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arxiv: 2606.12348 · v1 · pith:25ZIPZM5new · submitted 2026-06-10 · 🧮 math.NA · cs.NA

MATLAB-Based Layerwise Self-Adaptive Physics-Informed Neural Network in Applications to Multidimensional Coupled Burgers' Equations with High Reynolds Numbers

Harish P. Bhatt , Xi Chen , Jingsai Liang This is my paper

Pith reviewed 2026-06-27 08:54 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords physics-informed neural networksBurgers' equationsself-adaptive weightinghigh Reynolds numbershock frontsnumerical PDEdual-phase optimization
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The pith

A layerwise self-adaptive weighting strategy in PINNs improves accuracy and shock capture for multidimensional coupled Burgers' equations at high Reynolds numbers.

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

The paper develops a MATLAB-based physics-informed neural network that applies layerwise self-adaptive weighting to dynamically adjust penalties on the physics residual, initial conditions, and boundary conditions during training. It pairs this with a dual-phase optimization procedure intended to produce more stable convergence. Numerical tests on the target equations show lower relative L2 error than standard PINNs and better resolution of the steep fronts that form at high Reynolds numbers. Mesh-based methods such as finite differences and finite elements lose stability and become strongly mesh-dependent in the same regime. A reader would care because the equations model nonlinear transport phenomena where accurate long-time shock tracking is required.

Core claim

The proposed framework employs a layerwise self-adaptive weighting strategy that dynamically adjusts the penalty weights for the physics residual, initial conditions, and boundary conditions throughout training. Moreover, the framework uses a dual-phase optimization strategy to achieve more stable convergence. Numerical results exhibit that the proposed framework achieves higher accuracy in terms of relative L2-error norm than the standard PINN and is able to capture the development of sharp shock fronts as time evolves in the solution.

What carries the argument

The layerwise self-adaptive weighting strategy that dynamically adjusts penalty weights for physics residual, initial conditions, and boundary conditions, together with dual-phase optimization.

If this is right

  • Higher relative L2-error accuracy than standard PINN with or without L-BFGS optimization.
  • Successful tracking of sharp shock fronts that develop over time.
  • More stable convergence during training on the target equations.
  • Applicability to both one- and multi-dimensional versions of the coupled system.

Where Pith is reading between the lines

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

  • The same weighting mechanism could be tested on other nonlinear hyperbolic or convection-dominated PDEs that develop discontinuities.
  • Because the implementation is in MATLAB, it may lower the barrier for engineers already using that environment to try physics-informed networks.
  • The dual-phase schedule might interact with network depth or activation choice in ways that warrant separate ablation on larger problems.

Load-bearing premise

The layerwise self-adaptive weighting and dual-phase optimization produce stable, generalizable accuracy gains and shock capture for high-Re cases without requiring problem-specific retuning or introducing new instabilities.

What would settle it

A test run on a fresh high-Reynolds-number multidimensional coupled Burgers' problem in which the proposed method yields equal or larger relative L2 error than a standard PINN or visibly fails to resolve the emerging shock fronts.

Figures

Figures reproduced from arXiv: 2606.12348 by Harish P. Bhatt, Jingsai Liang, Xi Chen.

