Update Quasi-Newton Algorithm for Training ANN

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A limited memory BFGS algorithm is presented for solving large-scale symmetric nonlinear equations, where a line search technique without derivative information is used. On each iteration, the updated approximations of Hessian matrix satisfy the quasi-Newton form, which traditionally served as the basis for quasi-Newton methods. On the basis of the quadratic model used in this article, we add a new update of quasi-Newton form. One innovative features of this form's is its ability to estimate the energy function's or performance function with high order precision with second-order curvature while employ the given function value data and gradient. The global convergence of the proposed algorithm is established under some suitable conditions. Under some hypothesis the approach is established to be globally convergent. The updated approaches can be numerical and more efficient than the existing comparable traditional methods, as illustrated by numerical trials. Numerical results show that the given method is competitive to those of the normal BFGS methods. We show that solving a partial differential equation can be formulated as a multi-objective optimization problem, and use this formulation to propose several modifications to existing methods. Also the proposed algorithm is used to approximate the optimal scaling parameter, which can be used to eliminate the need to optimize this parameter. Our proposed update is tested on a variety of partial differential equations and compared to existing methods. These partial differential equations include the fourth order three dimensions nonlinear equation, which we solve in up to four dimensions, the convection-diffusion equation, all of which show that our proposed update lead to enhanced accuracy." } { "@context": "http://schema.org", "@type": "BreadcrumbList", "itemListElement": [ { "@type": "ListItem", "position": "1", "item": { "@id": "https://f1000research.com/", "name": "Home" } }, { "@type": "ListItem", "position": "2", "item": { "@id": "https://f1000research.com/browse/articles", "name": "Browse" } }, { "@type": "ListItem", "position": "3", "item": { "@id": "https://f1000research.com/articles/15-71", "name": "Update Quasi-Newton Algorithm for Training ANN" } } ] } Home Browse Update Quasi-Newton Algorithm for Training ANN ALL Metrics - Views Downloads Get PDF Get XML Cite How to cite this article Ghazi F, Tawfiq L and Kareem Z. Update Quasi-Newton Algorithm for Training ANN [version 1; peer review: 1 approved with reservations] . F1000Research 2026, 15 :71 ( https://doi.org/10.12688/f1000research.172826.1 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. Close Copy Citation Details Export Export Citation Sciwheel EndNote Ref. Manager Bibtex ProCite Sente EXPORT Select a format first Track Share ▬ ✚ Research Article Update Quasi-Newton Algorithm for Training ANN [version 1; peer review: 1 approved with reservations] Farah Ghazi 1 , Luma Tawfiq https://orcid.org/0000-0001-5778-4983 1 , Zainab Kareem 2 Farah Ghazi 1 , Luma Tawfiq https://orcid.org/0000-0001-5778-4983 1 , Zainab Kareem 2 PUBLISHED 16 Jan 2026 Author details Author details 1 Mathematics, University of Baghdad, Baghdad, Iraq 2 Ministry of Education, Directorate General of Education, KARKH II, Baghdad, Iraq Farah Ghazi Roles: Data Curation, Resources, Software, Validation Luma Tawfiq Roles: Conceptualization, Formal Analysis, Funding Acquisition, Investigation, Methodology, Supervision, Writing – Review & Editing Zainab Kareem Roles: Methodology, Project Administration, Visualization, Writing – Original Draft Preparation OPEN PEER REVIEW DETAILS REVIEWER STATUS This article is included in the Fallujah Multidisciplinary Science and Innovation gateway. Abstract The proposed design of neural network in this article is based on new accurate approach for training by unconstrained optimization, especially update quasi-Newton methods are perhaps the most popular general-purpose algorithms. A limited memory BFGS algorithm is presented for solving large-scale symmetric nonlinear equations, where a line search technique without derivative information is used. On each iteration, the updated approximations of Hessian matrix satisfy the quasi-Newton form, which traditionally served as the basis for quasi-Newton methods. On the basis of the quadratic model used in this article, we add a new update of quasi-Newton form. One innovative features of this form's is its ability to estimate the energy function's or performance function with high order precision with second-order curvature while employ the given function value data and gradient. The global convergence