FCNN: Five-point stencil CNN for solving reaction-diffusion equations

01/04/2022
by   Yongho Kim, et al.
Max Planck Society
10

In this paper, we propose Five-point stencil CNN (FCNN) containing a five-point stencil kernel and a trainable approximation function. We consider reaction-diffusion type equations including heat, Fisher's, Allen-Cahn equations, and reaction-diffusion equations with trigonometric functions. Our proposed FCNN is trained well using few data and then can predict reaction-diffusion evolutions with unseen initial conditions. Also, our FCNN is trained well in the case of using noisy train data. We present various simulation results to demonstrate that our proposed FCNN is working well.

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1 Introduction

To express diverse natural phenomena such as sound, heat, electrostatics, elasticity, thermodynamics, fluid dynamics, and quantum mechanics mathematically, various partial differential equations (PDEs) have been derived and numerical methods can be applied to solve these PDEs. Representative numerical methods for solving PDEs are the finite difference method, finite element method, finite volume method, spectral method, etc. We focus on the finite difference method (FDM) which is to divide a given domain into finite grids and find an approximate solution using derivatives with finite differences

PZ2012

. This method uses each and its neighbor points to predict the corresponding point at the next time step. Likewise, in convolutional neural networks (CNNs)

cnn , convolution operators extract each pixel of an output by using the corresponding pixel and its neighbor pixels of an input. Also, the convolution operator is basically immutable. Hence, well-structured convolutional neural networks have a potential to solve partial differential equations numerically. Therefore, we propose Five-point stencil CNN (FCNN) containing a five-point stencil kernel and a trainable approximation function to obtain numerical solutions of the PDEs. Among the various PDEs representing natural phenomena, we deal with reaction-diffusion type equations. The reaction-diffusion model has been applied and used in various fields such as biology NB1986 ; PBH2016 ; DYYMDJJJ2017 , chemistry BAG2009 ; ISBBDL2012 ; GHRR2013 , image segmentation HBB1995 ; SYHR2006 ; ZYMQS2020

, image inpainting

MBet2000 ; YDJSJ2015 ; JJS2016 , medical EQ2015 ; HYJ2015 ; MCYSA2021 , and so on. In this paper, we use second order reaction-diffusion type equations: heat, Fisher’s, Allen–Cahn (AC) equation, and reaction-diffusion equations with trigonometric functions terms.

In recent years, neural networks have been widely applied to solve PDEs. Physics-informed neural networks (PINNs) pinns1

based on multi-layer perceptron (MLP) models approximate solutions by the optimization of a loss function consisting of given physics laws. The biggest benefit of PINNs is that solutions can be inferred without any iterative process such as a recurrence equation with respect to time. Furthermore, it is used for diverse applications such as Hidden Fluid Mechanics

hfm that extracts hidden variables of a given equation using a PINN and observations. However, it is hard to optimize model parameters when we deal with complicated PDEs and their coefficients. In order to improve the training ability, combinations of PINNs and numerical methods have been developed, or other neural networks such as CNNs are selected. M. Raissi et al. pinns1 added Runge-Kutta methods to a PINN model for solving AC equation. Aditi et al. pinns2

proposed transfer learning and curriculum regularization which start training PINNs on a specific safe domain and then transfer to a target domain. Hao Ma et al.

unetpde proposed a U-shape CNN so-called U-net unet and they showed that the usage of target data in a loss function significantly improves optimization. Elie Bretin et al. meancur used convolutional neural networks derived from a semi-implicit approach to learn phase field mean curvature flows of the AC equation.

We propose data-driven models that approximate the solution of explicit finite difference scheme to solve second order reaction-diffusion type equations numerically. Our contributions are as follows:

  1. We propose a five-point stencil convolution operator to solve reaction-diffusion type equations.

  2. Our proposed model is trained using two consecutive snapshots to solve a given equation.

  3. We demonstrate the robustness of our method using five reaction-diffusion type equations and noisy data.

The remainder of this paper is organized as follows. In Section 2, we present how to create training data using explicit FDM, explain the FCNN concept, training process, and numerical solutions. In Section 3, we compare the prediction results using our proposed FCNN and the evolution of PDEs results using the FDM method and show the robustness of our FCNN. Finally conclusions are drawn in Section 4.

