1. Introduction
As a universal physical model, the focusing nonlinear Schrödinger (NLS) equation
$\begin{eqnarray}{\rm{i}}{\partial }_{t}\psi +\displaystyle \frac{1}{2}{\partial }_{x}^{2}\psi +| \psi {| }^{2}\psi =0,\quad x,t\in {{\mathbb{R}}}^{2},\end{eqnarray}$
where the amplitude envelope ψ = ψ(x, t) is a complex function defined on the spatial location x and the evolution time t, can be used to describe nonlinear wave phenomena of many conservative physical systems (e.g. fiber optics, plasma physics, marine science, Bose–Einstein condensates, finance, etc) [1–4]. It is also often referred to as the Gross–Pitaevskii (GP) equation in Bose–Einstein condensates [3]. Despite the simple expression form of the NLS equation, it contains a wealth of nonlinear wave structures [5, 6] (e.g. solitons, breathers, rouge waves, periodic solutions, etc). And the soliton-state transitions in the NLS equation can be realized by varying the parameters of explicitly exact solutions [7].In recent years, due to the rise of the parity-time (${ \mathcal P }{ \mathcal T }$) symmetry [8], the NLS equation can also be extended to study the nonlinear waves in nonconservative systems [8–19]. And the nearly-integrable NLS equation with complex ${ \mathcal P }{ \mathcal T }$-symmetric potentials can be written as [11–19]2 ). Thus the ${ \mathcal P }{ \mathcal T }$-symmetry generalizes the concept of solitons, which were previously thought to exist only in conservative systems. Due to the complexity of the ${ \mathcal P }{ \mathcal T }$-symmetric potentials, equation (2 ) also exists in the soliton-state transitions when the soliton parameters were varied in the ${ \mathcal P }{ \mathcal T }$-symmetric NLS equation [16].
$\begin{eqnarray}{\rm{i}}{\partial }_{t}\psi +\displaystyle \frac{1}{2}{\partial }_{x}^{2}\psi +| \psi {| }^{2}\psi +[V(x)+{\rm{i}}{W}(x)]\psi =0,\end{eqnarray}$
where the real-valued functions V(x) and W(x) denote the potential well and gain-loss distribution, respectively. The ${ \mathcal P }{ \mathcal T }$-symmetry of the potential requires V(x) = V(−x) and W(x) = −W(−x). Due to the constraints of the dispersion term, the nonlinear term and the potential, clusters of solitons can also be found in equation (To study wave propagations of ultra-short optical pulses in Kerr nonlinear media, the focusing third-order NLS equation (alias Hirota equation) [20]1 ). The Hirota equation (3 ) and its extensions can also be used to describe the strongly dispersive ion-acoustic wave in plasma physics [21] and the broader-banded waves in the deep ocean [22]. Particularly, equation (3 ) reduces to equation (1 ) at β = 0. The Hirota equation (3 ) is also completely integrable since it can be derived from the compatibility condition of the Lax pair [23, 24]. And it has been verified to admit the bright solitons [25–27], breathers [28, 29], and rouge waves [30–34] by means of the inverse scattering transform, Darboux transform, and numerical method. The state transitions from rouge waves to W-shaped solitons, breathers to the periodic waves, and breathers to anti-dark solitons have been investigated in [32, 33], which means that the solution parameters can be used to modulate the distinct wave structures of solitons.
$\begin{eqnarray}\begin{array}{l}{\rm{i}}{\partial }_{t}\psi +\displaystyle \frac{1}{2}{\partial }_{x}^{2}\psi +| \psi {| }^{2}\psi -{\rm{i}}\beta \left({\partial }_{x}^{3}\psi +6| \psi {| }^{2}{\partial }_{x}\psi \right)=0,\\ \quad \beta \in {\mathbb{R}},\end{array}\end{eqnarray}$
was presented, which is also a fundamental physical model, and the third-order dispersion and higher-order nonlinear term are taken into account in the NLS equation (Recently, deep learning (DL) [35] has attracted extensive attention due to the powerful fitting capability of neural networks. Meanwhile, the improvement of computer hardware (GPU) and the continuous updating of DL libraries (e.g. Pytorch [36], Tensorflow [37], Julia [38]) have also greatly promoted the development of DL. It is the continuous development of hardware and software that enables neural networks to process large amounts of data simultaneously and extract features for prediction. Nowadays, DL has impacted various areas, such as image recognition [39, 40], target detection [41], image generation [42], machine translation [43], speech recognition [44], natural language processing [45], and etc [46, 47]. The universal approximation theorem [48, 49] for neural networks has also created a boom in scientific machine learning (SciML) [50]. Due to its powerful approximation ability, it is natural to apply it to the numerical approximation of ordinary differential equations and partial differential equations (PDEs).
