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arxiv: 1906.09524 · v2 · pith:7B6L2V4Gnew · submitted 2019-06-23 · 💻 cs.NE · cs.LG· eess.SP

Fractional-order Backpropagation Neural Networks: Modified Fractional-order Steepest Descent Method for Family of Backpropagation Neural Networks

Yi-Fei PU , Jian Wang This is my paper

Pith reviewed 2026-05-25 18:11 UTC · model grok-4.3

classification 💻 cs.NE cs.LGeess.SP
keywords fractional calculusbackpropagation neural networkssteepest descent methodglobal optimizationfunction approximationfractional-order convergence
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The pith

Modified fractional-order steepest descent trains backpropagation networks for superior global optimization.

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

The paper proposes a modified fractional-order steepest descent method to train backpropagation neural networks and generalize them into fractional-order versions. It seeks to use long-term memory and nonlocality properties of fractional calculus to improve optimization over standard first-order approaches. A sympathetic reader would care because this could yield more reliable convergence to global solutions in training without altering network architecture. The work provides mathematical proofs of convergence plus experiments on function approximation and real data to support the advantage.

Core claim

A modified fractional-order steepest descent method based fractional-order backpropagation neural network has fractional-order global optimal convergence and fractional-order multi-scale global optimization, giving it a more efficient optimal searching capability to determine the global optimal solution than a classic first-order backpropagation neural network.

What carries the argument

Modified fractional-order steepest descent method that performs reverse incremental search in the negative directions of approximate fractional-order partial derivatives of the square error.

If this is right

  • The fractional-order network shows improved performance in example function approximation tasks.
  • It delivers fractional-order multi-scale global optimization in comparative tests.
  • Real data experiments demonstrate advantages over standard backpropagation neural networks.
  • The method generalizes classic first-order backpropagation using fractional calculus.

Where Pith is reading between the lines

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

  • The same fractional-order update rule could apply to other gradient descent variants in machine learning.
  • It might handle non-convex error surfaces more effectively due to the memory property.
  • Similar modifications could appear in control or signal processing applications that already use fractional operators.

Load-bearing premise

The network structure enables fractional-order global optimal convergence and fractional-order multi-scale global optimization analysis.

What would settle it

An experiment on a multimodal test function or real dataset where the classic first-order backpropagation neural network reaches a better or equal global optimum than the modified fractional-order version.

Figures

Figures reproduced from arXiv: 1906.09524 by Jian Wang, Yi-Fei PU.

Figure 1
Figure 1. Figure 1: displays the model of a BPNN, which is represented by abbreviated symbols denoting its three layers [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example of function approximation of a FBPNN trained by an improved FSDM. In [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Function to be approximated. The improved FSDM based FBPNN described in [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Mean squared error of a FBPNN trained by an improved FS [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Adaptive kernel function of fractional-order v k . From [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of iterative search processes of a FBPNN trained by an improved FSDM and a first-order BPNN: (a) Convergence trajectories; (b) Convergence patterns of squared error of th k iteration; (c) Fractional-order v of FBPNN. In [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of iterative search process of a FBPNN trained by an improved FSDM and a first-order BPNN: (a) Convergence trajectories; (b) Local magnification for convergence trajectories; (c) Convergence patterns of squared error of th k iteration; (d) Fractional-order v of FBPNN. In Figs. 7(a) and (b), the convergence trajectory of a first-order BPNN illustrates the manner in which it can converge to a loca… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of iterative search process of a FBPNN trained by an improved FSDM and a first-order BPNN: (a) Convergence trajectories; (b) Convergence patterns of squared error of th k iteration; (c) Fractional-order v of FBPNN. In [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of iterative search process of a FBPNN trained by an improved FSDM and a first-order BPNN: (a) Convergence trajectory of BPNN; (b) Local magnification for (a); (c) Convergence trajectory of FBPNN; (d) Local magnification for (c); (e) Convergence patterns of squared error of th k iteration of BPNN; (f) Convergence patterns of squared error of th k iteration of FBPNN; (g) Fractional-order v of FBP… view at source ↗
Figure 10
Figure 10. Figure 10: Mean squared error of a FBPNN trained by an improved F [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of iterative search process of a FBPNN trained by an improved FSDM and a first-order BPNN: (a) Convergence trajectories ( 108 1 w1,1  and 116 1 w11,1  ); (b) Convergence patterns of squared error of th k iteration ( 108 1 w1,1  and 116 1 w11,1  ); (c) Fractional-order v of FBPNN ( 108 1 w1,1  and 116 1 w11,1  ); (d) Convergence trajectories ( 110 1 w1,1   and 106 1 w11,1   ); (e) Conv… view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of iterative search process of a FBPNN trained by an improved FSDM and a first-order BPNN: (a) Convergence trajectory of BPNN; (b) Local magnification for (a); (c) Convergence trajectory of FBPNN; (d) Convergence patterns of squared error of th k iteration [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Responses of a FBPNN trained by an improved FSDM and a first-order BPNN for the convergence parameters: (a) Convergence parameters of [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
read the original abstract

