Journal of information and communication convergence engineering 2024; 22(4): 316-321
Published online December 31, 2024
https://doi.org/10.56977/jicce.2024.22.4.316
© Korea Institute of Information and Communication Engineering
Correspondence to : Donghyun Kim (E-mail: guru25@kistep.re.kr)
Center for Regional S&T Innovation Policy, Korea Institute of S&T Evaluation and Planning (KISTEP)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
The Vlasov-Poisson (VP) equation plays an important role in plasma physics. Most numerical methods for the VP equation are based on the finite difference method (FDM) or finite element method (FEM), where the computational costs are high. However, this study focuses on the efficient reconstruction of solutions to the VP equation. We begin by generating short-term solutions to the VP equation using an FDM-type algorithm. Among various versions of FDM schemes, we employ backward semi-Lagrangian-based methods with weighted, essentially non-oscillatory schemes for interpolation. Subsequently, a stable dataset without spurious oscillations is obtained. The spatiotemporal patterns within these snapshot solutions are then analyzed via dynamic mode decomposition (DMD). By projecting solution spaces onto the DMD modes, we efficiently extend the solution to unobserved future time steps. Experimental results indicate that the time cost for the DMD prediction is within one second, showing the efficiency of the proposed algorithm.
Keywords Data-driven, Dynamic mode decomposition, Long-term prediction, Vlasov-Poisson equation
The Vlasov-Poisson (VP) equation is a fundamental continuum model that describes the distribution function of charged particles within plasma under the zero-magnetic field assumption [1-3]. This model captures the dynamics induced by an electric field applied to charged particles across a specified domain. Because the VP equation is widely used in plasma physics, various attempts have been made to approximate its solution. The most popular approaches for solving the VP equation are based on the finite difference method (FDM) or finite element method (FEM) [4-9]. Let us look at some of the difficulties in numerically solving the VP equation based on FDM or FEM. Because the electric field is obtained by solving the Poisson equation for the electric potential variable at every time step, some time-consuming steps arise in the algorithms, namely, solving the algebraic system of form Ax = b at each time step. In addition, stability conditions must be considered for the VP equation to avoid spurious oscillations in numerical solutions. Therefore, typically, several time steps are required for the long-term prediction of the VP equation. Finally, shock phenomena occur, which may lead to nonphysical oscillations. To prevent oscillations, conservative numerical schemes have been designed or upwind-type techniques have been added to numerical algorithms. These challenges render it virtually impossible to achieve real-time simulation results using traditional FDM/FEM-based approaches. Because of the computational challenges of conventional methods, it is natural to find alternative approaches, such as data-driven algorithms.
Recently, there have been extensive developments in the machine-learning community, including supervised learning, generation algorithms, and large language models [10-12]. In addition, numerous data-driven analyses have been conducted for partial differential equation (PDE)-based fluid problems. Among them, dynamic mode decomposition (DMD) has achieved significant success in PDE operators and in forecasting solutions for PDE [13-17]. DMD operates by capturing a solution “snapshot” at a finite set of time points, leveraging these snapshots to model dynamic behavior efficiently. By decomposing these snapshots into spatial modes and their corresponding temporal dynamics, DMD identifies the dominant patterns that capture the system’s evolution. This approach allows for efficient forecasting of the solution behavior by reconstructing future states from the extracted modes.
In this study, we propose a new method for long-term forecasting solutions to the VP equation by employing the DMD method. To generate the snapshot data required for DMD, we first obtain a short-term solution to the VP equation using a conventional FDM-based method. Among the various FDM-based methods, we employ the method introduced in [18], where high-order polynomial interpolation is written in conservative form in the framework of backward semi-Lagrangian methods. In addition, weighted essentially non-oscillatory schemes are applied to ensure that the solutions remain free of spurious oscillations. Once these shortterm solutions to the VP equations are obtained, we decompose the spatiotemporal modes within the framework of the DMD. By projecting the snapshot solution onto the DMD mode space, we can efficiently extend the solutions in the time domain, enabling the reconstruction of the long-term solution to the VP equation. In the Results section, we report the performance of the proposed method. The DMD-based method effectively captures the dynamics of solutions with a short computation time.
