Deep learning models for global coordinate transformations that linearise PDEs

We develop a deep autoencoder architecture that can be used to find a coordinate transformation which turns a non-linear partial differential equation (PDE) into a linear PDE. Our architecture is motivated by the linearising transformations provided by the Cole–Hopf transform for Burgers’ equation and the inverse scattering transform for completely integrable PDEs. By leveraging a residual network architecture, a near-identity transformation can be exploited to encode intrinsic coordinates in which the dynamics are linear. The resulting dynamics are given by a Koopman operator matrix K. The decoder allows us to transform back to the original coordinates as well. Multiple time step prediction can be performed by repeated multiplication by the matrix K in the intrinsic coordinates. We demonstrate our method on a number of examples, including the heat equation and Burgers’ equation, as well as the substantially more challenging Kuramoto–Sivashinsky equation, showing that our method provides a robust architecture for discovering linearising transforms for non-linear PDEs.

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Burgers' equation by non-local shot noise data

Journal of Applied Probability ◽

10.2307/3214968 ◽

1994 ◽

Vol 31 (A) ◽

pp. 351-362 ◽

Cited By ~ 9

Author(s):

Donatas Surgailis ◽

Wojbor A. Woyczynski

Keyword(s):

Shot Noise ◽

Random Fields ◽

Burgers Equation ◽

Initial Conditions ◽

Scaling Limit ◽

Linear Partial Differential Equation ◽

Local Case ◽

Non Linear ◽

Noise Data ◽

Non Local

We study the scaling limit of random fields which are solutions of a non-linear partial differential equation, known as the Burgers equation, under stochastic initial conditions. These are assumed to be of a non-local shot noise type and driven by a Cox process. Previous work by Bulinskii and Molchanov (1991), Surgailis and Woyczynski (1993a), and Funaki et al. (1994) concentrated on the case of local shot noise data which permitted use of techniques from the theory of random fields with finite range dependence. Those are not available for the non-local case being considered in this paper.Burgers' equation is known to describe various physical phenomena such as non-linear and shock waves, distribution of self-gravitating matter in the universe, and other flow satisfying conservation laws (see e.g. Woyczynski (1993)).

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Burgers' equation by non-local shot noise data

Journal of Applied Probability ◽

10.1017/s0021900200107181 ◽

1994 ◽

Vol 31 (A) ◽

pp. 351-362 ◽

Cited By ~ 2

Author(s):

Donatas Surgailis ◽

Wojbor A. Woyczynski

Keyword(s):

Shot Noise ◽

Random Fields ◽

Burgers Equation ◽

Initial Conditions ◽

Scaling Limit ◽

Linear Partial Differential Equation ◽

Local Case ◽

Non Linear ◽

Noise Data ◽

Non Local

We study the scaling limit of random fields which are solutions of a non-linear partial differential equation, known as the Burgers equation, under stochastic initial conditions. These are assumed to be of a non-local shot noise type and driven by a Cox process. Previous work by Bulinskii and Molchanov (1991), Surgailis and Woyczynski (1993a), and Funaki et al. (1994) concentrated on the case of local shot noise data which permitted use of techniques from the theory of random fields with finite range dependence. Those are not available for the non-local case being considered in this paper. Burgers' equation is known to describe various physical phenomena such as non-linear and shock waves, distribution of self-gravitating matter in the universe, and other flow satisfying conservation laws (see e.g. Woyczynski (1993)).

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Using a cubic B-spline method in conjunction with a one-step optimized hybrid block approach to solve nonlinear partial differential equations

Computational and Applied Mathematics ◽

10.1007/s40314-021-01729-7 ◽

2021 ◽

Vol 41 (1) ◽

Author(s):

Higinio Ramos ◽

Anurag Kaur ◽

V. Kanwar

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Burgers Equation ◽

Block Method ◽

Step Size ◽

Time Step ◽

Partial Differential ◽

Time Step Size ◽

Linear Partial Differential Equations ◽

Non Linear

AbstractIn this paper, we develop an optimized hybrid block method which is combined with a modified cubic B-spline method, for solving non-linear partial differential equations. In particular, it will be applied for solving three well-known problems, namely, the Burgers equation, Buckmaster equation and FitzHugh–Nagumo equation. Most of the developed methods in the literature for non-linear partial differential equations have not focused on optimizing the time step-size and a very small value must be considered to get accurate approximations. The motivation behind the development of this work is to overcome this trade-off up to much extent using a larger time step-size without compromising accuracy. The optimized hybrid block method considered is proved to be A-stable and convergent. Furthermore, the obtained numerical approximations have been compared with exact and numerical solutions available in the literature and found to be adequate. In particular, without using quasilinearization or filtering techniques, the results for small viscosity coefficient for Burgers equation are found to be accurate. We have found that the combination of the two considered methods is computationally efficient for solving non-linear PDEs.