The use of DL-based methods to solve differential equations (DEs) can be broadly divided into two genres. One is the neural finite element method, which replaces the finite element with a global neural network as the base of the approximation solution function of DE, such as the physics-informed neural networks (PINNs) method [51–53], the deep Ritz method [54]; the other is the DL architecture that learns the mapping (i.e. solution operator) between infinite-dimensional function spaces, such as the DeepOnet [55], Fourier neural operator (FNO) [56, 57], and other neural operator methods [58, 59]. The approximation errors of the two infinite-dimensional neural operators are proved theoretically in the latter papers [60, 61]. The most important feature of PINNs is the incorporation of the PDEs into the loss functions, which greatly increases the accuracies of the prediction solutions, but it requires retraining of the network for any given new instance of the function coefficients or initial conditions (see e.g, [62–69] and references therein). However, the prediction solutions via the FNO can be quickly given for any given new parameters once the networks were trained, while we do not need to know the underlying PDEs. In other words, we can unitedly learn the PDE families via the FNO. Thus, FNO has been extended to various fields, such as weather forecasting [70], the multi-phase flow [71], and heterogeneous material modeling [72]. If the structure of the temporal domain or spatial domain looks particularly complicated (high frequency) features to the frequency domain can be approximated with a small number of basis functions, then FNO can do convolution in the frequency domain with a small amount of data to learn out the operators between infinite dimensional spaces. These operators maintain the characteristics of Fourier approximation of high frequencies and neural network approximation of low frequencies [73, 74], so the FNO performs well in the approximation problems.
As mentioned above, the parameters of exact solutions can modulate the wave structures in some nonlinear integrable and nearly-integrable systems. In other words, we can obtain abundant nonlinear wave structures by changing the solution parameters. A natural and interesting question is how to use a deep neural network to uniformly learn these soliton-state transitions. To the aim, we combine the FNO with these soliton structures, and by training the FNO network we can obtain the different types of soliton solutions for a given initial condition, although it has a strong influence on the types of solitons. Moreover, we find that the soliton-state transitions of nonlinear wave equations can be well learned via the FNO network. To the best of our knowledge, the FNO was used to study the real-valued PDEs, but it was not extended to learn soliton-state transitions of the complex-valued nonlinear wave equations.
The rest of this paper is arranged as follows. In section 2 , we specifically give the neural network framework of the FNO method, and the choice of some hyper-parameters. In section 3 , we learn the different types of solitons in the NLS equation (1 ) via the data-driven FNO deep learning. In section 4 , we learn the different types of solitons in the Hirota equation (3 ) via the data-driven FNO deep learning. The soliton state transitions of the NLS equation with ${ \mathcal P }{ \mathcal T }$-symmetric potential given by equation (2 ) are discussed in section 5 . Finally, we give some conclusions and discussions in section 6 .
2. Deep learning-based FNO methodology
The FNO method [56, 57] aims to learn a nonlinear mapping ${ \mathcal H }:{ \mathcal A }\to { \mathcal U }$ between two infinite-dimensional function space ${ \mathcal A }$ and ${ \mathcal U }$ from the finite input-output data observations located in the bounded and open spatial-temporal domain $D={\{({x}_{i},{t}_{i})\}}_{i\,=\,1}^{n}\subset {{ \mathcal R }}^{2}$, where ${ \mathcal A }$ and ${ \mathcal U }$ represent the parameter/initial condition spaces and (time-varying) solutions of a PDE, respectively. To approximate the operator, the FNO parameterizes ${ \mathcal H }$ to a parametric mapping ${{ \mathcal H }}_{{\rm{\Theta }}}$ via the neural network, which means
$\begin{eqnarray}{{ \mathcal H }}_{{\rm{\Theta }}}:{ \mathcal A }\times {\rm{\Theta }}\to { \mathcal U }\end{eqnarray}$
with ${\rm{\Theta }}:{ \mathcal R }\to { \mathcal R }$ denoting the trainable finite-dimensional parameter space in the network. The input function $a(x)\in { \mathcal A }$ is first transformed into a higher-dimensional representation v0 = P(a(x)) through a fully-connected shallow neural network P( · ), then the frequency domain is learned by the following Fourier layers $\begin{aligned}v_{\tau+1}(x) & =\sigma\left(W v_{\tau}(x)+\left(\mathcal{K}(a ; \Theta) v_{\tau}\right)(x)\right) \\\tau & =0,1, \ldots, T-1,\end{aligned}$
where σ denotes a nonlinear activation function, W is the linear transform (i.e. weight) and ${ \mathcal K }(a;{\rm{\Theta }})$ represents the kernel integral operator (i.e. kernel function) parameterized by the Θ, and can be defined as follow $\begin{eqnarray}\left({ \mathcal K }(a;{\rm{\Theta }}){v}_{\tau }\right)(x):= {\int }_{D}\kappa (x,y,a(x),a(y);{\rm{\Theta }}){v}_{\tau }(y){\rm{d}}y,\end{eqnarray}$
