Can't one use a Sinc interpolator directly for all the function approximation tasks?

Sinc interpolator with a fixed $h$ can be used for all the function interpolation tasks, but it won't guarantee the theorem 1 and 2 if $h$ is not the optimal one. In our SincKAN, as the optimal $h$ is unknown, the performance of a single Sinc interpolator is poor so we change to use several Sinc interpolators. Furthermore, several Sinc interpolators are theoretically more accurate than a singular interpolator, as we discussed on line 243 in our manuscript.

The experiments of section 3.1 do not seem so relevant for practical applications. It seems that what might be most interesting would be to take a selection of complicated PDE/ODE data (using any off the shelf differential equation solver) and use these instead. This would demonstrate the value of SincKANs in practical circumstances, and in situations of more than 1 dimension. As it stands, these benchmarks feel artificial and irrelevant for real world application.

The experiments of section 3.1 are not manufactured and impractical, sin-low and sin-high are from RICHARDSON et.al (2011), double-exponential and multi-sqrt is from Sugihara et.al (2003), piece-wise is from ChebyKAN, spectral-bias is from Rahaman et.al (2019).

We also did experiments on high-dimensional functions,

For approximation

let's consider the function:$f(\boldsymbol{x})=e^{-\alpha ||\boldsymbol{x}||^2_2}, \boldsymbol{x}\in [-1,1]^d$ from DAS-PINNs. Here we choose $\alpha=0.001, d=100.$

The results are:

For PDE:

We consider two different problems: the fractional PDEs, and high-dimensional Poisson. For fractional PDEs, we utilize fPINN on the spatial fractional order problem:

$

-\Delta^su+\gamma u=f, \quad \boldsymbol{x}\in (-1,1)^d

$

if $f=1+\gamma u$, the exact solution is

$

u(x)=\frac{2^{-2s}\Gamma(d/2)}{\Gamma(d/2+s)\Gamma(1+s)}(1-\|x\|^2)^s,

$

where $s\in \mathbb{R}_+$ is a hyperparameter. In our experiments, $s=0.95,d=1$.

The results are:

For high-dimensional Poisson equations. we consider

$

\Delta u =f, \quad \boldsymbol{x}\in [0,1]^d.

$

where $f=2\alpha(2\alpha ||\boldsymbol{x}||^2_2-d) e^{-\alpha ||\boldsymbol{x}||_2^2}$, the exact solution is $u(\boldsymbol{x})=e^{-\alpha ||\boldsymbol{x}||^2_2}$. In our experiments, $d=100, \alpha=10/d$.

The results are:

More scrutiny of the physics-informed losses would be beneficial. Some plots of solutions and errors across the poorer performing methods might help understand why they are performing badly. Is it that boundary conditions are not being adhered to? Maybe there are regions of high PDE loss in the resulting solution? Perhaps small changes (e.g. weighing boundary conditions effectively) might lead to improved performance.

We have already plotted the solutions to the poor performance in Figure 4 (c). The figure shows that the boundary condition is strictly obeyed for every network because we use weight=100 for boundary loss and weight=1 for residual loss. Besides, we did experiments on larger weights of boundary conditions to have a more strict boundary condition: we keep the weight of residual loss, and weight=1000 for boundary loss in the poor performance experiments i.e. $\epsilon=1000$ in Table 3. Additionally, we release the boundary condition by setting weight=10 to further explore the performance. Here are the results:

The manufactured solutions used in the learning for approximation secion (3.1) are arbitrary and seemingly not very relevant to real world applications.

Many are in terms of sins and exponentials, that it stands to reason are well approximated by Sincs.

1. interpolation is an important composition of KANs, but not all.