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Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Mathematics ◽

10.3390/math9131483 ◽

2021 ◽

Vol 9 (13) ◽

pp. 1483

Author(s):

Shanqin Chen

Keyword(s):

Large Time ◽

Hyperbolic Equations ◽

Strong Stability ◽

Matrix Exponential ◽

Linear Component ◽

Nonlinear Hyperbolic Equations ◽

Time Step ◽

Integrating Factor ◽

The Matrix ◽

Runge Kutta

Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

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Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws

Mathematics ◽

10.3390/math9151799 ◽

2021 ◽

Vol 9 (15) ◽

pp. 1799

Author(s):

Irene Gómez-Bueno ◽

Manuel Jesús Castro Díaz ◽

Carlos Parés ◽

Giovanni Russo

Keyword(s):

High Order ◽

Collocation Methods ◽

Quadrature Formulas ◽

Balance Laws ◽

Source Terms ◽

Time Step ◽

Systems Of Balance Laws ◽

Reconstruction Operator ◽

Non Linear ◽

Linear Problems

In some previous works, two of the authors introduced a technique to design high-order numerical methods for one-dimensional balance laws that preserve all their stationary solutions. The basis of these methods is a well-balanced reconstruction operator. Moreover, they introduced a procedure to modify any standard reconstruction operator, like MUSCL, ENO, CWENO, etc., in order to be well-balanced. This strategy involves a non-linear problem at every cell at every time step that consists in finding the stationary solution whose average is the given cell value. In a recent paper, a fully well-balanced method is presented where the non-linear problems to be solved in the reconstruction procedure are interpreted as control problems. The goal of this paper is to introduce a new technique to solve these local non-linear problems based on the application of the collocation RK methods. Special care is put to analyze the effects of computing the averages and the source terms using quadrature formulas. A general technique which allows us to deal with resonant problems is also introduced. To check the efficiency of the methods and their well-balance property, they have been applied to a number of tests, ranging from easy academic systems of balance laws consisting of Burgers equation with some non-linear source terms to the shallow water equations—without and with Manning friction—or Euler equations of gas dynamics with gravity effects.

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An automated Verlet neighbor list algorithm with a multiple time-step approach for the simulation of large systems

Computer Physics Communications ◽

10.1016/0010-4655(92)90108-b ◽

1992 ◽

Vol 70 (3) ◽

pp. 467-477 ◽

Cited By ~ 5

Author(s):

Ariel A. Chialvo ◽

Pablo G. Debenedetti

Keyword(s):

Multiple Time ◽

Time Step ◽

Multiple Time Step ◽

Neighbor List ◽

Large Systems

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Near Real-Time Global Solar Radiation Forecasting at Multiple Time-Step Horizons Using the Long Short-Term Memory Network

Energies ◽

10.3390/en13143517 ◽

2020 ◽

Vol 13 (14) ◽

pp. 3517 ◽

Cited By ~ 1

Author(s):

Anh Ngoc-Lan Huynh ◽

Ravinesh C. Deo ◽

Duc-Anh An-Vo ◽

Mumtaz Ali ◽

Nawin Raj ◽

...

Keyword(s):

Deep Learning ◽

Solar Radiation ◽

Short Term Memory ◽

Global Solar Radiation ◽

Multiple Time ◽

Short Term ◽

Time Step ◽

Term Memory ◽

Multiple Time Step ◽

Long Short Term Memory

This paper aims to develop the long short-term memory (LSTM) network modelling strategy based on deep learning principles, tailored for the very short-term, near-real-time global solar radiation (GSR) forecasting. To build the prescribed LSTM model, the partial autocorrelation function is applied to the high resolution, 1 min scaled solar radiation dataset that generates statistically significant lagged predictor variables describing the antecedent behaviour of GSR. The LSTM algorithm is adopted to capture the short- and the long-term dependencies within the GSR data series patterns to accurately predict the future GSR at 1, 5, 10, 15, and 30 min forecasting horizons. This objective model is benchmarked at a solar energy resource rich study site (Bac-Ninh, Vietnam) against the competing counterpart methods employing other deep learning, a statistical model, a single hidden layer and a machine learning-based model. The LSTM model generates satisfactory predictions at multiple-time step horizons, achieving a correlation coefficient exceeding 0.90, outperforming all of the counterparts. In accordance with robust statistical metrics and visual analysis of all tested data, the study ascertains the practicality of the proposed LSTM approach to generate reliable GSR forecasts. The Diebold–Mariano statistic test also shows LSTM outperforms the counterparts in most cases. The study confirms the practical utility of LSTM in renewable energy studies, and broadly in energy-monitoring devices tailored for other energy variables (e.g., hydro and wind energy).

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