2 Methods and numerical solutions

FDM is to divide a given domain into finite grids and find an approximate solution using derivatives with finite differences PZ2012 . We use explicit FDM to create training data with random initial conditions. We use only two consecutive FDM results, the initial and next time step results, as training data for each equation. To create training data, a computational domain is defined using a uniform grid of size and for and . Here, and are mesh sizes on the computational domain . Let be approximations of and is temporal step size. The boundary condition is zero Neumann boundary condition. Laplacian of a function is calculated using a five-point stencil method, the Laplacian can be approximated as follows:

(1)

In this way, the first and second derivatives of at each point (e.g., , , and ) can be approximated within the 3 3 local area centered . This concept can be equivalent to 3 3 convolution kernels. The 3 3 kernels following properties:

1. for any (element-wise summation)

2. for any (element-wise multiplication)

3. for any (element-wise division)

4. for any and any real numbers

The second-order PDEs can be solved numerically using combinations of 3 3 kernels. Therefore, if we build a proper CNN as the form of recurrence Eq. (4), we can solve a PDE (2) numerically. A previous study about AC equation cnnallencahn shows that FDM can be expressed as CNN.

(2)
(3)
(4)

To solve second order reaction-diffusion type equations

(5)

where is a diffusion coefficient, is a reaction coefficient, and is a smooth function to present reaction effect, we propose FCNN as a recurrence relation:

(6)
Figure 1: Computational graph of FCNN for second order reaction-diffusion type equations

As a CNN,

containing a 5-point stencil filter and a pad satisfying given boundary conditions solves

. In order to approximate terms, we define a trainable polynomial function as follows:

(7)

with model parameters for any and a real value . Let be a FCNN. Then, the inference is as follows:

(8)

where is a set of model parameters. Figure 1 shows the computational graph of our explicit model FCNN containing model parameters for any in a filter. Furthermore, represents the diffusion term on the uniform grid of and axises, so we set up and to cut down on training time. When the five-point stencil filter is used and is a -th order polynomial function, the number of model parameters is only . Thus, the set-up enables to learn physical patterns from few data. In Algorithm 1, an initial image and the prediction at the next time are used with train data and to train a model . The objective function is the mean square error function as follows:

(9)

where , and are the number of pixels in an output image, a prediction and its target respectively.

Set an initial value , a small constant
Initialize with model parameters
while  do
     
     Compute loss
     Update
end while
Algorithm 1 Training Procedure

2.1 Reaction-diffusion type equations

To demonstrate the robustness of FCNN, we consider reaction-diffusion type equations: heat, Fisher’s, AC equation, reaction-diffusion equations with trigonometric functions. The reaction and diffusion coefficients used in each formula are arranged in Table 1.

Heat Fisher’s AC Sine Tanh
1 1 1 0.1 0.5
0 20 40 10
Table 1: Diffusion () and reaction () coefficients for the simulations.

For the AC equation, where is the thickness of the transition layer and which value is cnnallencahn . For the other equations, we select arbitrary coefficients. For all the following equations, the continuous equations and the discretized equations are described in turn, and the zero Neumann boundary condition is used.

2.1.1 Heat equation

(10)
(11)

2.1.2 Fisher’s equation

(12)
(13)

2.1.3 AC equation

(14)
(15)

2.1.4 Reaction-diffusion equation with trigonometric function:

(16)
(17)

2.1.5 Reaction-diffusion equation with trigonometric function:

(18)
(19)

where and each discretized equation ((11), (13), (15), (17), (19)) is implemented based on the model structure proposed in cnnallencahn . When and , all the equations show the almost similary evolution so we use different reaction coefficient much bigger than diffusion coefficient to check diverse evolutions as shown in Table 1.

3 Simulation results

Assume that we observe a reaction-diffusion pattern and investigate the pattern rule under the constraint meaning that the observations and predictions follow the same PDE. Our proposed FCNN is trained using only two consecutive data which are the initial and next time step results for each equation. Then, we evaluate the model using diverse unseen initial values.

