eric. (2016). Deriving Global Parameter Estimates for the Noah Land Surface Model using FLUXNET and Machine Learning: Improving Noah LSM Parameters. Journal of Geophysical Research: Atmospheres. 121. 10.1002/2016JD024821.</p><p> </p><p> </p>

scholarly journals Applied Koopman Theory for Partial Differential Equations and Data-Driven Modeling of Spatio-Temporal Systems

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
J. Nathan Kutz ◽  
J. L. Proctor ◽  
S. L. Brunton
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Data Driven ◽  
Nonlinear Pdes ◽  
Dynamic Mode ◽  
Partial Differential ◽  
Koopman Operator ◽  
Dimensional Approximation ◽  
Spatio Temporal ◽  
The Impact

We consider the application of Koopman theory to nonlinear partial differential equations and data-driven spatio-temporal systems. We demonstrate that the observables chosen for constructing the Koopman operator are critical for enabling an accurate approximation to the nonlinear dynamics. If such observables can be found, then the dynamic mode decomposition (DMD) algorithm can be enacted to compute a finite-dimensional approximation of the Koopman operator, including its eigenfunctions, eigenvalues, and Koopman modes. We demonstrate simple rules of thumb for selecting a parsimonious set of observables that can greatly improve the approximation of the Koopman operator. Further, we show that the clear goal in selecting observables is to place the DMD eigenvalues on the imaginary axis, thus giving an objective function for observable selection. Judiciously chosen observables lead to physically interpretable spatio-temporal features of the complex system under consideration and provide a connection to manifold learning methods. Our method provides a valuable intermediate, yet interpretable, approximation to the Koopman operator that lies between the DMD method and the computationally intensive extended DMD (EDMD). We demonstrate the impact of observable selection, including kernel methods, and construction of the Koopman operator on several canonical nonlinear PDEs: Burgers’ equation, the nonlinear Schrödinger equation, the cubic-quintic Ginzburg-Landau equation, and a reaction-diffusion system. These examples serve to highlight the most pressing and critical challenge of Koopman theory: a principled way to select appropriate observables.

scholarly journals A Deep Neural Network Algorithm for Semilinear Elliptic PDEs with Applications in Insurance Mathematics

Risks ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 136
Author(s):  
Stefan Kremsner ◽  
Alexander Steinicke ◽  
Michaela Szölgyenyi
Keyword(s):  
Neural Network ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Deep Neural Network ◽  
Risk Measure ◽  
Elliptic Partial Differential Equations ◽  
High Dimensional ◽  
Control Problems ◽  
Partial Differential ◽  
Neural Network Algorithm

In insurance mathematics, optimal control problems over an infinite time horizon arise when computing risk measures. An example of such a risk measure is the expected discounted future dividend payments. In models which take multiple economic factors into account, this problem is high-dimensional. The solutions to such control problems correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In the present paper we propose a novel deep neural network algorithm for solving such partial differential equations in high dimensions in order to be able to compute the proposed risk measure in a complex high-dimensional economic environment. The method is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with unbounded random terminal time. In particular, backward stochastic differential equations—which can be identified with solutions of elliptic partial differential equations—are approximated by means of deep neural networks.

Systematic Construction of Neural Forms for Solving Partial Differential Equations Inside Rectangular Domains, Subject to Initial, Boundary and Interface Conditions

2020 ◽  
Vol 29 (05) ◽  
pp. 2050009
Author(s):  
Pola Lydia Lagari ◽  
Lefteri H. Tsoukalas ◽  
Salar Safarkhani ◽  
Isaac E. Lagaris
Keyword(s):  
Neural Network ◽  
Heat Conduction ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Systematic Approach ◽  
Thermal Contact ◽  
Initial Boundary ◽  
Spatial Domain ◽  
Interface Conditions ◽  
Partial Differential

A systematic approach is developed for constructing proper trial solutions to Partial Differential Equations (PDEs) of up to second order, using neural forms that satisfy prescribed initial, boundary and interface conditions. The spatial domain considered is of the rectangular hyper-box type. On each face either Dirichlet or Neumann conditions may apply. Robin conditions may be accommodated as well. Interface conditions that induce discontinuities, have not been treated to date in the relevant neural network literature. As an illustration a common problem of heat conduction through a system of two rods in thermal contact is considered.

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