2. Sinc function has different features with sins and exponentials, it is more like a 'quasi' delta function rather than sins and exponentials. For example let's interpolate $\sin(2\pi x), x\in[0,1]$, by @chebfun and @sincfun tools in MATLAB, like what we did in Figure1. The results are:

degree: 1.00e+01,sinc: 9.98e-01, cheby: 8.58e-06, cubic: 2.24e-03, 

degree: 2.00e+01,sinc: 8.95e-01, cheby: 7.71e-16, cubic: 4.37e-05, 

degree: 5.00e+01,sinc: 1.73e-01, cheby: 3.72e-16, cubic: 4.99e-07, 

degree: 8.00e+01,sinc: 1.65e-02, cheby: 4.59e-16, cubic: 6.83e-08, 

degree: 1.00e+02,sinc: 2.95e-03, cheby: 4.04e-16, cubic: 2.73e-08, 

degree: 1.50e+02,sinc: 2.94e-05, cheby: 5.14e-16, cubic: 5.27e-09, 

degree: 2.00e+02,sinc: 2.18e-07, cheby: 5.07e-16, cubic: 1.65e-09, 

degree: 2.50e+02,sinc: 1.33e-09, cheby: 5.43e-16, cubic: 6.74e-10, 

degree: 3.00e+02,sinc: 7.10e-12, cheby: 4.23e-16, cubic: 3.24e-10, 

degree: 4.00e+02,sinc: 9.84e-16, cheby: 3.96e-16, cubic: 1.02e-10, 

degree: 5.00e+02,sinc: 1.03e-15, cheby: 5.67e-16, cubic: 4.17e-11, 

Herein, the cubic and Chebyshev are better than Sinc for interpolating $\sin$ if degree<300.

We conducted experiments on lpmv, ellipj, sph-harm from KAN, the results are shown in the new revision (Table 1 and 5).

What exactly is f(jh) in equation (6)? Is it simply the value of f at jh? Is equation (6) interpolating f(x) using those values? If so, do we need to know those values precisely to approximate f(x) with this formulation?

In equation (6), f(jh) is exactly the value of f at the point x=jh. And yes, equation (6) interprets f(x) using those points. But for approximation problems, it is not necessary to know those values although those points are the theoretical choice if the loss function is exactly equation (6). Let's consider a simple approximation example to further explain it:

$

\min_{\{c_{-N},\cdots,c_N\}} \sum_{i=1}^M \left(f(x_i) - \sum_{j=-N}^N c_j S(j, h)(x_i)\right)^2

$

where $\{x_1,x_2,\cdots,x_N\}$ are given points. Obviously, if $M=2N+1$, and $x_i=(i-1-N)h$, the problem has an optimal solution $c_j=f(jh)$ because $S(j, h)(ih)=1 \text{ if } i = j, \text{ otherwise}, S(j, h)(ih)=0.$ But for other cases, including our case, that is $M\neq 2N+1$ and $x_i$ are randomly sampled, we can also solve this problem by, for example, the least squares method. Herein, we don't need to know those values $f(jh)$ precisely to approximate f(x) with this formulation.

Could other problems with singularities be tested? Does the proposed method also show significant advantages and accurately approximate solutions in all cases where singular phenomena appear? Could it be applied to other boundary layer problems or higher-dimensional problems beyond two dimensions?

We tested our methods on other problems including fractional PDEs and high-dimensional Poisson problems.

For fractional PDEs, we utilize fPINN on the spatial fractional order problem:

$

-\Delta^su+\gamma u=f, \quad \boldsymbol{x}\in (-1,1)^d

$

if $f=1+\gamma u$, the exact solution is

$

u(x)=\frac{2^{-2s}\Gamma(d/2)}{\Gamma(d/2+s)\Gamma(1+s)}(1-\|x\|^2)^s,

$

where $s\in \mathbb{R}_+$ is a hyperparameter. In our experiments, $s=0.95,d=1$.

The results are:

For high-dimensional Poisson equations. we consider

$

\Delta u =f, \quad \boldsymbol{x}\in [0,1]^d.

$

where $f=2\alpha(2\alpha ||\boldsymbol{x}||^2_2-d) e^{-\alpha ||\boldsymbol{x}||_2^2}$, the exact solution is $u(\boldsymbol{x})=e^{-\alpha ||\boldsymbol{x}||^2_2}$. In our experiments, $d=100, \alpha=10/d$.

The results are:

For line 140, it might be better to use the \paragraph function for terms like “Convergence theorem” or “On a general interval (a, b)” to save space.

Thanks for your valuable suggestions! We are considering how to add additional experiments to the main paper.