where a kernel function κ denotes a neural network parameterized by Θ, and can be learned via data. Thus by supposing κ(x, y; Θ) = κ(x − y; Θ) and convolution theorem, ${ \mathcal K }$ can be realized as $\begin{eqnarray}\left({ \mathcal K }({\rm{\Theta }}){\nu }_{k}\right)(x)={{ \mathcal F }}^{-1}\left({R}_{{\rm{\Theta }}}\cdot \left({{ \mathcal F }}_{{\nu }_{k}}\right)\right)(x),\end{eqnarray}$
where RΘ denotes the Fourier transform of a periodic function κ, and can be learned 1 × 1 convolution kernel (equivalent to the fully-connected layer between channels), ${ \mathcal F }$ and ${{ \mathcal F }}^{-1}$ denote the Fourier transform and its inverse, respectively: $\begin{eqnarray*}\begin{array}{rcl}{ \mathcal F }[f(x,t)](k) & = & {\displaystyle \int }_{D}f(x,t){{\rm{e}}}^{-2\pi {\rm{i}}{k}{x}}{\rm{dx}},\\ {{ \mathcal F }}^{-1}[{ \mathcal F }[f](k)] & = & {\displaystyle \int }_{D}{ \mathcal F }[f](k){{\rm{e}}}^{2\pi {\rm{i}}{k}{x}}{\rm{d}}{k}.\end{array}\end{eqnarray*}$
And in the realization, ${ \mathcal F },{{ \mathcal F }}^{-1}$ can be obtained by the fast Fourier transform (FFT) of finite modes. It is the presence of the Fourier layer that enables the FNO to more quickly and accurately learn the high-frequency information, which in physical space corresponds to complex solutions. Finally, with a two-layer fully-connected neural network, the output can be obtained: ${u(x,{t}_{k})}_{k={t}_{1}}^{T}=Q({\nu }_{T}(:))$. That is, the output of the FNO can be written as the form $\begin{eqnarray}{{ \mathcal H }}_{{\rm{\Theta }}}(a(:))=Q\circ {F}_{T}\circ ...\circ {F}_{1}\circ P(a).\end{eqnarray}$
It is noted that we learn the mapping from the system parameters to the (time-varying) solution of the PDE in this paper. However, the constraint of FNO is that the inputs and outputs need to be defined on the same set of grids (${\{({x}_{i},{t}_{i})\}}_{i\,=\,1}^{n}$), except for the channels, thus we embody the changed parameters in the initial conditions of the PDE, i.e. a(x) = u(x, t0).Training Data.—Since FNO is data-driven supervised learning, different from PINNs [51], it is essential to obtain the parameters of the whole network by high-quality label data. One can access the data from the exact solution (if available) or the numerical solutions via the high-accuracy numerical methods (e.g. spectral method, finite element method, etc). Also, for other problems in real life, the data can be derived from experiments and observations. In the following experiments, the parameters are generated from the uniform distribution on (a, b), thus we can get the corresponding initial conditions. Though the parameters only vary from (a, b), the impact on the types of solitons is huge which can be found in the discussions. If not specified, the total number of samples is 1500, of which 1000 are used for training and 200 for the test set.
The neural network framework.—Figure 1 displays the framework of the FNO. The fully-connected layer P increases the channels of the input function to width 32, then applies four Fourier integral operator layers with R being the 1 × 1 kernel. Finally, a two-layer fully-connected neural network Q changes the channel into 128 and the desired channel, which depends on the problem at hand. As for the finite term truncation of the FFT, we choose it as 12 in all problems. To overcome the internal covariate shift, the batch normalization [75] is applied to each Fourier layer. And the GeLu function σ(x) = xΦ(x) is chosen as the nonlinear activation function with Φ(x) denoting the distribution function of the standard normal distribution (notice that one can also choose other nonlinear activation functions). The Adam optimizer with the initial learning rate η = 0.001 halved every 100 epochs in the total 500 epochs is used to minimize the loss function defined as5 ). It is worth noting that since the FNO is used to study complex nonlinear wave equations in this paper, we must add another channel in the implementation.
$\begin{eqnarray}L(\psi ,\hat{\psi })=\displaystyle \frac{1}{{N}_{{\rm{train}}}}\displaystyle \sum _{{\ell }=1}^{{N}_{{\rm{train}}}}\displaystyle \frac{\parallel {\psi }_{{\ell }}(x,t)-{\hat{\psi }}_{{\ell }}(x,t){\parallel }_{2}}{\parallel {\psi }_{{\ell }}(x,t){\parallel }_{2}},\end{eqnarray}$
with ${\psi }_{{\ell }}(x,t),{\hat{\psi }}_{{\ell }}(x,t)$ representing the ground truth solution and the prediction solution of the ℓth sample in train sets, respectively. And the batch size is chosen as 20 for the mini-batch stochastic gradient descent method. To observe the training process better, the mean squared error (MSE) and relative L2 error on the test set are also recorded. It is worth noting that we treat the time t as a part of the spatial dimensions as well and apply FFT to it in the Fourier layer, so we have to add padding if necessary to preserve the periodic boundary conditions for the FFT. However, this does not mean that the FNO can only handle periodic problems due to the presence of Wvτ in equation (
Figure 1. Framework of Fourier neural operator (FNO), where Linear operator + Nonlinear integral operator + Activation function → Highly learned nonlinear operator, and ${v}_{\tau +1}(x)=\sigma \left({{Wv}}_{\tau }(x)+\left({ \mathcal K }(a;{\rm{\Theta }}){v}_{\tau }\right)(x)\right)$, τ = 0, 1,…,T − 1, x ∈ D. |
Remark. This is also the first application of the FNO method in complex-valued nonlinear wave equations such as the NLS equation (1 ), the ${ \mathcal P }{ \mathcal T }$ NLS equation (2 ) and the Hirota equation (3 ). Of course, the method can also be extended to other complex-valued nonlinear wave equations.