The remainder of this paper is organized as follows: In Section II, we describe the DMD-based method for longterm prediction of the VP equation. The results are presented in Section III. B. Finally, in Section IV, the concluding remarks and discussion are presented.
In this section, we describe a DMD-based prediction method for the VP equation. In Subsection A, the model equation for the VP system is provided. In Subsection B, the DMD-based solution-reconstruction approach is explained.
We consider one-dimensional VP equations, given by
where
To apply the DMD, a snapshot dataset of the solution to the VP equation is needed. To achieve this, we restrict our consideration to the finite domain. Because f is the probability density function, we assume
Subsequently, (1)-(6) are used to determine the unique solution. We use the numerical method in [18] to generate a snapshot dataset by employing a spatial grid with N nodes. Snapshots were collected at a discrete time step tn = n (n = 0, 1, ..., m) as follows:
The choice of parameter N, m will appear in the Results section.
In this subsection, we introduce a DMD-based method for reconstructing solutions to the VP equation. We begin by defining certain notation. Let sn represent the snapshot solution at time step tn as shown in (7). We reshape sn into the vector form. In the DMD framework, we assume there exists a matrix A that satisfies
Then, matrix A satisfies the relation AX = Y. Although matrix A can be found using the pseudo inverse of X, i.e., A = YX+, this process is computationally demanding because of the dense matrix multiplication. Instead, the DMD algorithm determines the eigenvectors and eigenvalues of A in an alternative manner, with these eigenvectors referred to as DMD modes [13].
Let us describe the DMD algorithm with r-number of DMD modes. Here, r is chosen so that r‹‹m. First, we compute the truncated SVD of X:
where
Using (9), reduced matrix A is obtained. Because A is an r×r matrix, the diagonalization process of
Here, the columns of W are eigenvectors of
Let Φi denote the i-th column of Φ and λi the i-th diagonal entry of Λ. Then, Φi represents the i-th DMD mode satisfying AΦi = λiΦi When the DMD modes Φ are determined, we can capture the dominant components of the VP equation easily.
We now propose a DMD reconstruction of the long-term solution to the VP equation. Our goal is to find a solution as follows:
for integer L > m, using sm as the final snapshot data along with the DMD modes and eigenvalues obtained in the previous steps. This approach allows us to predict the evolution of solutions beyond the observed data. First, we reconstruct sm in the DMD mode space:
The coefficients are determined by solving a least square problem:
When ci in (12) is determined using (13), the solution can be extended to tL > tm . Suppose k = L − m. By multiplying Ak into (12), we obtain
If λi is a complex number,
where
Here, we summarize our DMD-based method. First, to obtain the snapshot data at ti (i = 1 , .., m), we apply the FDM-type algorithm to the VP equation [18]. Next, we compute the DMD using Equations (8)-(8). Then, by representing the last snapshot as a linear combination of the DMD modes in (12), we extend the solution to unobserved time using (14).
In this section, we present the DMD-based reconstruction method for solving the VP equation. First, we apply the FDM-type method [18] to generate snapshot solutions, where a uniform mesh grid for spatio-variables (x, v) with N = 251 × 251 nodes is employed. Snapshot solutions are generated up to m = 50 (t1, t2, ..., t50). Using (14), the solution is extended to L > m. The L2 errors are measured against a reference solution generated by FDM [18].
In this study, we set the initial condition of the VP equation as
The number of DMD modes is chosen as r = 21. Solutions are generated up to t = 50 using the FDM-based method in [18], and snapshot data are arranged from t0 = 0 to t50 = 50 in the following form:
The computational time required for generating these solutions was approximately 71 s on an Intel® Core i7-13700 processor.
Using the dataset (15), we computed the DMD modes and eigenvalues of the VP system. It is worth noting that the computation of DMD modes takes only 0.06 s, which is negligible compared to the time required for generating the snapshot data. The first five DMD modes are shown in Fig. 1.