If sincKAN does not consistently outperform across all datasets, as seen in Table 2, what advantages does sincKAN offer over using standard MLP or modified MLP to train Physics-informed Loss? While sincKAN appears to perform better on boundary layer problems, as shown in Table 3, the paper’s title and focus could have been more aligned with boundary layer issues if that is its primary strength.

In our opinion, it is not necessary to be the best model for all datasets. Our work is to explore the implementations of KANs in PINNs and propose a powerful network that is helpful in PINNs. As Sinc interpolation is not as good as Chebyshev and splines when dealing with smooth functions, it is acceptable when we solving smooth PDEs, but SincKAN is not the best. But as the results are good enough, SincKAN excels in handling singular problems. We still believe that SincKAN is a competitive network. We agree that it is a good suggestion that our title should align with our results.

Are there recent studies related to PINNs specifically aimed at solving boundary layer problems? What would happen if epsilon increased to a larger value, such as $10^8$?

There are several methods proposed to solve boundary layer problems.

1. DAS-PINNs: they use sampling methods to make more points concentrate on the singular part;
2. C-PINN: they embed Chien’s composite expansion method into the loss function;
3. BL-PINN: they decompose the boundary-layer problems into two parts and train them individually.

We proposed a novel network that can approximate the boundary layer problems better than MLP. Thus, except for BL-PINN which uses two MLP networks to learn two non-singular parts, we believe SincKAN can be combined with C-PINN and DAS-PINNs as they don't change the network and SincKAN has the better capability of representing singular functions.

We tested SincKAN on $\epsilon=10^8$. Unfortunately, the performance is poor, $10^8$ is a pretty large number for neural networks.

It might also help to include a basic explanation of KAN’s underlying mechanism for readers unfamiliar with the model, as certain concepts could be difficult to grasp without this context.

Thanks for your valuable suggestion, we put the explanation of KANs in Appendix, which probably is not clear enough. We will add more explanations on the rebuttal version and mention it in the main paper.

Thank you for your detailed responses to my review comments. While I appreciate the effort you have put into addressing the concerns raised, I find that several critical issues remain unresolved, and your responses have not been reflected in the revision of the paper. This raises questions about why the proposed changes have only been addressed in the rebuttal but not incorporated into the manuscript itself. I provide further clarification of my concerns below

• SincKAN’s Key Strengths and Limitations

You argue that SincKAN does not need to be the best model across all datasets, focusing instead on its strength in solving boundary layer problems. However, this does not sufficiently address why results on datasets where SincKAN does not perform well were included in the first place. If boundary layer problems are the primary strength of SincKAN, the experiments and title should focus more explicitly on this. While you have stated that the title will be revised, no changes have been made to the manuscript, which undermines the credibility of this commitment. Additionally, why were the results included if they do not directly support the central claims of your paper?

• Boundary Layer Problems and ε Scaling

The lack of robust performance for larger values of ε (e.g., ) is acknowledged in your response, but no viable alternatives or in-depth analysis of this limitation were provided. Furthermore, your response did not address whether other methods, such as those presented in Finite Element Operator Network for Solving Parametric PDEs (which handled ) or Component Fourier Neural Operator for Singularly Perturbed Differential Equations (focused on singular problems), have achieved better performance in similar cases. If boundary layer problems are the central focus of your paper, as implied, comparisons with these studies would strengthen your argument. Including such comparisons and emphasizing singular boundary problem results in the experiments section should be prioritized in a revision.

• Additional Experiments and Explanation

While I appreciate the additional results provided for fractional PDEs and high-dimensional Poisson equations, visualizations (e.g., solution plots) would help to better convey the advantages of SincKAN. This could clarify the contexts where SincKAN excels and where it does not. Could you incorporate these visualizations into the main paper? Additionally, the explanation of KAN’s underlying mechanism is still insufficient in the main text, despite earlier feedback highlighting this as a critical issue. Readers unfamiliar with the model may struggle to understand your contributions without a clear and concise explanation included in the manuscript itself.