Figure 1
Figure 1. Figure 1: DNN Φθ : R d+1 → R where • First Hidden Layer 1: The initial mapping a1 = H1(z0) = W1z0 + b1, where W1 ∈ R n×(d+1) , b1 ∈ R n project the input z0 into a high-dimensional feature space of width n. • Intermediate Hidden Layers: The network processes the features through k − 1 additional hidden layers ai = Hi(zi−1) = Wizi−1 + bi ,Wi ∈ R n×n, bi ∈ R n, i = 2, 3, · · · , k. These layers enable the network to u… view at source ↗
Figure 2
Figure 2. Figure 2: The schematics of the proposed LSAAL-PINN for the 3D Burgers’ equation [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The comparison of exact solution (dots) vs. predicted solution (solid lines) u(x, t) at various time levels for 1D Burgers’ equation. Relative L2−errors for each temporal snapshot are included in the legend. The spatio-temporal solution profiles of the 1D Burgers’ equation predicted via all three PINN variants are compared with the exact solution using the surface plots in [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figure 4
Figure 4. Figure 4: The comparison between exact and predicted solution of 1D Burgers’ equation In the case of 1D Burgers’ equation, the trend of the training convergence behaviors of the three variants is depicted in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: throughout the training process. From [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of the exact solution (dots) and the predicted solution (solid lines) at various time levels for 2D CBEs. Relative L2−errors for each temporal snapshot are included in the legend. From top to bottom, the first row presents the solution for u, while the second row shows the solution for v at y = 2 3 with high Reynolds number. The spatio-temporal solution profiles of the component u obtained via t… view at source ↗
Figure 7
Figure 7. Figure 7: The spatio-temporal solution profile of u with ν = 0.0001 until time t = 1 obtained by three PINN variants. In the case of 2D CBEs, the trend of the training convergence behaviors of the three variants is depicted in [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The comparison of the total training loss values produced by three PINN variants for Example 2. Example 3 For the final test problem, we considered the following 3D CBEs on domain [−1, 1]3 : ut + uux + vuy + wuz = ν∆u, vt + uvx + vvy + wvz = ν∆v, wt + uwx + vwy + wwz = ν∆w, with the initial and boundary conditions extracted from the exact solution [22]: u(x, y, z, t) = − 2 Re  ϕx ϕ  , v(x, y, z, t) = − 2… view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the exact solution (dots) and the predicted solution (solid lines) at various time levels for 3D CBEs. Relative L2−errors for each temporal snapshot are included in the legend. From top to bottom, the first row shows the solution for u, the second for v, and the third for w at y, z = 0.5. In the case of 3D CBEs, the performance differences across the three PINN variants are depicted in [PITH… view at source ↗
Figure 10
Figure 10. Figure 10: The comparison of the total training loss values produced by three PINN variants for Example 3. To check the performance of the proposed LSAAL-PINN for solving 3D CBEs with high Reynolds number Re = 100, a numerical experiment ran on Example 3 with Ncol = 8500, Nic = 1500, Nbc = 1500, Adam-Iter = 4500, L-BFGS Iter = 2500, same network architecture and learning rate provided in [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of the exact solution (dots) and the predicted solution (solid lines) obtained via LSAAL-PINN at various time levels for 3D CBEs with Re = 100. Relative L2−errors for each component are included in the legend. In the [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The exact and LSAAL-PINN predicted solution of 3D CBEs with Re = 100 at t = 0.01. 4 Conclusion This study proposed a layerwise self-adaptive PINN framework with hybrid optimization strategies and compared it with a standard PINN with and without hybrid optimization strategies for solving MCBEs with high Reynolds numbers. Numerical results across 1D, 2D, and 3D domains confirmed that the LSAAL-PINN framewo… view at source ↗
read the original abstract

This paper presents an improved physics-informed neural network for simulating the spatio-temporal solution profile of the multidimensional coupled Burgers' equations with high Reynolds numbers. As time evolves, the sharp shock fronts emerge in the solution, creating significant computational challenges for the conventional mesh-based numerical methods. In particular, numerical methods such as finite differences and finite elements suffer from poor stability and strong mesh dependency when resolving the steep solution gradients. To address these challenges, the proposed framework employs a layerwise self-adaptive weighting strategy that dynamically adjusts the penalty weights for the physics residual, initial conditions, and boundary conditions throughout training. Moreover, the framework uses a dual-phase optimization strategy to achieve more stable convergence. To check the effectiveness and accuracy of the proposed framework, a set of numerical experiments is conducted to compare it with the standard Physics-Informed Neural Network (PINN) with and without Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimization. Numerical results exhibit that the proposed framework achieves higher accuracy in terms of relative $L_2-$ error norm than the standard PINN and is able to capture the development of sharp shock fronts as time evolves in the solution.

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

1 major / 2 minor

Summary. The paper proposes a MATLAB-based layerwise self-adaptive physics-informed neural network (PINN) with dual-phase optimization for multidimensional coupled Burgers' equations at high Reynolds numbers. It employs dynamic adjustment of penalty weights for physics residuals, initial conditions, and boundary conditions, claiming superior relative L2 accuracy and improved capture of sharp shock fronts compared to standard PINN (with and without L-BFGS).