of the proposed algorithm is established under some suitable conditions. Under some hypothesis the approach is established to be globally convergent. The updated approaches can be numerical and more efficient than the existing comparable traditional methods, as illustrated by numerical trials. Numerical results show that the given method is competitive to those of the normal BFGS methods. We show that solving a partial differential equation can be formulated as a multi-objective optimization problem, and use this formulation to propose several modifications to existing methods. Also the proposed algorithm is used to approximate the optimal scaling parameter, which can be used to eliminate the need to optimize this parameter. Our proposed update is tested on a variety of partial differential equations and compared to existing methods. These partial differential equations include the fourth order three dimensions nonlinear equation, which we solve in up to four dimensions, the convection-diffusion equation, all of which show that our proposed update lead to enhanced accuracy. READ ALL READ LESS Keywords Robust quasi-Newton methods, Convergence analysis, Numerical experiments, ANNs. unconstrained optimization. Corresponding Author(s) Luma Tawfiq ( [email protected] ) Close Corresponding author: Luma Tawfiq Competing interests: No competing interests were disclosed. Grant information: The author(s) declared that no grants were involved in supporting this work. Copyright: © 2026 Ghazi F et al . This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The author(s) is/are employees of the US Government and therefore domestic copyright protection in USA does not apply to this work. The work may be protected under the copyright laws of other jurisdictions when used in those jurisdictions. How to cite: Ghazi F, Tawfiq L and Kareem Z. Update Quasi-Newton Algorithm for Training ANN [version 1; peer review: 1 approved with reservations] . F1000Research 2026, 15 :71 ( https://doi.org/10.12688/f1000research.172826.1 ) First published: 16 Jan 2026, 15 :71 ( https://doi.org/10.12688/f1000research.172826.1 ) Latest published: 16 Jan 2026, 15 :71 ( https://doi.org/10.12688/f1000research.172826.1 ) 1. Introduction In recent years, some authors have used neural networks (ANNs) as an important technique to solve many real-world problems because of their specifications. Some authors have used ANNs to solve different types of differential equations, such that 1 , 3 first proposed the concept of solving differential equations using ANNs by formulating a trial solution of the differential equation. The authors tested the applicability and accuracy of their developed method not only for differential equations but also for systems of coupled differential equations. Furthermore, the authors compared their results with those obtained using other numerical methods and reported that the developed ANN was superior in terms of memory requirements and accuracy. 4 – 6 For this reason, the authors aimed to develop this technique to obtain the best results. One of these developments is the training rules, particularly the quasi-Newton method, because it is a second-order convergence. Many authors such 7 – 12 have proposed modifications for the training algorithm. Others such 13 – 20 suggest some rules for the speed of convergence. Several attempts have been made to solve different types of differential equations by using feed forward neural networks. In, 21 reported a hybrid method was reported that combines optimization techniques with neural networks to solve high-order differential equations. The quasi-Newton method is the most useful method for minimizing a smooth n variable function. (1) minimize f ( x ) , x ∈ R n where f : R n → R 1 is continuously differentiable. 22 In contrast to utilizing the real value of the Hessian or its inverse, in the proposed update, we use a symmetric positive definite estimate of the Hessian (H) or its inverse ( inv H). The following is the form: (2) x k + 1 = x k + α k d k , d k = − g k H k = − g k H k − 1 If H is not an invertible matrix, then the pseudoinverse of H. Wolfe conditions are used to determine the step length ( α k ) and search direction ( d k ), as follows: (3) f ( x k + α k d k ) ≤ f ( x k ) + δ α k g k T d k (4) d k T g ( x k + α k d k ) ≥ σ d k T g k where 0 < δ < σ < 1 was typically used. For more details, refer to. 23 The parameter α k is computed using a line - search in the following form: (5) α k = − g k T d k / d k T Q d k For more details, please refer to. 24 Its direction is computed by solving: (6) B k d k + g k = 0 For each iteration, B k is the updated Hessian estimate. The Broyden Fletcher Goldfarb-Shanno (BFGS) approach, proposed by