3. The deep learning soliton-state transitions of the NLS equation
In this section, we mainly use the data-driven FNO method to learn the soliton-state transitions of the NLS equation with the initial-value condition:
$\begin{eqnarray}\left\{\begin{array}{l}{\rm{i}}{\partial }_{t}\psi +\displaystyle \frac{1}{2}{\partial }_{x}^{2}\psi +| \psi {| }^{2}\psi =0,\quad x\in (-6,6),\,t\in (-6,6),\\ \psi (x,-6)={\psi }_{0}(x),\quad x\in [-6,6].\end{array}\right.\end{eqnarray}$
The NLS equation (10 ) admits the soliton solution with one parameter [5, 6]11 ) denotes the Akhmediev breather (AB), the spatial-periodic, and temporal-local soliton, while it describes the Kuznetsov Ma (KM) breather when a > 0.5. And the rouge wave can be obtained from equation (11 ) by the limit of a → 0.5:11 ) with a being located nearby 0.5.
$\begin{eqnarray}\psi (x,t)={{\rm{e}}}^{{\rm{i}}{t}}\left[1-\displaystyle \frac{2(1-2a)\cosh ({bt})+{\rm{i}}{b}\sinh ({bt})}{\cosh ({bt})-\sqrt{2a}\cos (\omega x)}\right],\end{eqnarray}$
where $b={\left[8a(1-2a)\right]}^{1/2}$ and ω = 2(1 − 2a)1/2 with $a\in {\mathbb{R}}$. The only parameter a determines the different wave structures of this solution. As 0 < a < 0.5, equation ( $\begin{eqnarray}\psi (x,t)={{\rm{e}}}^{{\rm{i}}t}\left[1-\frac{4(1+2{\rm{i}}t)}{4\left({x}^{2}+{t}^{2}\right)+1}\right].\end{eqnarray}$
One can also observe the rouge wave-like solution in equation (The spatial domain [−6, 6] is split into 256 modes, while the temporal domain [−6, 6] is split into 25 modes. Based on the problem, the 1500 samples are generated by selecting the uniform distribution of the parameter a on (0, 1), then 1500 different initial conditions can be deduced. Here, we aim to learn the nonlinear operator mapping the initial condition ψ(x, t0) to the solution of some later time T > t0, ${ \mathcal H }:{L}^{2}\left([-6,6],t={t}_{0};{\mathbb{C}}\right)\to {L}^{2}\left(x\in [-6,6],t\in ({t}_{0},T];{\mathbb{C}}\right)$ defined by
$\begin{eqnarray}{ \mathcal H }:\,\psi (x,t){| }_{\{[-\mathrm{6,6}],t={t}_{0}\}}\to \psi (x,t){| }_{[-\mathrm{6,6}]\times ({t}_{0},T]}.\end{eqnarray}$
In the implementation, the complex function ψ(x, t) should be written into the real and imaginary parts, which is the first application of the FNO in complex nonlinear wave equations.After the 500 training epochs, the parameters Θ in the neural networks can be obtained. And the total parameters in the neural networks is 2368 162. Finally, the MSE and the relative L2 error on the train (loss) and test sets arrive 3.16e-3, 1.93e-3, 1.97e-3, respectively. And the dynamic change versus epoch is depicted in figure 2(a1). As it can be seen that the loss function decreases rapidly in the first 50 epochs and then decreases at a very slow rate. It can be observed that the FNO performs well in the test set, though the initial condition is a new instance for the neural networks from figure 2(b). It is noted that the predicted L2 error is 10−3 units displayed in figure 2(a2).
Figure 2. The training and testing progress for the problem ( |
The different types of data-driven solutions via FNO belonging to the different parameters are exhibited in what follows (see figure 3):
Figure 3. The data-driven soliton-state transitions of the NLS equation ( |
Case 1. As for the Kuznetsov–Ma breather, we choose a = 0.683 in the test set. The corresponding initial condition is presented in figure 3(a1). It can be concluded that the learned soliton via FNO matches well with the exact solution from figures 3(b1), (c1), (d1). It is worth mentioning that we find that the approximation error of PINNs is similar in magnitude to that of FNOs, however, the FNO can learn the entire family of solutions under a given distribution, while the PINNs can only learn some specific solution each time. This illustrates the superiority of the neural operator method.
