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arxiv: 2606.04125 · v1 · pith:5A6LZU6Fnew · submitted 2026-06-02 · ⚛️ physics.app-ph · math-ph· math.MP· physics.comp-ph

A Systematic Benchmark of Physics-Informed Neural Network Architectures for the Stiff Poisson-Nernst-Planck System: Adaptive LossWeighting and Multi-Scale Resolution

David Pankaczy , Conrard Giresse Tetsassi Feugmo This is my paper

Pith reviewed 2026-06-28 07:06 UTC · model grok-4.3

classification ⚛️ physics.app-ph math-phmath.MPphysics.comp-ph
keywords physics-informed neural networksPoisson-Nernst-Planckstiff systemsadaptive loss weightingneural tangent kernelbenchmarklithium symmetric cellelectric double layer
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The pith

The balanced residual decay rate scheme matches neural tangent kernel accuracy on concentration fields for stiff Poisson-Nernst-Planck problems while cutting wall-clock time.

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

The paper establishes the first systematic benchmark of eleven physics-informed neural network configurations grouped into four strategy families on a one-dimensional Poisson-Nernst-Planck model of a lithium symmetric cell. It reports that the balanced residual decay rate scheme reaches neural tangent kernel performance on concentration profiles yet requires less mean computation time, with loss-landscape geometry aligning with the root-mean-square error ordering. A sympathetic reader would care because the Poisson-Nernst-Planck system is a canonical stiff coupled problem whose extreme coefficient ratios and thin electric-double-layer boundary layers defeat both conventional meshes and standard neural training due to spectral bias and multi-task loss imbalance. The work supplies an open-source implementation inside PhysicsNeMo Sym for reuse on similar stiff electrokinetic systems.

Core claim

Among the tested architectures the balanced residual decay rate scheme matches neural tangent kernel performance for concentration fields while reducing mean wall-clock time, making it the preferable strategy under compute constraints; root-mean-square errors vary across the eleven configurations and loss-landscape geometry corroborates the ranking.

What carries the argument

The balanced residual decay rate (BRDR) scheme, which dynamically reweights individual loss terms according to the observed decay rates of their residuals to counteract multi-task imbalance during training of physics-informed networks on stiff coupled PDEs.

If this is right

  • BRDR becomes the strategy of choice when wall-clock time is the binding constraint for concentration-field accuracy.
  • Loss-landscape geometry supplies an independent diagnostic that tracks RMSE rankings across architectures.
  • The released PhysicsNeMo Sym implementation can be applied directly to other stiff coupled PDE problems in computational mechanics.
  • Adaptive loss-weighting strategies mitigate the multi-task imbalance that otherwise limits PINN accuracy on stiff PNP systems.

Where Pith is reading between the lines

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

  • The time advantage of BRDR may extend to other multi-physics stiff systems whose loss terms decay at mismatched rates.
  • A two-dimensional or three-dimensional version of the same benchmark would test whether the observed ranking survives increased spatial complexity.
  • Open release of the code lowers the threshold for testing PINNs on electrokinetic transport in batteries, membranes, and biological ion channels.

Load-bearing premise

The eleven PINN configurations organized into four strategy groups, the one-dimensional physically parametrised PNP model for a lithium symmetric cell, and the finite volume method reference are representative enough to rank architectures for general stiff PNP problems.

What would settle it

A repeat of the eleven-configuration benchmark on a two-dimensional PNP geometry or a materially different physical parameter set in which the BRDR scheme no longer matches NTK accuracy on concentrations or loses its wall-clock advantage.

Figures

Figures reproduced from arXiv: 2606.04125 by Conrard Giresse Tetsassi Feugmo, David Pankaczy.

Figure 1
Figure 1. Figure 1: Schematic of the one-dimensional lithium symmetric cell. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: PhysicsNeMo Sym computational pipeline. Halton collocation points are fed to the neural network architecture (left); automatic differentiation (AD) through the Graph computes PDE, boundary condition (BC), and initial condition (IC) residuals (centre-left); Neural Tangent Kernel (NTK) or balanced residual decay rate (BRDR) adaptive weighting rescales individual loss components before aggregation to total lo… view at source ↗
Figure 3
Figure 3. Figure 3: NTK-weighted PINN validation against the FVM reference. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: NTK-weighted PINN training loss. Total loss (left) and individual loss components (right), averaged over ten independent runs. Individual loss components (PDE: partial differential equation, BC: boundary condition, IC: initial condition) converge at comparable rates throughout training, consistent with the balanced weighting reported in Section 4.2 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Wall-clock time distributions. Box plots of wall time for the NTK configuration over ten runs (left) and across all architectures (right). The red line marks the median; box edges denote the first and third quartiles; whiskers extend to 1.5× the interquartile range (IQR). All runs performed on one NVIDIA H100 GPU. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Loss landscape geometry for all PINN configurations. [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: NTK training loss for three collocation densities. [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
read the original abstract