This paper offers a novel mathematical approach, the modified Fractional-order Steepest Descent Method (FSDM) for training BackPropagation Neural Networks (BPNNs); this differs from the majority of the previous approaches and as such. A promising mathematical method, fractional calculus, has the potential to assume a prominent role in the applications of neural networks and cybernetics because of its inherent strengths such as long-term memory, nonlocality, and weak singularity. Therefore, to improve the optimization performance of classic first-order BPNNs, in this paper we study whether it could be possible to modified FSDM and generalize classic first-order BPNNs to modified FSDM based Fractional-order Backpropagation Neural Networks (FBPNNs). Motivated by this inspiration, this paper proposes a state-of-the-art application of fractional calculus to implement a modified FSDM based FBPNN whose reverse incremental search is in the negative directions of the approximate fractional-order partial derivatives of the square error. At first, the theoretical concept of a modified FSDM based FBPNN is described mathematically. Then, the mathematical proof of the fractional-order global optimal convergence, an assumption of the structure, and the fractional-order multi-scale global optimization of a modified FSDM based FBPNN are analysed in detail. Finally, we perform comparative experiments and compare a modified FSDM based FBPNN with a classic first-order BPNN, i.e., an example function approximation, fractional-order multi-scale global optimization, and two comparative performances with real data. The more efficient optimal searching capability of the fractional-order multi-scale global optimization of a modified FSDM based FBPNN to determine the global optimal solution is the major advantage being superior to a classic first-order BPNN.

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

2 major / 0 minor

Summary. The manuscript proposes a modified fractional-order steepest descent method (FSDM) for training backpropagation neural networks, generalizing them to fractional-order BPNNs (FBPNNs). It asserts mathematical proofs of fractional-order global optimal convergence and multi-scale global optimization under an 'assumption of the structure', and reports comparative experiments claiming superior global search performance over classic first-order BPNNs on function approximation and real data tasks.

Significance. If the proofs hold and the structural assumption is independently justified, the approach could advance the use of fractional calculus for neural network optimization by providing non-local memory effects that improve global convergence properties.

major comments (2)
  1. [Abstract] Abstract: The central superiority claim rests on 'the mathematical proof of the fractional-order global optimal convergence, an assumption of the structure, and the fractional-order multi-scale global optimization', yet no derivation steps, error bounds, or justification for the assumption of the structure are supplied. This assumption is load-bearing for the argument that the modified FSDM enables better global optimality than first-order BPNNs.
  2. [Theoretical analysis section] Theoretical analysis section: The reverse incremental search is described as following 'negative directions of the approximate fractional-order partial derivatives', but without explicit update rules, convergence analysis, or demonstration that the fractional-order modification does not reduce to a reparameterized first-order method under the stated assumption, the proof cannot be evaluated for circularity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and the recommendation of major revision. The comments correctly identify areas where additional clarity on the proofs and assumption would strengthen the manuscript. We address each point below and will make the requested expansions in the revised version.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central superiority claim rests on 'the mathematical proof of the fractional-order global optimal convergence, an assumption of the structure, and the fractional-order multi-scale global optimization', yet no derivation steps, error bounds, or justification for the assumption of the structure are supplied. This assumption is load-bearing for the argument that the modified FSDM enables better global optimality than first-order BPNNs.

    Authors: The abstract is a high-level summary; the full derivation steps, error bounds, and justification of the structural assumption appear in the Theoretical Analysis section. We agree that the abstract does not sufficiently signpost these elements or the assumption's role. In revision we will expand the abstract to reference the key proof components and add an explicit justification subsection for the assumption together with error-bound discussion, making its load-bearing status transparent. revision: yes

  2. Referee: [Theoretical analysis section] Theoretical analysis section: The reverse incremental search is described as following 'negative directions of the approximate fractional-order partial derivatives', but without explicit update rules, convergence analysis, or demonstration that the fractional-order modification does not reduce to a reparameterized first-order method under the stated assumption, the proof cannot be evaluated for circularity.

    Authors: Explicit update rules using the fractional-order partial derivatives are stated in the section, and convergence is shown via the subsequent theorems under the structural assumption. We acknowledge that a direct demonstration that the method does not collapse to a reparameterized first-order scheme is missing and could raise circularity concerns. In the revision we will insert the complete update-rule equations, expand the convergence proof, and add a subsection with a counter-example illustrating the non-local memory effect that produces distinct trajectories from standard first-order BPNNs. revision: yes

Circularity Check

1 steps flagged

Convergence and multi-scale optimization claims reduce to posited 'assumption of the structure'

specific steps
  1. self definitional [Abstract]
    "Then, the mathematical proof of the fractional-order global optimal convergence, an assumption of the structure, and the fractional-order multi-scale global optimization of a modified FSDM based FBPNN are analysed in detail."

    The proof of convergence and multi-scale optimization is explicitly bundled with 'an assumption of the structure' that enables those properties. This makes the claimed global optimality advantage equivalent to the assumption by construction rather than an independent derivation from first principles or external benchmarks.

full rationale

The abstract states that the paper provides 'the mathematical proof of the fractional-order global optimal convergence, an assumption of the structure, and the fractional-order multi-scale global optimization'. This phrasing indicates the central superiority claim (more efficient global search over first-order BPNN) is analyzed under an assumption of structure that is not shown to be independently derived or verified; the claimed fractional-order advantages are therefore conditional on that modeling choice by construction. No equations or self-citations are quoted that would allow further reduction, but the load-bearing role of the assumption matches the self-definitional pattern at the level of the derivation chain presented.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the unstated structure assumption for convergence proofs and on the fractional order as a tunable parameter whose selection is not detailed.

free parameters (1)
  • fractional order
    The non-integer order of the derivative in the modified steepest descent update is a parameter whose value must be chosen or tuned for each application.
axioms (1)
  • domain assumption Fractional calculus properties (long-term memory, nonlocality, weak singularity) can be directly transferred to improve the optimization dynamics of first-order gradient descent in neural networks.
    Invoked in the abstract to motivate the entire generalization from BPNNs to FBPNNs.

pith-pipeline@v0.9.0 · 5856 in / 1167 out tokens · 45387 ms · 2026-05-25T18:11:37.677226+00:00 · methodology

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