Broyden, Fletcher, Goldfarb, and Shanno, is now one of the most effective training methods. Using the following formula, matrix B k + 1 in the BFGS technique can be updated: (7) B k + 1 BFGS = B k − B k s k s k T B k T s k T B k s k + y k y k T s k T y k Let H k be the inverse of B k . Undoubtedly, the suggested update in (8) is publicly known as (8) H k + 1 BFGS = H k − H k y k s k T + s k y k T H k s k T y k + [ 1 + y k T H k y k s k T y k ] s k s k T s k T y k See 25 , 26 for further details. For the update process, we let: (9) B k + 1 s k = y k where s k = x k + 1 − x k = α k d k and y k = g k + 1 − g k (see 27 ). The numerical experiment showed that the BFGS technique outperformed all the other training approaches. Convex minimization using the update approach has been extensively investigated; for example, see. 1 , 2 , 28 To demonstrate that the update approach using the Wolfe line search may not succeed for non-convex functions, Dai created an example with six cycling points. 29 Many improvements have been suggested, including changes in the regular BFGS technique, and a modified BFGS algorithm (MBFGS) has been devised to improve and speed the global convergence of the BFGS method. 30 , 31 They demonstrated that the approach converged worldwide for nonconvex optimization problems. To determine whether a novel quasi-Newton methodology has global convergence and outperforms the BFGS method in terms of computation, see. 32 , 33 In practice, the modified BFGS technique is typically preferred to efficiently compute matrix H (or H −1 ) using a symmetric positive definite matrix. While the standard method employs only gradient values, the modified approach uses both. Without making any convexity assumptions about the goal function, global convergence was demonstrated. 34 2. Derivation of suggested update A new additional update was derived using a quadratic model of the goal function. Consequently, the quadratic model of the objective function is given as (10) f k + 1 = f k + s k T g k + 1 2 s k T Q ( x k ) s k where Q ( x k ) is the Hessian matrix. The first derivative of the above equation can be written as: (11) ∇ f k + 1 = g k + Q ( x k ) s k Thus, the curvature information in Eq. (10) can be approximated by (12) s k T Q ( x k ) s k = 2 3 ( f k − f k + 1 ) + 2 3 s k T Q ( x k ) s k Because the updated B k + 1 is supposed to approximate the Q ( x k ) , it is reasonable to have (13) s k T B k + 1 s k = 2 3 ( f k − f k + 1 ) + 2 3 s k T Q ( x k ) s k Using (11) in (13), we obtain: (14) s k T B k + 1 s k = 2 3 s k T y k + 2 3 ( f k − f k + 1 ) The new quasi-Newton (QN-) equation is given by: (15) s k T y k ~ = 2 3 s k T y k + 2 3 ( f k − f k + 1 ) From the above equation, the different gradients can be written as (16) B k + 1 s k = y k ~ , y k ~ = 2 3 y k + 2 / 3 ( f k − f k + 1 ) s k T u k u k where u k is a vector such that s k T u k ≠ 0 . The BFGS update is modified based on the revised quasi-Newton equation. Alternatively, the vector u k choices in Equation (16) can be expressed as: (i) For u k = y k , Equation (16) becomes: y k ~ = 2 3 y k + 2 / 3 ( f k − f k + 1 ) s k T y k y k . (ii) For u k = g k , Equation (16) becomes: y k ~ = 2 3 y k + 2 / 3 ( f k − f k + 1 ) s k T g k g k . (iii) For u k = g k + 1 , Equation (16) becomes: y k ~ = 2 3 y k + 2 / 3 ( f k − f k + 1 ) s k T g k + ` 1 g k + ` 1 . From the above explanation of the results, we can write the algorithm as follows: Stage 1: Let x 0 ∈ R n , k = 0 and H 0 = I Stage 2: If ‖ g k ‖ = 0 , stop. Stage 3: Evaluate d k = − H k g k . Stage 4: Determine the optimal learning rate (step - size) by α k using Eqs. (4) & (5) . Stage 5: Let x k + 1 = x k + α k d k . Update H k + 1 by using Equations (9) and (16) if s k T y k ~ > 0 ; otherwise, leave H k + 1 = H k . Stage 6: Take k = k + 1 , and then go to Stage 2. The following theorem illustrates the theoretical benefits of the new quasi-Newton Equation (16) . To ensure that the matrix B k + 1 is positive definite, we need only prove that s k T y k ~ > 0 holds. Theorem 1. Let matrix sequence B k + 1 be generated using Equation (6) . Thus, the sequence B k + 1 is positive- definite. Proof. From the different gradient definitions, we have: (17) s k T y k ~ = 2 3 y k + 2 3 ( f k − f k + 1 ) By applying Wolfe's condition to the previous equation, we obtain: (18) s k T y k ~ ≥ 2 3 ( s k T y k − δ g k T s k ) Because s k T y k > 0 and − δ g k T s k > 0 , Eq. (18) , we obtain (19) s k T y k ~ ≥ 0 Therefore, B k + 1 is positive -definite. 3. Convergent analysis We provide a global convergence of innovative approaches under circumstances that are comparatively understated. 1. The level was set to L 0 = { x ∈ R n : f ( x ) ≤ f ( x 0 ) } be convex. 