Significance. If the numerical improvements hold under scrutiny, the layerwise self-adaptive weighting and dual-phase strategy could offer a practical enhancement for training stability in PINNs applied to convection-dominated PDEs with steep gradients, where mesh-based methods often fail.

major comments (1)
  1. [Abstract] Abstract: the central claim that 'numerical results exhibit that the proposed framework achieves higher accuracy in terms of relative L2-error norm' is presented without any specific error values, tables, baseline comparisons, network sizes, or training details, which is load-bearing for assessing whether the method actually outperforms standard PINN.
minor comments (2)
  1. [Abstract] The abstract introduces 'layerwise self-adaptive weighting strategy' and 'dual-phase optimization strategy' without equations or pseudocode, making it hard to understand the precise implementation even at a high level.
  2. [Abstract] No mention of the specific form of the multidimensional coupled Burgers' equations (e.g., the exact system solved) or the range of Reynolds numbers tested.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and constructive feedback. We address the major comment below and agree that revisions to the abstract are warranted to better support our claims with quantitative details from the full manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that 'numerical results exhibit that the proposed framework achieves higher accuracy in terms of relative L2-error norm' is presented without any specific error values, tables, baseline comparisons, network sizes, or training details, which is load-bearing for assessing whether the method actually outperforms standard PINN.

    Authors: We agree that the abstract would be strengthened by including specific quantitative results. The full manuscript contains detailed comparisons in the numerical experiments section, including tables of relative L2 errors for the proposed method versus standard PINN (with and without L-BFGS), along with network architectures, training details, and Reynolds number cases. To address this point, we will revise the abstract to incorporate key error values and baseline comparisons from those experiments, ensuring the central claim is supported by specifics. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper describes an empirical method (layerwise self-adaptive weighting plus dual-phase optimization) for PINNs applied to high-Re Burgers' equations and validates it via direct numerical comparisons of relative L2 error against standard PINN on the same test problems. No derivation chain exists that reduces a claimed result to its own fitted parameters or to a self-citation; the reported improvements are external benchmarks, not quantities defined in terms of the method itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted. The self-adaptive weights are implicitly fitted during training but their number and initialization are unspecified.

pith-pipeline@v0.9.1-grok · 5755 in / 1098 out tokens · 17824 ms · 2026-06-27T08:54:34.147016+00:00 · methodology

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Reference graph

Works this paper leans on

22 extracted references · 2 canonical work pages · 2 internal anchors

  1. [1]

    Sanchez-Lengeling, A

    B. Sanchez-Lengeling, A. Aspuru-Guzik, A gentle introduction to deep learning in molecules, Science 361 (2018) 360-365

    2018

  2. [2]

    Hinton, L

    G. Hinton, L. Deng, D. Yu, G. E. Dahl, A-r. Mohamed, N. Jaitly, A. Senior, V . Vanhoucke, P. Nguyen, T. N. Sainath, B. Kingsbury, Deep neural networks for acoustic modeling in speech recognition: The shared views of four research groups, IEEE Signal Process. Mag. 29 (2012) 82-97

    2012

  3. [3]

    Krizhevsky, I

    A. Krizhevsky, I. Sutskever, G. Hinton, ImageNet classification with deep convolutional neural networks, Adv. Neural Inf. Process. Syst. 25 (2012), 1097–1105

    2012

  4. [4]

    FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators

    J. Pathak, S. Subramanian, P. Harrington, S. Raja, A. Chattopadhyay, M. Mardani, T. Kurth, D. Hall, Z. Li, K. Azizzadenesheli, et al., FourCastNet: A global data-driven high-resolution weather model using adaptive Fourier neural operators, arXiv preprint arXiv:2202.11214 (2022)

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  5. [5]

    Y . Wu, M. Schuster, Z. Chen, Q. V . Le, M. Norouzi, W. Macherey, M. Krikun, Y . Cao, Q. Gao, K. Macherey, et al., Google’s neural machine translation system: Bridging the gap between human and machine translation, arXiv preprint arXiv:1609.08144 (2016)

