Summary
This paper considers numerical methods for the Fokker-Plank equation and its "fractional Laplacian" version. These partial differential equations describe the evolution of densities of certain random processes: in the case of standard FP, the process is a regular diffusion (an SDE driven by Brownian motion), whereas the fractional version corresponds to a process driven by $\alpha$-stable noise. Accordingly, the authors focus on initial conditions given by probability densities.
The idea behind the paper is to adapt the fairly recent methodology of positive semidefinite models (PSD) to this PDE problem. PSD models estimate nonnegative functions via feature maps $\phi:X\to H$, where $X$ is the domain of the function and $H$ is a Hilbert space. Given $\phi$, the nonnegative function of interest is estimated as $\langle \phi, M\phi\rangle$, where the PSD linear operator $M$ is chosen from the data. The whole method is based on the same idea as the well-known kernel trick: there is no need to explicitly deal with the Hilbert space $H$; only the kernel matrix over data points will matter.
In a nutshell, the paper gives a method for approximately solving the two aforementioned PDEs via PSD models. The salient features of this approach are as follows.
1) The PSD estimator can be obtained via a semidefinite program (as shown in previous work).
2) The method will output probability densities, and one can sample from them via methods from previous work.
3) The complexity of the problem grows like $\approx \epsilon^{-d/\beta}$, where $\epsilon$ is the target accuracy, $d$ is the dimension and $\beta$ measures the degree of smoothness of the solution. The exponent is not quite right, but the point is that sufficient smoothness will avoid the curse of dimensionality.
One may note that this method has rigorous error guarantees. This sets it apart from other PDE solvers based on machine learning ideas, such as Physics Inspired Neural Nets (PINNs).
Strengths
The paper gives strong convergence guarantees for a new PDE solving method. Although several ingredients come from previous work, they are combined with new ideas into an impressive package. As mentioned above, this is a ML-based PDE solver with rigorous guarantees. Importantly, these guarantees do not require coercivity or hypercontractivity properties that are usually required from Langevin samples for solutions.
Weaknesses
* No experiments are performed to compare the method with other PDE solvers (though this is understandable in a heavily mathematical paper).
* The writing is a bit sloppy in places: I found quite a few typos and have several minor comments on writing (I could probably find some more typos if I kept looking). See the list below.
NB: were it not for the sloppiness, I would have given this paper a higher score.
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_List of typos and comments on writing_
(Line numbers refer to the full paper in the SM.)
Lines 146/7: "It was" and "It has" should be "They were" and "They have".
Line 180: why define $j=\sqrt{-1}$ if you could have just written $\sqrt{-1}$ in the exponent? (You never use this notation again, do you?)
Line 185: Shouldn't the condition for infinite $p$-th moment be $p\geq \alpha$?
Line 201 and elsewhere: I find it a bit odd that the transpose notation is used for vectors in Hilbert space. If you are going to use it, at least explain here what is going on.
Line 251: why is this sensence needed here?
Line 205: $\eta$ was treated as a vector before, and will be treated as a vector later in the text. Here, however, it seems to be a scalar.
Line 223: "The cost of the algorithm is..." -- per sample point, right?
Line 254: "Hence, $p^\star(x,t)$"
Line 255 should end with ":".
Line 270: the definition of $\psi$ should be in a numbered display for easier refererence.
Display (8): if $p^*$ is the true solution, why do we have it in here? (Its contribtion vanishes.) Indeed, the definition of the loss later does not involve $p^*$; see eg. (9). (On the other hand, $p^\star$ shows up again in the display right after line 989.)
Line 294 and elsewhere: when you are mentioned a numbered theorem or assumption, start the corresponding word with a capital letter.
Line 295: $\mathcal{H}_X\otimes\mathcal{H}_T$
Line 318: "by Gaussian kernel approximation" -- should this be here?
Line 365: "Similarly to the previous section, two steps are required to obtain"
Line 327 and elsewhere: I think $\mathbb{E}_t$ was not defined anywhere.
Line 352: "If the kernel satisfies the Bochner..."
Line 355 and elsewhere: this is a matter of taste, but couldn't you replace "utilize" with "use".
Line 376: "The major difficulty"
Line 391 (taste): "does require" could be "require".
Line 395: I think "approximated" should be "approximating".
Proposition 4 is also in Evans's book, which was cited previously (I am saying this from memory, so please check. $W_2$ is replaced by $H$ in that book.)
Many spots of the SM: the inequality $\|f\star g\|_p\leq \|f\|_p\|g\|_1$ should be mentioned, as it is used in several steps.
Lemma 2: it seems that $\tilde{p}$ is replaced by $f$ in the statement of the Lemma and at several steps of the proof. (Eg. the display right after 665.) This also seems to take place in the next Lemma.
Line 680: Since you are repeating a definition from the preceding proof, please say so.
Line 682: "We use the result of Lemma 1"
Line 746: Should "The following result holds" be here?
Display (34): this "fill distance" is the smallest value of $h$ such that $\tilde{X}$ is an $h$-net of $[-R,R]^d$.
Display below line 777: how should I interpret the norm in the RHS?
Lines 873/4: why are you spaeking of conditional probabilities here?
Line 891: what is $f(\delta)$?
Line 910: "redefine" should be "define again" (they are not the same).
Line 928: capitalize "Holder".
Questions
* Have you attempted experiments against other ML approaches, or other methods from the literature?
* Methods have been proposed recently to speed up SDPs, including randomized sketches (https://math.paperswithcode.com/paper/scalable-semidefinite-programming). Could these methods be applicable in the present setting?
Rating
7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
Confidence
4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
Limitations
They have not addressed limitations explicitly, but I do not think that that would have been necessary.