2. Because the gradient satisfies the Lipschitz continuity, there is a positive constant called L > 0 : (20) ( ∇ f ( x - ) − ∇ f ( x + ) ) ≤ L ‖ x - − x + ‖ , ∀ x - , x + ∈ L 0 . The series { x k } generated by a new algorithm is evident in S because { f k } is a decreasing series, and there is a constant f ∗ that results in (21) lim k → ∞ f k = f ∗ 3. Let Q be a matrix from the 2 nd derivatives of the f . Then, there exist constants R and r , such that: (22) r ‖ z ‖ 2 ≤ z T Qz ≤ R ‖ z ‖ 2 for all z ∈ R n , for more details see. 12 – 14 Theorem 2. If { x k } is generated using the proposed algorithm. Then we have: (23) r ‖ s k ‖ 2 ≤ s k T y k ~ ≤ R ‖ s k ‖ 2 . and (24) ‖ y k ~ ‖ ≤ ( L + R ) ‖ s k ‖ . Proof: By different gradient definitions y k ~ and combining Equations (10) with (16) , we obtain: (25) s k T y k ~ = s k T Q ( x k ) s k = 2 3 s k T y k + 2 3 ( f k − f k + 1 ) = 2 ( f k + 1 − f k ) − 2 s k T g k . Utilizing the mean value theorem and Taylor series, we obtain: (26) f k + 1 = f k + s k T g k + 1 2 s k T Q ( η k ) s k where ξ ∈ ( 0 , 1 ) and η k = x k + ξ ( x k + 1 − x k ) . As such by Eqs. (25) and (26) , as follows: (27) s k T y k ~ = 2 ( s k T g k + 1 2 s k T Q ( η k ) s k ) − 2 s k T g k = 2 s k T g k + s k T Q ( η k ) s k − 2 s k T g k = s k T Q ( η k ) s k Meeting Assumption 3, it is simple to surmise: (28) r ‖ s k ‖ 2 ≤ s k T y k ~ ≤ R ‖ s k ‖ 2 Then, we obtain different gradient definitions of y k ~ by direct calculations: (29) ‖ y k ~ ‖ = ‖ 2 3 y k + [ 2 / 3 ( f k − f k + 1 ) ] s k T u k u k ‖ ≤ 2 3 ‖ y k ‖ + | [ s k T Q ( η k ) s k − 2 / 3 ( s k T y k ) ] | ‖ δ k ‖ ‖ u k ‖ ‖ u k ‖ ≤ 4 3 ‖ y k ‖ + | [ s k T Q ( η k ) s k ] | ‖ s k ‖ ≤ 4 / 3 L ‖ s k ‖ + R ‖ s k ‖ ≤ ( 4 / 3 L + R ) ‖ s k ‖ The proof is finished. Theorem 3. If the constants a 1 > 0 and a 2 > 0 exist, then the following inequality holds: (30) s k T B k s 2 ≥ a 2 ‖ s k ‖ 2 , and ‖ B k s k ‖ ≤ a 1 ‖ s k ‖ for any k . The sequence { x k } is obtained using the new algorithm, and we obtain: (31) lim k → ∞ inf ‖ g k ‖ = 0 . Proof: The proof is straightforward, similar to the proof of Theorem 3 in. 6 In this study, we prove a global convergence theorem for non-convex problems and suggest a cautious updating strategy that is comparable to that mentioned previously. We state a Powell-related lemma for motivational purposes. 15 Lemma 1. A smooth function f that is limited below can be treated using the BFGS technique if a constant M > 0 exists, which makes the inequality: (32) ‖ y k ~ ‖ 2 / s k T y k ~ ≤ M then: (33) lim k → ∞ inf ‖ g k ‖ = 0 . Theorem 4. If these Assumptions hold, { x k } is generated by the new algorithm. Then Eq. (32) holds. Proof: If Eq.(33) fails to hold, then there exists a constant ε > 0 , such that: (34) ‖ g k ‖ ≥ ε . Therefore, a constant r > 0 exists, such that: (35) r ‖ s k ‖ 2 ≤ s k T y k ~ . So, combining Eqs. (29) and (35) imply that: (36) ‖ y k ~ ‖ 2 / s k T y k ~ ≤ M . The proof is finished. 4. Numerical experiments In this section, we present a numerical comparison of QN -techniques and suggest modifications for solving 4 th order nonlinear partial differential equations. Example 1: Consider the nonlinear 4 th order has the form; ʯ xt − ʯ xxxy − 2 ʯ xx ʯ y − 4 ʯ x ʯ xy = 0 ; ʯ ( x , y , 0 ) = 1 2 sech 2 ( 1 2 ( x + y ) ) and , exact solution ʯ ( x , y , z , t ) = tanh ( 1 2 ( x + y − t ) ) The results of solving the above equation at different times t are presented in Table 1 . The neural solution for this equation is shown in Figure 1 . We stopped utilizing the algorithms by employing Himmeblau's law 18 : If | f ( x k ) | > 1 0 − 5 , then | f ( x k ) − f ( x k + 1 ) | | f ( x k ) | = 1 . Otherwise, | f ( x k ) − f ( x k + 1 ) | = 1 . For every problem, if ‖ g k ‖ < ε is used, the program is terminated. Quasi-Newton approaches perform better when an appropriate quasi-Newton equation is employed. The performance of the new update with u k = g k + 1 was the best of the three methods, whereas the normal performance of the new update with u k = y k and u k = g k was somewhat better than that of the BFGS technique. As a result, among the QN -procedures for unconstrained problems, the new update with u k = g k + 1 is the most well -organized. Example 2: Consider the nonlinear 4 th order has the form: ʯ tt − ʯ xx − ʯ xxxx − ʯ yy − ʯ zz − 3 ( ʯ 2 ) xx = 0 ʯ ( x , y , z , 0 ) = 1 2 sech 2 ( 1 2 ( x + y + z ) ) , u t = tanh ( 1 2 ( x + y + z ) ) sech 2 ( 1 2 ( x + y + z ) ) Exact solution: ʯ ( x , y , z , t ) = 1 2 sech 2 ( 1 2 ( x + y + z − 2 t ) ) The neural solution for this equation is shown in Figure 2 when z = -0.5. The accuracy for epochs and time is presented in Table 2 , and Table 3 , illustrates the results of the neural solution of the equation. Table 1. The results of suggested algorithm for different values of time t. X = y ti exact Suggested update t = 0.001 t = 0.01 t = 0.05 t = 0.25 t = 0. 