Case 2. For the rouge waves, figure 3(a2) displays the parameter and initial condition, which differs from the KM breather. Though the coefficients/initial conditions have a strong influence on the type of solution, FNO still demonstrates its strong approximation capability. It can be seen that the learned FNO rouge wave is very close to that of the exact solution from figures 3(b2), (c2), (d2).
From the above results, we can conclude that the FNO performs well in the soliton state transitions of the NLS equation (10 ) by varying the parameters/initial conditions. Also, the FNO achieves state-of-the-art (SOTA) in terms of prediction accuracy of the solutions although the dynamical behaviors of the solitons are more complicated.
4. The deep learning soliton-state transitions of the Hirota equation
In this section, we mainly use the data-driven FNO method to deep learn the soliton-state transitions of the Hirota equation (3 ) with the initial-value condition:
$\begin{eqnarray}\left\{\begin{array}{l}{\rm{i}}{\partial }_{t}\psi +\frac{1}{2}{\partial }_{x}^{2}\psi +| \psi {| }^{2}\psi -{\rm{i}}\beta \left({\partial }_{x}^{3}\psi +6| \psi {| }^{2}{\partial }_{x}\psi \right)=0,\quad x\in (-8,8),\,t\in (-6,6),\\ \psi (x,-6)={\psi }_{0}(x),\quad x\in [-8,8].\end{array}\right.\end{eqnarray}$
4.1. Leaning an operator from rouge waves to W-shaped solitons
The solutions of the Hirota equation (14 ) can be deduced as15 ) admits two types of rouge waves with different dynamic behaviours (q = 0, qs/2). However, in the case q = qs, equation (15 ) reads the W-shaped waves with one peak and two valleys17 ) can also be obtained by choosing q → 1 in equation (16 ).
$\begin{eqnarray}\displaystyle \begin{array}{l}\psi (x,t)={\kappa }_{0}\\ \quad \left[\frac{4+8{\rm{i}}{a}^{2}\left(1-q/{q}_{s}\right)\tau }{1+4{a}^{2}{\left(\xi -v\tau \right)}^{2}+4{a}^{4}{\left(1-q/{q}_{s}\right)}^{2}{\tau }^{2}}-1\right],\end{array}\end{eqnarray}$
with κ0 being a non-zero background solution $\begin{eqnarray}{\kappa }_{0}=a{{\rm{e}}}^{{\rm{i}}\theta },\quad \theta ={qx}+\left[{a}^{2}-{q}^{2}/2+\beta (6{{qa}}^{2}-{q}^{3})\right]t\end{eqnarray}$
by means of Darboux transformation method, where v = q + (2a2 − q2)/(2qs), ξ = x − xc, τ = t − tc, and (xc, tc) determines the center of the solution. And a and q denote the amplitude and wavenumber of the background solution, respectively. The state transitions induced by the higher-order effects and background frequency have been analyzed via the modulation instability [32]. For the case q ≠ qs, equation ( $\begin{eqnarray}{\psi }_{s}(x,t)=a{{\rm{e}}}^{{\rm{i}}{\theta }_{s}}\left[\frac{4}{1+4{a}^{2}{\left(\xi -{v}_{s}\tau \right)}^{2}}-1\right],\end{eqnarray}$
with ${\theta }_{s}=\theta {| }_{q={q}_{s}},{v}_{s}=v{| }_{q={q}_{s}}$. The W-shaped soliton (We devote ourselves to researching the state-transition from the rouge wave to a W-shaped soliton in the Hirota equation (14 ) by modulating the parameter. Thus, the FNO can be used to learn the state transitions in equation (15 ). Without loss of generality, we choose qs = 1, tc = 3, xc = 0, a = 1, and β = 0.1 in the following discussions. The spatial domain [−8, 8] is split into 256 modes, while the temporal domain [−6, 6] is discrete into 49 modes for the input-output observations. It is worth mentioning that we have added a fine time split to capture the structure of the solution more precisely here. Considering the effect of q/qs on the solution pattern, we select 1500 samples uniformly distributed on (0, 1) and substitute them into equation (15 ) to obtain the corresponding initial conditions. Similarly to the previous case, although the fluctuation of q varies tinily, it has a huge influence on the initial condition and the localized waves at later moments. Our goal here is to use the FNO to learn the mapping from the initial condition ψ(x, t0) to the solution of some later time T > t0, ${ \mathcal H }:{L}^{2}\left([-8,8];t={t}_{0};{\mathbb{C}}\right)\to {L}^{2}\left([-8,8];t\in ({t}_{0},T];{\mathbb{C}}\right)$ defined by
$\begin{eqnarray}{ \mathcal H }:\,\psi (x,t){| }_{\{[-\mathrm{8,8}],t={t}_{0}\}}\to \psi (x,t){| }_{[-\mathrm{8,8}]\times ({t}_{0},T]}.\end{eqnarray}$
The real and imaginary parts need to be separated in order to carry out the output of the input of the network.The FNO can be trained by minimizing the relative L2 error for 500 epochs via the mini-batch gradient descent using the Adam optimizer with a decay learning rate. It is worth mentioning that the time of each epoch increases to about 4.1s due to the increase of the time discretization. Once the network is trained, the network can give the corresponding solution $\psi (x,t)\in { \mathcal U }$ in a very short time for a new input instance $a\in { \mathcal A }$. During the training process, we record the mean squared error, the relative L2 error on train, and test sets. The dynamic changes versus epoch is depicted in figure 4(a1). The values arrive 2.49e-3, 3.33e-3, 3.44e-3, respectively. The values of the loss functions decrease faster in the first 300 epochs, and then slower in the last 200 epochs. We can find that the FNO also performs very well on the test set from figure 4(a2). The relative L2 errors of the learned solutions are mostly below 10−2.