Therefore, while your responses demonstrate some acknowledgment of the points raised, they have not been adequately incorporated into the paper. The lack of revisions addressing these concerns, particularly those related to experimental focus, visualization, and model explanation, prevents me from reevaluating the contribution of the paper. As such, now, I will stay my initial score.

Thanks for your further discussion. We planned to submit the final manuscript at the end of this discussion, as our new experiments haven't obtained the error bar. Now we submit the version as you required.

The explanation of KAN’s underlying mechanism is still insufficient in the main text

Due to the limitation of the main paper, we think we can only mention it and guide readers to read it in our Appendix. But it is pretty thoughtful and concise for a reader who doesn't know KANs (See G.3).

Why were the results included if they do not directly support the central claims of your paper?

In Table 1, the approximation experiments including the singular case and smooth cases show that SincKAN has the ability to be the best network. However, due to the limitations discussed in the Conclusion section, SincKAN can't be applied in PINNs for every case. Thus, we want to determine the extent to which those restrictions impair SincKAN's performance in complex but non-single cases. That is the reason we have results in nonlinear functions, Burgers' equations, and Navier-Stokes equations: we want to show that although SincKAN has those limitations, SincKAN is still good enough for general cases. Thus, we agree with changing the title since approximation problems are not very practical and we pay more attention on the PINNs.

Boundary Layer Problems

1. The FEONet is a hybrid framework that combines neural networks and traditional methods. Such a framework is always better than pure PINNs because it inherits the accuracy of traditional methods. But there are several drawbacks including complexity and grid generation. The most significant drawback is that it can't solve high-dimensional problems which PINNs are famous for, because the basis functions of finite element methods have to decompose the whole domain so that the complexity increases exponentially with the dimension. Herein FEONet also inherits the drawback of suffering the curse of dimensionality. So we don't need to compare with this paper.

2. The ComFEO solving boundary layer problems by Neural operators which are usually supervised learning while PINNs are unsupervised learning. This paper generates solutions from traditional numerical methods (see the second paragraph of Experiments in ComFEO). So we don't need to compare with this paper.

For the papers we mentioned above, the largest $\epsilon$ is $10^4$, and the experiments we did on it show that SincKAN can obtain $5.27e-03$\pm$1.29e-03$ which is also good enough, we added this experiments on our paper. Besides, we think the performance can be better if SincKAN is equipped with these mentioned approaches.

Could you incorporate these visualizations into the main paper?

Yes, we plot the errors and training loss of fractional PDE into the main paper (Figure 4)and plot the solutions in the Appendix (Figure 7).

And for high-dimensional Poisson equations, as it is not practical to plot the solution, so we plot the relative error of every point. Our testing points are composed of 80% interior points and 20% boundary points. To show our points are in a reasonable distribution, we also plot the location in a 2D profile (Figure 5).

Dear reviewer,

We are sorry about the lack of revision, that make you stay the score. We believe that we have already answered all questions and enriched the manuscript by tables, figures and explanation. Is there anything else that we haven't answered or haven't answered fully? We are glad to have further discussion with you or get a positive feedback from you.

The PDEs selected for the experiments are generally unsuitable to demonstrate the benefits of PIKANs. The chosen problems are extremely small and these problems can be solved using any numerical PDE solver. Since PINNs / PIKANs don't provide any practical advantage in small problems, it would be interesting to see if the benefits proposed here can be demonstrated in real problems.

Dear reviewer, this is a quick comment on the chosen experiments.

Our motivation of proposing the SincKAN is to solve the boundary layer problems and singularity problems which have difficulties in PINNs. Those problems represent a kind of problems called perturbed problems that have wild applications e.g. chemical reactors: Turian et al. (1973), and fluid dynamics: Gordon et al. (2000). Additionally, although numerical PDE solvers can solve perturbed problems, those approaches often require expert knowledge and specific tricks.

To further show the advantages of SincKAN, we are going to add more numerical examples, including the high-dimensional PDEs, which the traditional numerical methods have the curse of dimensionality, and the fractional PDEs which the traditional numerical methods still face challenges due to their non-locality and singularity. We believe those problems can show the benefits of PIKANs and convince you to recognize our contributions.

For MLP architectures, how were the architectures chosen? Since the problems are small and toy-ish, over-parameterization may cause a higher final loss.

