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arxiv: 2606.28184 · v1 · pith:VSJ3PJ7Dnew · submitted 2026-06-26 · 🧮 math.NA · cs.NA· physics.comp-ph

A fast sum-of-Gaussians algorithm for the high-dimensional fractional Fokker-Planck equation

Shidong Jiang , Dong Wang , Qi Zhou This is my paper

Pith reviewed 2026-06-29 02:56 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.comp-ph
keywords fractional Fokker-Planck equationsum-of-Gaussiansfundamental solutionhigh-dimensional computationtensor-product gridcomplete monotonicityheat kernelfractional operator
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The pith

A sum-of-Gaussians approximation turns the fundamental solution of the fractional Fokker-Planck equation into a sum of separable heat kernels.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs an algorithm that approximates the Fourier symbol of the fractional operator by a sum of Gaussians whose number grows only logarithmically with the desired accuracy. Because each Gaussian term factorizes into a product of one-dimensional heat kernels, the entire fundamental solution on a tensor grid can be stored and evaluated with work and memory linear in the dimension rather than exponential. The same separated form lets any initial condition that itself admits a tensor-product representation be evolved exactly by one-dimensional convolutions alone. The approach is justified by the complete monotonicity of the stretched exponential, which supplies an exact Laplace-transform representation that is then discretized.

Core claim

The fundamental solution is represented as a separated sum of heat kernels obtained by applying a sum-of-Gaussians quadrature to the nonseparable stretched exponential that appears in the Fourier symbol; each term factorizes across coordinates, so that on a tensor-product grid the representation is assembled in O(M d N) operations where M grows only logarithmically with inverse tolerance and accuracy is preserved up to dimension 10^5.

What carries the argument

Sum-of-Gaussians quadrature applied to the Laplace-transform representation of the stretched exponential (itself the Fourier symbol of the fractional operator), justified by its complete monotonicity as the transform of a one-sided α-stable density.

If this is right

  • Any initial datum expressible as a finite sum of tensor products can be evolved in closed form using only one-dimensional convolutions.
  • Storage and arithmetic cost on a tensor grid scale linearly with dimension rather than exponentially.
  • An a-priori error bound holds for the pure-fractional case and supplies an explicit parameter-selection rule for given accuracy over chosen space-time ranges.
  • Accuracy exceeding ten digits is maintained in dimensions up to 10^5, exceeding the range reachable by radial quadrature on the same problem.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The separated representation supplies a concrete class of high-dimensional solutions whose error can be tracked analytically, offering a test bed for tensor-network or neural-network approximations to more general data.
  • Because the same complete-monotonicity argument applies to other Fourier multipliers whose symbols are completely monotone, the technique extends immediately to other fractional operators whose symbols admit Laplace representations.
  • The logarithmic growth of M with tolerance suggests the method remains practical for tolerances far below machine epsilon, provided the one-dimensional convolutions can be performed to matching precision.

Load-bearing premise

The stretched exponential arising from the fractional operator is completely monotone and therefore equals the Laplace transform of a one-sided α-stable density.

What would settle it

Compute the relative error of the sum-of-Gaussians fundamental solution against a direct Fourier inversion on a moderate-dimensional grid for a sequence of tolerances; the error should remain below the prescribed tolerance while the number of terms grows only logarithmically.

Figures

Figures reproduced from arXiv: 2606.28184 by Dong Wang, Qi Zhou, Shidong Jiang.

Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
read the original abstract

We present a fast, high-order algorithm for the free-space fractional Fokker-Planck equation (FFPE) in arbitrary spatial dimension. Its fundamental solution, corresponding to a Dirac-delta initial condition, is obtained from the explicit Fourier representation by applying a sum-of-Gaussians (SOG) approximation to the nonseparable stretched exponential, using its complete monotonicity as the Laplace transform of a one-sided $\alpha$-stable density. Each Gaussian term is an ordinary heat kernel and therefore factorizes across spatial coordinates. On a tensor-product grid, the separated form can be assembled in $O(MdN)$ work and storage, rather than forming all $O(N^d)$ grid values, where $M$ is the number of Gaussian terms and $N$ is the number of points per dimension. We prove an a~priori error estimate for the pure-fractional fundamental solution and give a parameter-selection procedure for prescribed accuracy over specified ranges of space and time. In numerical experiments the method achieves more than ten digits of relative accuracy, with $M$ growing only logarithmically in the inverse tolerance, and maintains this accuracy in dimensions up to $d=10^{5}$. This exceeds the dimensions reached in comparable radial-quadrature tests, where the integrand becomes increasingly oscillatory as the dimension grows. Because the method represents the fundamental solution as a separated sum of heat kernels, any initial datum given as a finite sum of tensor products can be evolved in closed form using only one-dimensional convolutions. This yields a computable class of high-dimensional solutions that is amenable to error analysis, and tensor neural networks provide one possible way to construct such separated representations for more general data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript presents a fast algorithm for the free-space high-dimensional fractional Fokker-Planck equation. The fundamental solution is obtained from its Fourier representation by applying a sum-of-Gaussians (SOG) approximation to the stretched exponential exp(−t|ξ|^α), justified by its complete monotonicity as the Laplace transform of a one-sided α-stable density. Each term is a separable heat kernel, enabling O(M d N) assembly and storage on tensor-product grids. The paper states an a priori error estimate for the pure-fractional case together with a parameter-selection procedure, and reports numerical results achieving more than ten digits of relative accuracy with M growing only logarithmically in the inverse tolerance, up to d = 10^5.

Significance. If the central claims hold, the work supplies a dimension-robust method for a class of fractional evolution equations whose direct discretization is prohibitive in high d. The separation into heat kernels, the a priori error control, and the logarithmic scaling of M constitute concrete strengths; the approach also yields an explicit class of high-dimensional solutions amenable to further analysis via one-dimensional convolutions.

minor comments (2)
  1. [Abstract and § on error estimate] The abstract and introduction should state the precise range of t and |x| over which the reported ten-digit accuracy and the a priori error bound are guaranteed; without this the parameter-selection procedure cannot be fully assessed.
  2. [SOG construction section] Clarify whether the quadrature rule used to discretize the Laplace-transform integral for the SOG coefficients is the same for all α ∈ (0,2] or whether its nodes/weights depend on α; this affects the claimed dimension independence.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the recognition of the dimension-robust separated representation, a priori error control, and logarithmic scaling of M. The recommendation for minor revision is noted. No specific major comments were provided in the report, so we address the overall evaluation below.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper begins from the known Fourier representation of the FFPE fundamental solution and invokes the standard fact that exp(−t |ξ|^α) is completely monotone (hence the Laplace transform of a one-sided α-stable density) to obtain an exact integral representation as a mixture of Gaussians. The subsequent finite SOG is produced by quadrature whose error is controlled independently of dimension. No equation reduces the target quantity to a fitted parameter defined by the same data, no load-bearing step rests on a self-citation, and the central construction is self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method rests on the Fourier representation of the FFPE (standard) and the complete monotonicity of the stretched exponential (domain fact from stable-process theory). The only free parameter introduced by the paper itself is the number of Gaussian terms M together with the quadrature nodes and weights chosen to meet a tolerance; these are selected by the supplied procedure rather than fitted to solution data.

free parameters (1)
  • M (number of Gaussian terms)
    Chosen via the parameter-selection procedure to achieve prescribed accuracy over given ranges of space and time; not fitted to solution values.
axioms (1)
  • domain assumption The Fourier symbol of the fractional operator yields a stretched exponential that is completely monotone and equals the Laplace transform of a one-sided α-stable density.
    Invoked in the abstract to justify replacing the multiplier by a sum of ordinary Gaussians.

pith-pipeline@v0.9.1-grok · 5848 in / 1516 out tokens · 37494 ms · 2026-06-29T02:56:00.392254+00:00 · methodology

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