Figure 4. The training and testing progress on equation ( |
To better visualize the type of travelling waves, we also choose three parameters/initial conditions (new instances do not appear in the training set) to demonstrate the processes of soliton state-transitions (see figure 5):
Figure 5. The data-driven soliton-state transitions of solutions ( |
Case 1. Firstly, we choose q = 0.065 to display the rouge waves, and the corresponding initial condition is exhibited in figure 5(a1). The dynamic behaviors of the learned solution, the exact solution, and the absolute error between them are demonstrated in figures 5(b1)–(d1). The learned solution captures the characteristics of the rouge wave well, and the maximum absolute error is only around 0.05.
Case 2. We then set q = 0.530 to explore the features of the elongated rouge waves (see figure 5(a2)). The representative visualizations of the predicted solutions for the input samples are shown in figures 5(b2)–(d2). As it can be seen that the prediction achieves a good agreement with exact soliton over the entire spatial-temporal domain. It is worth noting that the initial condition, in this case, is similar to that in figure 5(a1), but the solution at a later time is different. However, FNO still captures the two different types of solutions well, which is a good indication of the superiority of the FNO algorithm for the approximations of complex problems.
Case 3. For the case of W-shaped travelling waves, q = 0.973 and the initial condition are chosen as the input of the deep neural network. We can observe that the input for the network differs from the case of the above two cases from figure 5(a3). It can be seen that an excellent agreement can be realized between the FNO's learned solutions and the exact solitons in figures 5(b3)–(d3). Despite the strong influence of the parameters on the solution, the predicted solutions via FNO can be learned very well.
Based on the above results, the Fourier layer in the FNO plays a critical role in the network to capture the high frequency part. That is, the FNO is able to distinguish well between different frequencies although the inputs (initial values) are very similar. And this is what is missing in traditional neural networks [73, 74].
4.2. Leaning an operator from Akhmediev breathers to periodic waves
Here we study the state transition from the Akhmediev breather to the periodic wave and the related wave phenomena. Equation (3 ) also possesses the Akhmediev breather in the form [33]16 ), ${v}_{1}\,=\beta \left(2{a}^{2}+4{a}_{1}^{2}-{q}^{2}\right)-2q(q\beta +1/2),\kappa =2\eta {v}_{2}$, $\eta =\sqrt{{a}^{2}-{a}_{1}^{2}}$, ${v}_{2}={a}_{1}\left(1-q/{q}_{s}\right)$, qs = −1/(6β). Similarly, the shape of equation (19 ) is changed with q/qs as q ≠ qs (q = 0, q = qs/2). As for q = qs, the Akhmediev breather (19 ) is converted to the periodic wave19 ).