5 0 -0.000499999958333338 -0.000048659724380 -0.00499995832713615 -0.025004418506876 -0.124353001771672 -0.244918662401479 0.1 0.0991729368500791 0.099174522493650 0.0947152247011525 0.074859690643595 -0.0249947929685649 -0.148885033624227 0.2 0.196894751347250 0.196894751347288 0.192565398608004 0.173235732159165 0.0748596906873580 -0.0499583749589804 0.3 0.290854977351376 0.290854977351250 0.286730291373398 0.268271182008229 0.173235157834554 0.0499583749579298 0.4 0.379521061607639 0.379521061607816 0.375662661174346 0.358357398344881 0.268271160988048 0.148885033623492 0.5 0.461723842547565 0.461723842547454 0.458175852175461 0.442230453940485 0.358357335349861 0.244918662402002 0.6 0.536693682582613 0.536693686709420 0.533482128457157 0.519021833904887 0.442230290513323 0.336375352939167 0.7 0.604050311415608 0.604050311415511 0.601184473121516 0.588259256403465 0.519021833898177 0.421898609908564 0.8 0.663757149868171 0.663757149868364 0.661232203097477 0.649827607630977 0.588259256398005 0.500520211189160 0.9 0.716054324313046 0.716054380560282 0.713854553039899 0.703905603862037 0.649827419353020 0.571668985813867 1 0.761384088809508 0.761384088809337 0.759486275064505 0.750893283626045 0.703905603936521 0.635140845030389 Figure 1. Illustration the results using new algorithm for different time t. Figure 2. Solution for z = -1/2. Table 2. Properties of the proposed algorithm for solving Example 1 . Train Function “Trainbfg” Performance of train Epoch Time Msereg [t = 0.001] 4.72e-27 818 0:00:02 1.4903e-11 [t = 0.01] 7.27e-23 404 0:00:00 3.0524e-17 [t = 0.05] 9.34e-24 33 0:00:00 7.6100e-12 [t = 0.25] 2.64e-27 909 0:00:01 8.7302e-16 [t = 0. 5] 1.59e-24 593 0:00:01 5.4723e-12 Table 3. MSE and performance for training, validation, and testing for the solution of Example 2 . LM Suggested update BFG SCG RP Time 00:00:39 00:00: 8 00:00:44 00:00:12 Best Epoch 1000 810 1000 1000 MSE 2.61912e-12 6.9328543e-17 2.9424106e-07 5.9553091e-06 Best training perf 2.694601e-12 6.694813e-14 2.21545518e-07 5.9894044e-06 Best validation perf 2.334575e-12 7.2694735e-16 1.996644e-07 5.7156087e-06 Best test perf 2.5514463e-12 7.7070942e-15 2.254638e-07 6.0358983e-06 5. Conclusions In this study, we constructed improved BFGS quasi-Newton updating formulae by using the proposed robust QN -equation. Second-order information from Hessian’s Hessian objective function Hessian’s is used in this study to develop a novel quasi-Newton equation. Two nonlinear 4 th order example are provided to illustrate the accuracy of the suggested update, The results are consistent with the practical results and conform to the results that the suggested update, is globally convergent. Data availability No data were included in this study. References 1. Hassan MA: Estimate the Effect of Rainwaters in Contaminated Soil by Using Simulink Technique. JPCS. 2018; 1003 : 012057. 1-6. 2. Jamil HJ, Al-Noor TH: Hookah Smoking with Health Risk Perception of Different Types of Tobacco. JPCS. 2020; 1664 (1): 012127. 1-11. Publisher Full Text 3. Hussein NA: Solitary Wave Solution of Zakharov-Kuznetsov Equation. AIP Conf. Proc. 2022; 2398 (1): 060084. 1-6. 4. Ali MH: Design optimal neural network for solving unsteady state confined aquifer problem. Mathematical Modelling of Engineering Problems. 2023; 10 (2): 565–571. Publisher Full Text 5. Kareem ZH: Solving Three-Dimensional Groundwater Recharge Based on Decomposition Method. JPCS. 2020; 1530 : 012068. 1-8. Publisher Full Text 6. Al-Noor NH, Al-Noor TH: Estimate the rate of contamination in Baghdad soils by using numerical method. JPCS. 2019; 1294 (3): 032020. 1-10. 7. Khamas AH: New approach for calculate exponential integral function. Iraqi Journal of Science. 2023; 64 (8): 4034–4042. 8. Abood IN: Persons’ Camp Using Interpolation Method. JPCS. 2018; 1003 : 012055. 1-9. 9. Enadi MO: New approach for solving three-dimensional space partial differential equation. Baghdad Sci. J. 2019; 16 (3): 786–792. 10. Ghazi FF: New Approach for Solving Two- dimensional Spaces. PDE, JPCS. 2020; 1530 : 012066. 1-9. Publisher Full Text 11. 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Hussein NA: Efficient Approach for Solving (2+1) D- Differential Equations. Baghdad Sci. J. 2023; 20 (1): 166–174. 18. Kareem ZH: New soliton solution of nonlinear (3+1) D-mKd, (3+1)D-KP, and (3+1)D-gKP equation. AIP Conference Proceedings. 2024; 3036 (1): 040034. 1-12. 19. Khamas AH: Determine the Effect Hookah Smoking on Health with Different Types of Tobacco by using parallel processing technique. JPCS. 2021; 1818 : 012175. 