In this subsection, the long-term predictive capabilities of the DMD algorithm are demonstrated. We begin by representing the last snapshot using the DMD modes, as shown in (12). In Fig. 2, we compare the graphs of the last snapshot s50 with those of the reconstructed snapshot
Next, we generate a long-term DMD-based solution for times t = 100, 150 using (14). Because snapshot data were obtained up to t = 50, predicting the solution of t ≥ 100 represents a long-term forecast. To mitigate oscillations, we applied post-smoothing to the DMD-generated solutions. For the smoothing operator, we used a kernel average smoother [19]. The loss was reduced about one half after post-smoothing.
Fig. 3 shows graphs of the FDM-generated reference solution (Fig. 3a) and DMD-predicted solution (Fig. 3b) at t = 100. Overall, the two graphs match well. We report the relative L2 errors between the DMD-predicted solutions and reference solutions, and the CPU time for generating the DMD prediction (14) in Table 1. The relative L2 error is 0.0484 at t = 100 and 0.096 at t = 150. Notably, the time required for reconstruction was only 0.04 s, which is significantly shorter than the time required to generate snapshot data.
Table 1 . Relative L2 errors between DMD-predicted solutions and reference solutions and CPU time for computing (14).
T | Relative L2 errors | CPU time |
---|---|---|
100 | 0.0484 | 0.04 s |
150 | 0.096 | 0.04 s |
In this paper, we propose a novel method for predicting the long-term solution of the Vlasov-Poisson (VP) equation. Initially, short-term solutions were obtained using the finitedifference method (FDM). By arranging these snapshot solutions uniformly over time, we extracted the underlying spatiotemporal patterns using the DMD. Finally, by representing the solutions in the span of the DMD modes, we successfully reconstructed the solutions to the VP equation. This methodology can be applied to various situations, such as coupling the VP equation with Maxwell’s equations. Applying the DMD to the VP component significantly enhances the computational efficiency of the coupled system.
Donghyun Kim
joined Korea Institute of S&T Evaluation and Planning (KISTEP) in 2008 and is currently an Associate Research Fellow at Center for Regional S&T Innovation Policy. His research interests include government R&D investment, regional R&D data analysis, and regional policy directions.
Journal of information and communication convergence engineering 2024; 22(4): 316-321
Published online December 31, 2024 https://doi.org/10.56977/jicce.2024.22.4.316
Copyright © Korea Institute of Information and Communication Engineering.
1Center for Regional S&T Innovation Policy, Korea Institute of S&T Evaluation and Planning (KISTEP)
Correspondence to:Donghyun Kim (E-mail: guru25@kistep.re.kr)
Center for Regional S&T Innovation Policy, Korea Institute of S&T Evaluation and Planning (KISTEP)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
The Vlasov-Poisson (VP) equation plays an important role in plasma physics. Most numerical methods for the VP equation are based on the finite difference method (FDM) or finite element method (FEM), where the computational costs are high. However, this study focuses on the efficient reconstruction of solutions to the VP equation. We begin by generating short-term solutions to the VP equation using an FDM-type algorithm. Among various versions of FDM schemes, we employ backward semi-Lagrangian-based methods with weighted, essentially non-oscillatory schemes for interpolation. Subsequently, a stable dataset without spurious oscillations is obtained. The spatiotemporal patterns within these snapshot solutions are then analyzed via dynamic mode decomposition (DMD). By projecting solution spaces onto the DMD modes, we efficiently extend the solution to unobserved future time steps. Experimental results indicate that the time cost for the DMD prediction is within one second, showing the efficiency of the proposed algorithm.
Keywords: Data-driven, Dynamic mode decomposition, Long-term prediction, Vlasov-Poisson equation
The Vlasov-Poisson (VP) equation is a fundamental continuum model that describes the distribution function of charged particles within plasma under the zero-magnetic field assumption [1-3]. This model captures the dynamics induced by an electric field applied to charged particles across a specified domain. Because the VP equation is widely used in plasma physics, various attempts have been made to approximate its solution. The most popular approaches for solving the VP equation are based on the finite difference method (FDM) or finite element method (FEM) [4-9]. Let us look at some of the difficulties in numerically solving the VP equation based on FDM or FEM. Because the electric field is obtained by solving the Poisson equation for the electric potential variable at every time step, some time-consuming steps arise in the algorithms, namely, solving the algebraic system of form Ax = b at each time step. In addition, stability conditions must be considered for the VP equation to avoid spurious oscillations in numerical solutions. Therefore, typically, several time steps are required for the long-term prediction of the VP equation. Finally, shock phenomena occur, which may lead to nonphysical oscillations. To prevent oscillations, conservative numerical schemes have been designed or upwind-type techniques have been added to numerical algorithms. These challenges render it virtually impossible to achieve real-time simulation results using traditional FDM/FEM-based approaches. Because of the computational challenges of conventional methods, it is natural to find alternative approaches, such as data-driven algorithms.