$\begin{eqnarray}\psi (x,t)={\kappa }_{0}\left[\frac{2{\eta }^{2}\cosh (\kappa t)+2{\rm{i}}\eta {a}_{1}\sinh (\kappa t)}{\cosh (\kappa x)-{a}_{1}/a\cos \left[2\eta \left(x+{v}_{1}t\right)\right]}-1\right],\end{eqnarray}$
where κ0 is given by equation ( $\begin{eqnarray}\psi (x,t)={\kappa }_{0}\left[\displaystyle \frac{2{\eta }^{2}}{1-{a}_{1}/a\cos [2\eta (x+{vt})]}-1\right],\end{eqnarray}$
which can also be obtained by the limit q → qs in equation (To this end, the FNO is extended to learn the conversion from the Akhmediev breather to the periodic wave by varying q (initial condition) in equation (19 ). The parameters are chosen as a = 1, a1 = 0.7, β = 0.1 in the numerical experiments. Since the structure of the wave is more complicated at this point, we increase the number of divisions in the spatial-temporal domain. We discrete (x, t) ∈ [−10, 10] × [−6, 6] into 512 × 121 grids for the finite collections of input-output pairs. the FNO aims to learn the mapping from the initial condition ψ(x, t0) to the solution of some later time T > t0, ${ \mathcal H }:{L}^{2}\left([-10,10];t={t}_{0};{\mathbb{C}}\right)\to {L}^{2}$ $([-10,10];t\in ({t}_{0},T];{\mathbb{C}})$ defined by
$\begin{eqnarray}{ \mathcal H }:\,\psi (x,t){| }_{\{[-\mathrm{10,10}],t={t}_{0}\}}\to \psi (x,t){| }_{[-\mathrm{10,10}]\times ({t}_{0},T]}.\end{eqnarray}$
Since the neural network only supports the real-valued input and output, it is necessary to separate the initial conditions into real and imaginary parts for the input, and finally combine the real and imaginary parts of the output to get the prediction solutions.By training the network for 500 epochs with the Adam optimizer, the parameters Θ can be obtained by the gradient descent method and back propagation mechanisms. Due to the increase in the number of configured points in the spatial-temporal domain, the time to run each epoch also increases to about 17s. However, once the network is trained, it can make predictions in less than 1s for each instance on the test set, which is much faster than the traditional algorithms. To better visualize the training process, we record the MSE, relative L2-error on train, and test sets. These values eventually arrive 2.23e-3, 5.79e-3, 5.64e-3, respectively. It can be observed that the value of the loss function decreases faster in the first 200 epochs and slower in the last 300 epochs from figure 6(a1). The performance of FNO on the test set is also recorded in figure 6(a2). From the relative L2 error, we can see that the FNO still behaves well on 200 unseen samples.
Figure 6. The training and testing progress of equation ( |
In order to display the change of the soliton state, we choose a few specific parameters (initial conditions) located in the test set (see figure 7):
Figure 7. The data-driven soliton-state transitions of solutions ( |
Case 1. In the case of the Akhmediev breather, q = −10−3 is chosen to exhibit the initial condition in figure 7(a1). Figures 7(b1)–(d1) also demonstrate the FNO's learned soliton, ground truth, and the absolute error between the two. Despite the complicated structure of the Akhmediev breather, the FNO can still capture these features very well. Meanwhile, the absolute error of 0.1 is achieved only in a few positions.
Case 2. Then q = −0.830 is utilized to explore the decreased t-direction localization Akhmediev breather, the corresponding initial condition is depicted in figure 7(a2). As it can be seen that although the value of q does not fluctuate much, the effect on the initial value is more significant. The excellent agreement between the predicted soliton and the exact solution is displayed in figures 7(b2-d2) while the maximum error is only around 0.05. Surprisingly, the FNO is able to learn complicated structures for different initial values while ensuring the accuracy of the predicted solutions.
On the basis of the above results, we can draw the conclusion that for the initial conditional input $a\in { \mathcal A }$ with a given distribution, the FNO is able to extract the key features with only 1000 samples, although there is a large gap in the ground truth corresponding to these 1000 samples. Meanwhile, the network is able to give the output $\psi (x,t)\in { \mathcal U }$ for new samples quickly which greatly improves the generalization ability of FNO. Surprisingly, compared with the traditional neural network prediction model, the solutions learned by FNO also achieve SOTA in terms of accuracy.
5. The NLS equation with the complex ${ \mathcal P }{ \mathcal T }$-symmetric potential
In this section, we mainly consider the data-driven FNO deep learning of the soliton-state transitions of the NLS equation with ${ \mathcal P }{ \mathcal T }$-symmetric potential given by equation (2 ) with the initial-value condition:
$\begin{eqnarray}\begin{array}{c}{\rm{i}}{{\rm{\partial }}}_{t}\psi +\frac{1}{2}{{\rm{\partial }}}_{x}^{2}\psi +|\psi {|}^{2}\psi +[V(x)+{\rm{i}}W(x)]\psi =0,\\ \,x\in (-10,10),\,t\in (0,5),\\ \,\psi (x,0)={\psi }_{0}(x),\,x\in [-10,10],\end{array}\end{eqnarray}$
where the complex potential is chosen as the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II (δ(x)-Scarf-II) potential [16] $\begin{eqnarray}\begin{array}{rcl}V(x) & = & 2\alpha (\delta (x)-\tanh | x| )-2\,{{\rm{sech}} }^{2}x,\\ W(x) & = & -{W}_{0}{\partial }_{x}({\rm{sech}} x\,{{\rm{e}}}^{\alpha | x| }),\quad \alpha \lt 1,\end{array}\end{eqnarray}$