Thanks for the reminder, our hyperparameters are from the NSFNet, the most powerful model for solving the high-resolution dataset Johns Hopkins Turbulence Database. After your question, we think we should test some small MLP networks. To show the performance of small sizes of MLP, we run experiments on the 1D and 2D boundary layer problem with different sizes.

For 1D boundary layer problems

For 2D boundary layer problems

From the above results of 1D boundary layer problems, we found that the competitive size is $5\times 50$ and $5\times 100$,, herein, in 2D case, we only tested those two size, the results are

However, only in 1D case, the result is better than our SincKAN, otherwise, the results are still worse than SincKAN.

it would be interesting to see if the benefits proposed here can be demonstrated in real problems.

We consider two different problems: the fractional PDEs, and high-dimensional Poisson.

For fractional PDEs, we utilize fPINN on the spatial fractional order problem:

$

-\Delta^su+\gamma u=f, \quad \boldsymbol{x}\in (-1,1)^d

$

if $f=1+\gamma u$, the exact solution is

$

u(x)=\frac{2^{-2s}\Gamma(d/2)}{\Gamma(d/2+s)\Gamma(1+s)}(1-\|x\|^2)^s,

$

where $s\in \mathbb{R}_+$ is a hyperparameter. In our experiments, $s=0.95,d=1$.

The results are:

For high-dimensional Poisson equations. we consider

$

\Delta u =f, \quad \boldsymbol{x}\in [0,1]^d.

$

where $f=2\alpha(2\alpha ||\boldsymbol{x}||^2_2-d) e^{-\alpha ||\boldsymbol{x}||_2^2}$, the exact solution is $u(\boldsymbol{x})=e^{-\alpha ||\boldsymbol{x}||^2_2}$. In our experiments, $d=100, \alpha=10/d$.

The results are:

Finally, we want to clarify that our work is to explore the implementations of KANs in PINNs and propose a powerful network that is helpful in PINNs. We are interested in solving numerical PDEs rather than discovering physical laws (i.e. the interpretability). Herein we are focusing on whether KAN is more accurate than MLP as KAN claimed that can help PINNs solve PDEs, or which kind of version of KAN may be helpful to us.

Dear reviewer wJoX,

Reviewer mr4i asked about the experiments for larger $\epsilon$, so we conducted the experiments on $\epsilon=10^4$ for boundary layer problems and added them in Table 3. For your concerned questions, we also did experiments on small size MLP, the results are:

Dear reviewer,

Is there anything else that we haven't answered or haven't answered fully? We believe that we have answered the questions and weaknesses from you fully and comprehensively. And we argue that the rating misevaluates our manuscript so we kindly ask you to update your rating or have a further conservation with us.

Using more advanced basis functions for KANs is a meaningful research direction but currently it seems to me no single method can be seen as the best solution for all PDEs. In fact sincKAN may have better performance in the presence of singularities but may not be the best option otherwise. In the current version, I can't derive a concrete conclusion as which version of KANs to use for what type of problems (within PDE domain or even beyond).

In our opinion, there is no best choice for all PDEs. In traditional methods, there are finite difference methods, finite element methods, finite volume methods, spectral methods, etc. Every method has its advantages and disadvantages, for example, the spectral methods are faster and have exponential convergence but require specific conditions, and finite element methods are adaptive but can't hold the conservation law. Even for the basis functions in KANs, splines are good at smooth functions but sensitive to the interval, Chebyshev is faster and more accurate but requires specific interpolation points, Sinc is good at singular functions but can't achieve the same accuracy as Chebyshev on smooth functions.

Even for PINNs which was proposed in 2019, it is not easy to conclude which version of PINNs to use for which type of PDEs. But we believe that SincKAN is a good choice, especially for singular problems.

Finally, in "vanilla" KANs paper there are some arguments on the advantage of using B-splines, which enable them to use grid-related tricks including grid extension and grid update tricks. In other variants also there are some interesting merits; it's hard to say SincKAN still has all these merits and in addition to them provides more robustness in the case of singularities.