Recently, there have been extensive developments in the machine-learning community, including supervised learning, generation algorithms, and large language models [10-12]. In addition, numerous data-driven analyses have been conducted for partial differential equation (PDE)-based fluid problems. Among them, dynamic mode decomposition (DMD) has achieved significant success in PDE operators and in forecasting solutions for PDE [13-17]. DMD operates by capturing a solution “snapshot” at a finite set of time points, leveraging these snapshots to model dynamic behavior efficiently. By decomposing these snapshots into spatial modes and their corresponding temporal dynamics, DMD identifies the dominant patterns that capture the system’s evolution. This approach allows for efficient forecasting of the solution behavior by reconstructing future states from the extracted modes.
In this study, we propose a new method for long-term forecasting solutions to the VP equation by employing the DMD method. To generate the snapshot data required for DMD, we first obtain a short-term solution to the VP equation using a conventional FDM-based method. Among the various FDM-based methods, we employ the method introduced in [18], where high-order polynomial interpolation is written in conservative form in the framework of backward semi-Lagrangian methods. In addition, weighted essentially non-oscillatory schemes are applied to ensure that the solutions remain free of spurious oscillations. Once these shortterm solutions to the VP equations are obtained, we decompose the spatiotemporal modes within the framework of the DMD. By projecting the snapshot solution onto the DMD mode space, we can efficiently extend the solutions in the time domain, enabling the reconstruction of the long-term solution to the VP equation. In the Results section, we report the performance of the proposed method. The DMD-based method effectively captures the dynamics of solutions with a short computation time.
The remainder of this paper is organized as follows: In Section II, we describe the DMD-based method for longterm prediction of the VP equation. The results are presented in Section III. B. Finally, in Section IV, the concluding remarks and discussion are presented.
In this section, we describe a DMD-based prediction method for the VP equation. In Subsection A, the model equation for the VP system is provided. In Subsection B, the DMD-based solution-reconstruction approach is explained.
We consider one-dimensional VP equations, given by
where
To apply the DMD, a snapshot dataset of the solution to the VP equation is needed. To achieve this, we restrict our consideration to the finite domain. Because f is the probability density function, we assume
Subsequently, (1)-(6) are used to determine the unique solution. We use the numerical method in [18] to generate a snapshot dataset by employing a spatial grid with N nodes. Snapshots were collected at a discrete time step tn = n (n = 0, 1, ..., m) as follows:
The choice of parameter N, m will appear in the Results section.
In this subsection, we introduce a DMD-based method for reconstructing solutions to the VP equation. We begin by defining certain notation. Let sn represent the snapshot solution at time step tn as shown in (7). We reshape sn into the vector form. In the DMD framework, we assume there exists a matrix A that satisfies
Then, matrix A satisfies the relation AX = Y. Although matrix A can be found using the pseudo inverse of X, i.e., A = YX+, this process is computationally demanding because of the dense matrix multiplication. Instead, the DMD algorithm determines the eigenvectors and eigenvalues of A in an alternative manner, with these eigenvectors referred to as DMD modes [13].
Let us describe the DMD algorithm with r-number of DMD modes. Here, r is chosen so that r‹‹m. First, we compute the truncated SVD of X:
where
Using (9), reduced matrix A is obtained. Because A is an r×r matrix, the diagonalization process of
Here, the columns of W are eigenvectors of
Let Φi denote the i-th column of Φ and λi the i-th diagonal entry of Λ. Then, Φi represents the i-th DMD mode satisfying AΦi = λiΦi When the DMD modes Φ are determined, we can capture the dominant components of the VP equation easily.