1-10. Publisher Full Text 20. Ghazi FF: Design optimal neural network based on new LM training algorithm for solving 3D-PDEs. An International Journal of Optimization and Control: Theories & Applications. 2024; 14 (3): 249–260. Publisher Full Text 21. Hussein NA: Exact solutions for systems of nonlinear (2+1)D-Differential Equations. Iraqi Journal of Science. 2022; 63 (10): 4388–4396. 22. Helal MM: Double LA transform and their properties for solving partial differential equations. AIP Conference Proceedings. 2023; 2834 (1): 080140. 1-10. 23. Ibrahim AQ: Efficient method for solving fourth-order PDEs. JPCS. 2021; 1818 : 012166. 1-9. 24. Kareem ZH: Solving (3+1) D-New Hirota Bilinear Equation Using Tanh Method and New Modification of Extended Tanh Method. Advances in the Theory of Nonlinear Analysis and its Applications. 2023; 7 (4): 114–122. 25. Ali MH: Efficient design of neural networks for solving third-order partial differential equations. JPCS. 2020; 1530 (1): 012067. Publisher Full Text 26. Hussein NA: Exact soliton solution for systems of nonlinear (2+1)D-DEs. AIP Conference Proceedings. 2023; 2834 (1): 080137. 27. Kareem ZH: New modification of decomposition method for solving high-order strongly nonlinear partial differential equations. AIP Conference Proceedings. 2022; 2398 : 0600363. 1-9. 28. Mohammed MA, Tawfiq LNM: New accurate method for solving fourth-order two-dimension evolution equations. Journal of Interdisciplinary Mathematics. 2025; 28 (3): 1269–1276. Publisher Full Text 29. Hussein NA: New Approach for Solving (2+1)-Dimensional Differential Equation. JPCS. 2021; 1818 (1): 012182. Publisher Full Text 30. Ghazi FF: Modeling the Contamination of Soil Adjacent to Mohammed Al-Qassim Highway in Baghdad, Iraqi Journal of. Science. 2020; Vol. 61 (10): pp: 2663–2670. (2020). Publisher Full Text 31. Thirthar AA, Shah K, Abdeljawad T: Design an efficient neural network for solving steady state problems. J. Math. Comput. Sci. 2025; 36 (1): 84–98. Publisher Full Text 32. Ali MH: Efficient design of neural network based on new BFGS training algorithm for solving ZK-BBM equation. AIP Conference Proceedings. 2024; 3036 (1): 040009. 33. Abdul-Jabbar SA: Design Efficient Neural Network for Solving 2D-Nonlinear Wave-Like Equations. AIP Conference Proceedings. 2025; 3264 (1): 050055. 34. Ghazi FF: Solving 5th Order nonlinear 4D-PDEs Using Efficient Design of Neural Network. AIP Conference Proceedings. 2025; 3264 (1): 050042. Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 16 Jan 2026 ADD YOUR COMMENT Comment Author details Author details 1 Mathematics, University of Baghdad, Baghdad, Iraq 2 Ministry of Education, Directorate General of Education, KARKH II, Baghdad, Iraq Farah Ghazi Roles: Data Curation, Resources, Software, Validation Luma Tawfiq Roles: Conceptualization, Formal Analysis, Funding Acquisition, Investigation, Methodology, Supervision, Writing – Review & Editing Zainab Kareem Roles: Methodology, Project Administration, Visualization, Writing – Original Draft Preparation Competing interests No competing interests were disclosed. Grant information The author(s) declared that no grants were involved in supporting this work. Article Versions (1) version 1 Published: 16 Jan 2026, 15:71 https://doi.org/10.12688/f1000research.172826.1 Copyright © 2026 Ghazi F et al . This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The author(s) is/are employees of the US Government and therefore domestic copyright protection in USA does not apply to this work. The work may be protected under the copyright laws of other jurisdictions when used in those jurisdictions. Download Export To Sciwheel Bibtex EndNote ProCite Ref. Manager (RIS) Sente metrics Views Downloads F1000Research - - PubMed Central info_outline Data from PMC are received and updated monthly. - - Citations open_in_new 0 open_in_new 0 open_in_new SEE MORE DETAILS CITE how to cite this article Ghazi F, Tawfiq L and Kareem Z. Update Quasi-Newton Algorithm for Training ANN [version 1; peer review: 1 approved with reservations] . F1000Research 2026, 15 :71 ( https://doi.org/10.12688/f1000research.172826.1 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS track receive updates on this article Track an article to receive email alerts on any updates to this article. TRACK THIS ARTICLE Share Open Peer Review Current Reviewer Status: ? Key to Reviewer Statuses VIEW HIDE Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Version 1 VERSION 1 PUBLISHED 16 Jan 2026 Views 0 Cite How to cite this report: Wijakmatee T and Liu J. Reviewer Report For: Update Quasi-Newton Algorithm for Training ANN [version 1; peer review: 1 approved with reservations] . F1000Research 2026, 15 :71 ( https://doi.org/10.5256/f1000research.190584.r463394 ) The