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  6. [6]

    Jumper, R

    J. Jumper, R. Evans, B. Pritzel, T. Green, M. Figurnov, O. Ronneberger, K. Tunyasuvunakool, R. Bates, A. Žídek, A. Potapenko, et al., Highly accurate protein structure prediction with AlphaFold, Nature 596 (2021) 583-589

    2021

  7. [7]

    Raissi, P

    M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378 (2019) 686-707. 12 MATLAB-Based Layerwise Self-Adaptive Physics-Informed Neural Network in Applications to Multidimensional Coupled Burger...

    2019

  8. [8]

    Di Bella, D

    A. Di Bella, D. Santoro, M. Raissi, P. Roccaro, Physics-informed neural networks in water and wastewater systems: a critical review. Water Res. 288 (2026), 125449

    2026

  9. [9]

    Quarteroni, A

    A. Quarteroni, A. Valli, Numerical approximation of partial differential equations, Springer Science & Business Media, (2008)

    2008

  10. [10]

    A. D. Jagtap, E. Kharazmi, G. E. Karniadakis, Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems, Comput. Methods Appl. Mech. Eng. 365 (2020), 113028

    2020

  11. [11]

    X. Meng, Z. Li, D. Zhang, G. E. Karniadakis, PPINN: Parareal physics-informed neural network for time- dependent PDEs, Comput. Methods Appl. Mech. Eng. 370 (2020), 113250

    2020

  12. [12]

    Mattey, S

    R. Mattey, S. Ghosh, A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations, Comput. Methods Appl. Mech. Eng. 390 (2022), 114474

    2022

  13. [13]

    L. D. McClenny, U. M. Braga-Neto, Self-adaptive physics-informed neural networks. J. Comput. Phys. 474 (2023), 111722

    2023

  14. [14]

    R. D. Ortiz Ortiz, O. Martínez Núñez, A. M. Marín Ramírez, Solving Viscous Burgers’ Equation: Hybrid Approach Combining Boundary Layer Theory and Physics-Informed Neural Networks. Mathematics 12 (2024), 3430

    2024

  15. [15]

    Y . Song, H. Wang, H. Yang, M. L. Taccari, X. Chen, Loss-attentional physics-informed neural networks, J. Comput. Phys. 501 (2024), 112781

    2024

  16. [16]

    X. Wang, S. Yi, H. Gu, J. Xu, W. Xu, WF-PINNs: solving forward and inverse problems of the Burgers equation with steep gradients using weak-form physics-informed neural networks, Sci. Rep. 15(1) (2025), 40555

    2025

  17. [17]

    Zhang, C

    Z. Zhang, C. Ruan, Z. Liu, Two improved physics-informed Neural Networks for solving Burgers equation. J. Comput. Sci. 93 (2026), 102756

    2026

  18. [18]

    Glorot, Y

    X. Glorot, Y . Bengio, Understanding the difficulty of training deep feedforward neural networks, in: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings (2010), 249–256

    2010

  19. [19]

    Stein, Large sample properties of simulations using Latin hypercube sampling, Technometrics 29 (1987), 143–151

    M. Stein, Large sample properties of simulations using Latin hypercube sampling, Technometrics 29 (1987), 143–151

    1987

  20. [20]

    Basdevant, M

    C. Basdevant, M. Deville, P. Haldenwang, J.M. Lacroix, J. Ouazzani, R. Peyret, P. Orlandi, and A.T. Patera, Spectral and finite difference solutions of the Burgers equation, Comput. Fluids 14 (1986), 23-41

    1986

  21. [21]

    C. A. J. Fletcher, Generating exact solutions of the two-dimensional Burgers’ equations, Int. J. Numer. Methods Fluids 3 (1983) 213-216

    1983

  22. [22]

    H. S. Shukla, M. Tamsir, V . K. Srivastava, and M. M. Rashidi, Modified cubicβ-spline differential quadrature method for numerical solution of three-dimensional coupled viscous Burger equation, Mod. Phys. Lett. B 30 (2016), 1650110. 13

    2016