We now propose a DMD reconstruction of the long-term solution to the VP equation. Our goal is to find a solution as follows:
for integer L > m, using sm as the final snapshot data along with the DMD modes and eigenvalues obtained in the previous steps. This approach allows us to predict the evolution of solutions beyond the observed data. First, we reconstruct sm in the DMD mode space:
The coefficients are determined by solving a least square problem:
When ci in (12) is determined using (13), the solution can be extended to tL > tm . Suppose k = L − m. By multiplying Ak into (12), we obtain
If λi is a complex number,
where
Here, we summarize our DMD-based method. First, to obtain the snapshot data at ti (i = 1 , .., m), we apply the FDM-type algorithm to the VP equation [18]. Next, we compute the DMD using Equations (8)-(8). Then, by representing the last snapshot as a linear combination of the DMD modes in (12), we extend the solution to unobserved time using (14).
In this section, we present the DMD-based reconstruction method for solving the VP equation. First, we apply the FDM-type method [18] to generate snapshot solutions, where a uniform mesh grid for spatio-variables (x, v) with N = 251 × 251 nodes is employed. Snapshot solutions are generated up to m = 50 (t1, t2, ..., t50). Using (14), the solution is extended to L > m. The L2 errors are measured against a reference solution generated by FDM [18].
In this study, we set the initial condition of the VP equation as
The number of DMD modes is chosen as r = 21. Solutions are generated up to t = 50 using the FDM-based method in [18], and snapshot data are arranged from t0 = 0 to t50 = 50 in the following form:
The computational time required for generating these solutions was approximately 71 s on an Intel® Core i7-13700 processor.
Using the dataset (15), we computed the DMD modes and eigenvalues of the VP system. It is worth noting that the computation of DMD modes takes only 0.06 s, which is negligible compared to the time required for generating the snapshot data. The first five DMD modes are shown in Fig. 1.
In this subsection, the long-term predictive capabilities of the DMD algorithm are demonstrated. We begin by representing the last snapshot using the DMD modes, as shown in (12). In Fig. 2, we compare the graphs of the last snapshot s50 with those of the reconstructed snapshot
Next, we generate a long-term DMD-based solution for times t = 100, 150 using (14). Because snapshot data were obtained up to t = 50, predicting the solution of t ≥ 100 represents a long-term forecast. To mitigate oscillations, we applied post-smoothing to the DMD-generated solutions. For the smoothing operator, we used a kernel average smoother [19]. The loss was reduced about one half after post-smoothing.
Fig. 3 shows graphs of the FDM-generated reference solution (Fig. 3a) and DMD-predicted solution (Fig. 3b) at t = 100. Overall, the two graphs match well. We report the relative L2 errors between the DMD-predicted solutions and reference solutions, and the CPU time for generating the DMD prediction (14) in Table 1. The relative L2 error is 0.0484 at t = 100 and 0.096 at t = 150. Notably, the time required for reconstruction was only 0.04 s, which is significantly shorter than the time required to generate snapshot data.
Table 1 . Relative L2 errors between DMD-predicted solutions and reference solutions and CPU time for computing (14)..
T | Relative L2 errors | CPU time |
---|---|---|
100 | 0.0484 | 0.04 s |
150 | 0.096 | 0.04 s |
In this paper, we propose a novel method for predicting the long-term solution of the Vlasov-Poisson (VP) equation. Initially, short-term solutions were obtained using the finitedifference method (FDM). By arranging these snapshot solutions uniformly over time, we extracted the underlying spatiotemporal patterns using the DMD. Finally, by representing the solutions in the span of the DMD modes, we successfully reconstructed the solutions to the VP equation. This methodology can be applied to various situations, such as coupling the VP equation with Maxwell’s equations. Applying the DMD to the VP component significantly enhances the computational efficiency of the coupled system.
Table 1 . Relative L2 errors between DMD-predicted solutions and reference solutions and CPU time for computing (14)..
T | Relative L2 errors | CPU time |
---|---|---|
100 | 0.0484 | 0.04 s |
150 | 0.096 | 0.04 s |
Byung-Hyun Lim, Ismatov Akobir, and Ki-Il Kim, Member, KIICE
Journal of information and communication convergence engineering 2024; 22(3): 199-206 https://doi.org/10.56977/jicce.2024.22.3.199