They use grid extension because cubic interpolation is a local interpolation method that requires the interval, but our Sinc, Chebyshev, and wavelet are global interpolation methods thus it is not necessary to update the grids during training. We agree with the statement of Wav-KAN that those tricks are limitations instead of merits of vanilla KAN. On the other hand, SincKAN can implement the grid extension more smoothly if we set h to be invariant: recall the interpolation: we can keep the old coefficients and train for the new coefficients as the optimal value is f(jh), which is independent with $N$. Even if $h$ is changed, as $h$ is inversely proportional to $N$, $h$ will change slightly with $N$ increases, which means the loss won't have an evident jump. To demonstrate it intuitively, consider given $M,N$ with $M>N$ and $c_j, j=-N,..,N$ is already fully trained, suppose the interpolation before extension is

$

\sum_{j=-N}^{N}=c_jS(j,h)

$

After extension,

$

\sum_{j=-M}^{M}=c^\prime_jS(j,h)

$

we can set $c^\prime_j=c_j \text{ for } j=-N,\cdots,N$, the rest $c_j$ is initialized by zero so that the error after extension is equal to the error before extension.

Can sincKAN help with some pathological issues of PINNs? Would be interesting to explore; see for example Wang, S., Teng, Y., and Perdikaris, P. Understanding and Mitigating Gradient Flow Pathologies in Physics- Informed Neural Networks. SIAM Journal on Scientific Computing, 43(5):A3055–A3081, 2021a. or Wang, S., Sankaran, S., and Perdikaris, P. Respecting causal- ity is all you need for training physics-informed neural networks. arXiv preprint arXiv:2203.07404, 2022a. or the more recent paper Challenges in Training PINNs: A Loss Landscape Perspective by Rathore et al.

Thanks for your question about the exploration of SincKAN. As SincKAN is an alternative network of MLP, approaches to changing other compositions can be implemented in SincKAN. Let's consider the first two papers.

For your first paper, Wang, S (2021), proposed a novel network that has already been used in our experiments(Modified MLP) that can't be combined with SincKAN totally, and an algorithm called learning rate annealing algorithm. The algorithm can improve the performance to 1.99e-04 in perturbed compared with the result 1.88e-03 in Table 2.

In the second paper Wang, S (2022), proposed a novel formula for the loss function, we implement this formula with SincKAN in our NS problem. The novel loss can improve the performance a little, ns-tg-u: 7.07e-04, ns-tg-v: 4.58e-04 compared with the results ns-tg-u: 6.51e-04, ns-tg-v: 1.34e-03 in Table 2.

As for the pathological issues, the boundary layer problems we used in our experiments actually belong to this category because the high-frequency part causes the spectral bias, see Wang, S When and why PINNs fail to train: A neural tangent kernel perspective. Herein, thanks to the capability of handling singularity, SincKAN can help solve pathological issues by itself or by combining with other powerful technologies.

I still don't see the real advantages of theorems 1 and 2, since the optimal value of step is not uses in sincKAN.

Theorem 1 and 2 are for Sinc numerical methods, especially the Sinc interpolations. Those two theorems inspired us to use Sinc interpolation in KANs. However, because applying the Sinc interpolation is not a good idea as we discuss in our manuscript, we unfold this approach to meet the demands of machine learning. Thus, this theorem is our fundamental theorem for replacing the cubic interpolation in KANs. Furthermore, interpolation is one of the most important compositions in KANs but not all of KANs. Thus, Theorem 1 and 2 can't dominate SincKAN totally.

Also, if a function needs to be Hardy, does that mean the guarantees are only valid for differentiable functions f? It would be good to discuss the implementations.

Herein, in our SincKAN, it is not only valid for differentiable functions f, for example, piece-wise and spectral-bias are discontinuous functions.

Also, I advise authors formally add another theorem to Coordinate transformation section to prove for the composition, the theory still holds (it should be straightforward).

Thanks for your valuable suggestion, we will add the additional theorem on the Appendix, here is the theorem:

Theorem 3

Suppose $f(x), x\in \mathbb{R}$, satisfies **Theorem 1 **, that is,

(1) $f$ belongs to $H^1\left(\mathcal{D}_d\right)$, where $H^1$ is the Hardy space and $\mathcal{D}_d=\{z\in \mathbb{C} \ | \ \left|\Im z\right|<d\}$;

(2) $f$ decays exponentially on the real line, that is, $|f(x)| \leq \alpha \exp (-\beta|x|), \ \forall x \in \mathbb{R}$.