direct URL for this report is: https://f1000research.com/articles/15-71/v1#referee-response-463394 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 12 Mar 2026 Thossaporn Wijakmatee , Chemical Science and Engineering, Institute of Science Tokyo, Meguro, Tokyo, Japan Junjie Liu , Chemical Science and Engineering, Institute of Science Tokyo, Meguro, Tokyo, Japan Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.190584.r463394 The authors propose an updated Quasi-Newton algorithm designed to overcome the limitations of the BFGS algorithm, specifically for solving PDEs. While the topic is of interest, several points remain unclear and require significant clarification. A major revision is necessary to ... Continue reading READ ALL The authors propose an updated Quasi-Newton algorithm designed to overcome the limitations of the BFGS algorithm, specifically for solving PDEs. While the topic is of interest, several points remain unclear and require significant clarification. A major revision is necessary to address the following issues: 1. The transition and approximation assumptions between Equation 10 and Equation 12 are unclear. Please clarify the motivation for this specific formulation and explicitly state any necessary assumptions made. 2. The authors propose three alternative choices for the u k vector on pages 4-5. Please explain the underlying logic or theoretical consideration behind these choices and provide a clear motivation for each. 3. For Examples 1 and 2, the authors must specify the boundary conditions of the PDE problems and clearly define the computational domain. 4. The loss function used to train the neural network is not clearly defined. Please provide the explicit mathematical form of the loss function used in the optimization. 5. The methodology used to compare the updated algorithm with the conventional version is ambiguous. Consequently, the performance differences shown in Figures 1 and 2 are difficult to interpret. 6. The captions for figures and tables are too brief and require more detailed clarification to provide sufficient context. 7. Please clarify the meaning of 5 neuron, 7 neuron, and 11 neuron, and provide further details regarding the NN architecture, such as the number of hidden layers. 8. The conclusion states that the proposed update is “globally convergent”. Please provide a more robust theoretical justification or proof to support this claim. 9. To ensure reproducibility, the authors should provide details on the computational environment, including the software platform, specific libraries, and other relevant technical details. 10. The notation should be more clearly defined. I recommend summarizing symbols and variables in a table to improve readability. 11. The abstract mentions the convection-diffusion equation, but there is no further discussion in the main text regarding the motivation for this problem or its original references. 12. Please carefully revise the wording and sentence structure to ensure they align with the superscript reference style. For example, phrases such as “In, 21 reported...” or “Others such 13-20 suggest...” should be rewritten for clarity. 13. The grouping of references does not follow standard academic practice and should be revised. 14. The description of the BFGS approach on page 3 requires a formal reference. 15. The numerical order of the references must be rechecked; for instance, Reference 2 appears for the first time at the end of page 3, following higher-numbered citations. Is the work clearly and accurately presented and does it cite the current literature? Partly Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? Partly If applicable, is the statistical analysis and its interpretation appropriate? Partly Are all the source data underlying the results available to ensure full reproducibility? Partly Are the conclusions drawn adequately supported by the results? Partly Competing Interests: No competing interests were disclosed. Reviewer Expertise: machine learning, chemical engineering, quantum mechanics We confirm that we have read this submission and believe that we have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however we have significant reservations, as outlined above. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Wijakmatee T and Liu J. Reviewer Report For: Update Quasi-Newton Algorithm for Training ANN [version 1; peer review: 1 approved with reservations] . F1000Research 2026, 15 :71 ( https://doi.org/10.5256/f1000research.190584.r463394 ) The direct URL for this report is: https://f1000research.com/articles/15-71/v1#referee-response-463394 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 16 Jan 2026 ADD YOUR COMMENT Comment keyboard_arrow_left keyboard_arrow_right Open Peer Review Reviewer Status info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Reviewer Reports Invited Reviewers 1 Version 1 16 Jan 26 read Thossaporn Wijakmatee , Institute of Science Tokyo, Meguro, Japan Junjie Liu , Institute of Science Tokyo, Meguro, Japan Comments on this article All Comments (0) Add a comment Sign up for content alerts Sign Up You are now signed up to receive this alert Browse by related subjects keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Wijakmatee T et al. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 12 Mar 2026 | for Version 1 Thossaporn Wijakmatee , Chemical Science and Engineering, Institute of Science Tokyo, Meguro, Tokyo, Japan Junjie Liu , Chemical Science and Engineering, Institute of Science Tokyo, Meguro, Tokyo, Japan 0 Views copyright © 2026 Wijakmatee T et al. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved With Reservations info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions The authors propose an updated Quasi-Newton algorithm designed to overcome the limitations of the BFGS algorithm, specifically for solving PDEs. While the topic is of interest, several points remain unclear and require significant clarification. A major revision is necessary to address the following issues: 1. The transition and approximation assumptions between Equation 10 and Equation 12 are unclear. Please clarify the motivation for this specific formulation and explicitly state any necessary assumptions made. 2. The authors propose three alternative choices for the u k vector on pages 4-5. Please explain the underlying logic or theoretical consideration behind these choices and provide a clear motivation for each. 3. For Examples 1 and 2, the authors must specify the boundary conditions of the PDE problems and clearly define the computational domain. 4. The loss function used to train the neural network is not clearly defined. Please provide the explicit mathematical form of the loss function used in the optimization. 5. The methodology used to compare the updated algorithm with the conventional version is ambiguous. Consequently, the performance differences shown in Figures 1 and 2 are difficult to interpret. 6. The captions for figures and tables are too brief and require more detailed clarification to provide sufficient context. 7. Please clarify the meaning of 5 neuron, 7 neuron, and 11 neuron, and provide further details regarding the NN architecture, such as the number of hidden layers. 8. The conclusion states that the proposed update is “globally convergent”. Please provide a more robust theoretical justification or proof to support this claim. 9. To ensure reproducibility, the authors should provide details on the computational environment, including the software platform, specific libraries, and other relevant technical details. 10. The notation should be more clearly defined. I recommend summarizing symbols and variables in a table to improve readability. 11. The abstract mentions the convection-diffusion equation, but there is no further discussion in the main text regarding the motivation for this problem or its original references. 12. Please carefully revise the wording and sentence structure to ensure they align with the superscript reference style. For example, phrases such as “In, 21 reported...” or “Others such 13-20 suggest...” should be rewritten for clarity. 13. The grouping of references does not follow standard academic practice and should be revised. 14. The description of the BFGS approach on page 3 requires a formal reference. 15. The numerical order of the references must be rechecked; for instance, Reference 2 appears for the first time at the end of page 3, following higher-numbered citations. Is the work clearly and accurately presented and does it cite the current literature? Partly Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? Partly If applicable, is the statistical analysis and its interpretation appropriate? Partly Are all the source data underlying the results available to ensure full reproducibility? Partly Are the conclusions drawn adequately supported by the results? Partly Competing Interests No competing interests were disclosed. Reviewer Expertise machine learning, chemical engineering, quantum mechanics We confirm that we have read this submission and believe that we have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however we have significant reservations, as outlined above. reply Respond to this report Responses (0) Wijakmatee T and Liu J. Peer Review Report For: Update Quasi-Newton Algorithm for Training ANN [version 1; peer review: 1 approved with reservations] . 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