In the simulations, we use random initial value data with mesh so that the size of the input data is containing a pad as a boundary condition. Also, (Heat, Fisher’s, AC) or (Sine, Tanh) for is fixed depending on given equations and a

convolutional filter is used with the stride of

in Eq. (7). Hence, the filter has 10,000 () chances to learn the evolution of results images, so training a model using only two consecutive images are enough to optimize nine or thirteen model parameters (). As an optimizer, ADAM adam is used with a learning rate of 0.01 and without any regularization. Instead, we apply early stopping earlystopping based on a validation data to avoid overfitting. To demonstrate the approximation for non-polynomial functions , we additionally consider sine and tanh functions besides heat, Fisher’s, and AC equations.

For the evaluation, we implement FCNN and FDM respectively and then measure the averaged relative

error with 95% confidence interval over 100 novel random initial values as shown in Table

2.

Equations Relative error
Heat
Fisher’s
AC
Sine
Tanh
Table 2: Relative error between FCNN and FDM. The shows 95% confidence intervals over 100 different random initial values.

Furthermore, we validate the errors using different types of initial values for each equation as shown in Table 3. The initial conditions are described in the Appendix Section.

Initial shapes: circle star circles torus maze
Heat
Fisher’s
AC
Sine
Tanh
Table 3: Relative error between FCNN and FDM with diverse initial values.

Figures 2-6 show the time evolution results when unseen initial shapes (circle, star, three circles, torus, and maze) are given after learning with two training data (random initial condition and next time step result with FDM). We compare the predicted results from pretrained models to the FDM results.

(a)
(b)
(c)
(d)
(e)
Figure 2: Time evolution of a circle shape of (a) Heat, (b) Fisher’s, (c) AC, (d) Sine, and (e) Tanh equations.
(a)
(b)
(c)
(d)
(e)
Figure 3: Time evolution of a star shape of (a) Heat, (b) Fisher’s, (c) AC, (d) Sine, and (e) Tanh equations.
(a)
(b)
(c)
(d)
(e)
Figure 4: Time evolution of three circles of (a) Heat, (b) Fisher’s, (c) AC, (d) Sine, and (e) Tanh equations.
(a)
(b)
(c)
(d)
(e)
Figure 5: Time evolution of a torus shape of (a) Heat, (b) Fisher’s, (c) AC, (d) Sine, and (e) Tanh equations.
(a)
(b)
(c)
(d)
(e)
Figure 6: Time evolution of a maze shape of (a) Heat, (b) Fisher’s, (c) AC, (d) Sine, and (e) Tanh equations.
Relative Error
Table 4: Relative error with noise. The shows 95% confidence intervals over 100 different random initial values.
Figure 7: Inference using a contaminated model with noise rates ( 0, 5, and 30)

Data-driven models are sensitive to data noise. To investigate the noise effect, we inject Gaussian random noise to and then the model is trained using and for the AC equation. Table 4 shows that the model can be trained under the noise condition. Figure 7 displays the results of the inference using contaminated models.

4 Conclusions

In this paper, we proposed Five-point stencil CNN (FCNN) containing a five-point stencil kernel and a trainable approximation function. We considered reaction-diffusion type equations including heat, Fisher’s, Allen–Cahn equations, and reaction-diffusion equations with trigonometric functions. We showed that our proposed FCNN can be trained well using few data (used only two consecutive data) and then can predict reaction-diffusion evolution with unseen diverse initial conditions. Also, we demonstrated the robustness of our FCNN under the noise condition.

Acknowledgments

The corresponding author Y. Choi was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (NRF-2020R1C1C1A0101153712).

Appendix

In this appendix session, we describe the initial conditions used in the simulation results session 3. A detailed description of these initial conditions can be found in our previous research paper cnnallencahn .

(1) The initial condition of a circle shape

(20)

where is the initial radius of a circle.

(2) The initial condition of a star shape

(21)

where


(3) The initial condition of a torus shape

(22)

where and are the radius of major (outside) and minor (inside) circles, respectively. And, for simplicity of expression, .

(4) The initial condition of a maze shape

The initial condition of a maze shape is complicated to describe its equation, so refer to the codes which are available from the first author’s GitHub web page (https://github.com/kimy-de) and the corresponding author’s web page (https://sites.google.com/view/yh-choi/code).

(5) The initial condition of a random shape

(23)

here the function rand has a random value between and .

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