Then for given $\mu \geq 0, \sigma > 0$ and $\xi=\frac{x-\mu}{\sigma}$, we want to prove that $\tilde{f}(\xi)=f(\sigma\xi+\mu), \xi \in \mathbb{R}$ satisfies condition (1) and (2).

(1) As $\mu$ and $\sigma$ are real number,

$f(x)\in H^1\left(\mathcal{D}_d\right)$

$\Leftrightarrow \lim_{\varepsilon \rightarrow 0} \int_{\partial \mathcal{D}_d(\varepsilon)}|f(z)||\mathrm{d} z|<\infty$

$\Leftrightarrow\int_{-\infty}^{\infty}|f(x \pm id)|\mathrm{d} x<\infty$

For $\tilde{f}$

$\int_{-\infty}^{\infty}|\tilde{f}(\tilde{x}\pm i\tilde{d})|\mathrm{d} \tilde{x}$

$= \int_{-\infty}^{\infty}|f(\sigma \tilde{x}+\mu \pm i\sigma \tilde{d})|\mathrm{d} x$

$= \int_{-\infty}^{\infty}|f(x^\prime \pm id^\prime)|\mathrm{d} x^\prime, \text{ where } x^\prime=\sigma \tilde{x} +\mu,d^\prime = \sigma \tilde{d}$

If $d^\prime < d$, i.e. $\tilde{d}<\frac{d}{\sigma}$, then $\int_{-\infty}^{\infty}|f(x^\prime\pm id^\prime)|\mathrm{d} x^\prime < \infty \Rightarrow \tilde{f}(\xi)\in H^1\left(\mathcal{D}_{\tilde{d}}\right)$. As $\frac{d}{\sigma}>0$, $\tilde{d}$ exists, (1) is satisfied.

(2) As $|\tilde{f}(\xi)| \leq \alpha \exp (-\beta|\sigma\xi+\mu|), \ \forall \xi \in \mathbb{R}$, if there exists $\tilde{\alpha},\tilde{\beta}>0$ such that $\alpha \exp (-\beta|\sigma\xi+\mu|) \leq \tilde{\alpha}\exp (-\tilde{\beta}|\xi|)$, then (2) is satisfied:

$\alpha \exp (-\beta|\sigma\xi+\mu|) \leq \tilde{\alpha}\exp (-\tilde{\beta}|\xi|)$

$\Rightarrow \log \alpha -\beta|\sigma\xi+\mu| \leq \log \tilde{\alpha} -\tilde{\beta}|\xi|$

$\Rightarrow \tilde{\beta} \leq \frac{\log \frac{\tilde{\alpha}}{\alpha}}{|\xi|}+\beta\left|\sigma+\frac{\mu}{\xi}\right|, \text{ if } \xi\neq 0.$

Thus, if $\tilde{\alpha}>\alpha$, there exists a $\tilde{\beta}>0$ that satisfies the above inequality; on the other hand if $\xi=0$, obviously $\alpha \exp (-\beta|\sigma\xi+\mu|) \leq \alpha < \tilde{\alpha}$. Herein, $\tilde{\alpha}, \tilde{\beta}$ exists.

In total, as $\tilde{f}$ guarantees conditions (1) and (2), $\tilde{f}$ satisfies Theorem 1.

Thanks for the comment. We understand that the grid extension algorithm doesn't totally convince you, so we conducted experiments on spectral-basis and bl to show its performance. In Figure 10 and 11, you can find that the evident jump only occurs several times if we use grid extension.

In the experiments, we found that setting a degree = 100 for the total training is always better than increasing the degree from 8 to 100. Besides, because of the cost of changing the basis, increasing the degree is slightly faster than setting degree=100. However, grid extension can be used to find the best number of degrees. We think it is valuable to discuss this technology in our paper, so we added it in the Appendix, you can find all details in Appendix J.

Look forward to further discussion.