which becomes the ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential at α = 0 [9, 11].In particular, equation (22 ) with the ${ \mathcal P }{ \mathcal T }$ potential (23 ) can be shown to possess the solitons24 ) that equation (22 ) possesses:
$\begin{eqnarray}\begin{array}{l}\psi (x,t)=\phi (x){{\rm{e}}}^{{\rm{i}}\mu t},\quad \phi (x)=\frac{{W}_{0}}{3}{\rm{sech}} (x){{\rm{e}}}^{\alpha | x| }\\ \exp \,\left(-\frac{{\rm{i}}{W}_{0}}{3}{\int }_{0}^{x}{\rm{sech}} (s){{\rm{e}}}^{\alpha | s| }{\rm{d}}s\right),\end{array}\end{eqnarray}$
and μ = 1 + α2. It follows from solution (| • | The peakon solution when α < 0. |
| • | The smooth bright soliton for α = 0. |
| • | The double-hump peakon solution with one sharp valley if 0 < α < 1. |
Similarly, the soliton-state transitions can be modulated by the system parameter α, thus FNO can be applied to find the abundant structures in the ${ \mathcal P }{ \mathcal T }$-symmetric NLS equation. We utilize 256 modes and 6 modes to discrete the spatial-temporal domain (x, t) ∈ [−10, 10] × [0, 5] for the finite observations of ${a}_{j}(x)\in { \mathcal A }$ and ${u}_{j}(x)\in { \mathcal U }$, where j indicates the input-output pairs. Due to the effect of the parameter α on the type of solitons, we choose 1500 α uniformly distributed on (−1, 1) and generate 1500 different types of initial conditions through it as the FNO's input. We are interested in learning the operator mapping the initial condition ψ(x, 0) to the soliton up to some later time T > 0, ${ \mathcal H }:{L}^{2}$ $\left([-10,10];t=0;{\mathbb{C}}\right)\to {L}^{2}$ $\left([-10,10];t\in (0,T];{\mathbb{C}}\right)$ defined by
$\begin{eqnarray}\begin{array}{c}{ \mathcal H }:\,\psi (x,t){|}_{\left\{\left[-10,10\right],t=0\right\}}\to \psi (x,t){|}_{\left[-10,10]\times (0,T\right]}.\end{array}\end{eqnarray}$
Similar to the previous implementation, we add an extra channel to separate the real and imaginary parts of the complex function ψ(x, t).The parameters Θ can be obtained after training the FNO by minimizing the loss function defined by equation (9 ) for 500 epochs with the mini-batch gradient descent method. The results for the mean squared error, relative L2 error on train, and test sets in the process are presented in figure 8(a1). Throughout the training process, the Loss function decreases in a fast and then slow trend. And the values finally arrive 9.2e-4, 4.78e-3, 4.81e-3, respectively. The prediction L2 error is measured in new unseen instances that are not used during the training process which is exhibited in figure 8(a2). As can be seen, the errors on the 200 test samples are most less than 10−2.
Figure 8. The training and testing progress of equation ( |
Furthermore, we investigate the predicted solution, the ground truth as well as the absolute error under the different initial conditions for visualization (see figure 9, where W0 = 2):
Figure 9. The data-driven soliton-state transitions of equation ( |
Case 1. We first consider the peakon soliton given by equation (24 ) with α < 0, which is not differentiable at x = 0. α = −0.988 and the corresponding initial condition for the input sample is depicted in figure 9(a1). And figures 9(b1–d1) display the comparison between the exact soliton and the learned solution for the input. We can observe that the FNO's predicted solution achieves an excellent agreement with the ground truth.
Case 3. The double-hump soliton with one sharp valley is also investigated by setting α = 0.369 which is unseen in the training sets (see figure 9(a3)). Thanks to the transformation of high frequency information by the Fourier layer, it can be seen that the predicted solution of FNO captures the small structure of the complex solution very well (one sharp valley) depicted in figures 9(b3)–(d3) while the high accuracies are also maintained.
On the basis of these results, we can conclude that the FNO can be regarded as a powerful data-driven deep learning method that the generalization ability of the network is greatly enhanced compared to the PINNs [52], which the input parameters/initial conditions are given and fixed during the training and testing process. Besides, the FNO can predict solitons with more complicated wave structures.
6. Conclusions and discussions
In summary, we have effectively studied the soliton state-transitions in some soliton equations by means of the FNO method. We consider various types of nonlinear wave excitations in the NLS equation, Hirota equation, and NLS equation with ${ \mathcal P }{ \mathcal T }$-symmetric potential. Despite the complicated structures of the solutions, the FNO is still able to capture the key features and information. And for the given new samples, it can give more accurate prediction solutions. The results obtained in this paper are useful to further analyze the soliton-state transfers and neural operator networks solving complex-valued nonlinear PDEs.
In the case that the studied equation has no analytical solution, one can find its numerical solutions. Since FNO is a supervised deep learning model, the quality of supervised data will affect the approximation effect of FNO to a certain extent. Therefore, we need to obtain supervised data through some high-precision numerical algorithm as the input label. However, due to the limitations of numerical algorithms, it is sometimes difficult to obtain supervised data, so it is particularly important how to design an unsupervised learning operator method. This will be considered and discussed in our future work.