The Poisson Nernst Planck PNP system constitutes a canonical stiff coupled PDE problem where the charge density prefactor produces extreme coefficient ratios and the electric double layer imposes sharp boundary layers. Physics informed neural networks PINNs are appealing here because they require no mesh and differentiate through the physics automatically. Spectral bias and multi task loss imbalance however have limited their accuracy on stiff PNP systems. We present the first systematic data free benchmark of eleven PINN configurations organised into four strategy groups on a physically parametrised one dimensional PNP model for a lithium symmetric cell implemented within NVIDIA PhysicsNeMo Sym and validated against a finite volume method FVM reference. Root mean square errors RMSE span across architectures. The balanced residual decay rate BRDR scheme matches Neural Tangent Kernel NTK performance for concentration fields while reducing mean wall clock time making it the preferable strategy under compute constraints. Loss landscape geometry corroborates the RMSE ranking. We release an open source PhysicsNeMo Sym implementation for reuse on stiff coupled PDE problems in computational mechanics.

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 / 1 minor

Summary. The paper conducts the first systematic benchmark of eleven PINN configurations grouped into four strategy groups for solving the stiff one-dimensional Poisson-Nernst-Planck equations modeling a lithium symmetric cell. Implemented in NVIDIA PhysicsNeMo Sym and validated against a finite volume method reference, the study concludes that the balanced residual decay rate (BRDR) scheme achieves performance comparable to the Neural Tangent Kernel (NTK) approach for concentration fields while reducing mean wall-clock time, making it preferable under compute constraints. Loss landscape analysis supports the RMSE rankings, and the code is released openly.

Significance. If the empirical findings hold, this work provides valuable guidance on loss-weighting and multi-scale strategies for PINNs applied to stiff coupled PDEs like PNP systems. The open-source PhysicsNeMo Sym implementation is a clear strength, enabling reproducibility and extension to other computational mechanics problems.

major comments (2)
  1. [Abstract] Abstract: The abstract states that RMSE spans architectures and that BRDR matches NTK while lowering wall-clock time, but supplies no numerical values, error bars, data exclusion criteria, or validation details against the FVM reference. This absence makes it impossible to assess the magnitude or statistical reliability of the claimed match.
  2. [Abstract] Abstract: The recommendation that BRDR is the preferable strategy under compute constraints rests entirely on results from a one-dimensional physically parametrized lithium symmetric cell. No experiments or discussion address whether the relative performance of BRDR versus NTK persists when the electric double layer becomes a surface in 2D/3D or when stiffness regimes and collocation requirements change.
minor comments (1)
  1. The four strategy groups and eleven configurations would benefit from an explicit summary table listing each architecture, its loss-weighting or multi-scale component, and key hyperparameters.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The abstract states that RMSE spans architectures and that BRDR matches NTK while lowering wall-clock time, but supplies no numerical values, error bars, data exclusion criteria, or validation details against the FVM reference. This absence makes it impossible to assess the magnitude or statistical reliability of the claimed match.

    Authors: The abstract is intended as a concise overview. The manuscript provides full numerical RMSE values, standard deviations across runs, explicit comparison criteria against the FVM reference, and loss-landscape diagnostics in the results section and supplementary tables. To improve standalone readability of the abstract we will insert representative quantitative values (e.g., mean RMSE for concentration fields under BRDR and NTK) together with a brief statement on the validation protocol. revision: yes

  2. Referee: [Abstract] Abstract: The recommendation that BRDR is the preferable strategy under compute constraints rests entirely on results from a one-dimensional physically parametrized lithium symmetric cell. No experiments or discussion address whether the relative performance of BRDR versus NTK persists when the electric double layer becomes a surface in 2D/3D or when stiffness regimes and collocation requirements change.

    Authors: The study is deliberately scoped to a canonical one-dimensional stiff PNP problem to enable a controlled, systematic comparison of eleven architectures. The manuscript makes no claim of dimensional generality; the recommendation is explicitly tied to the 1-D lithium-symmetric-cell setting under the reported stiffness and collocation conditions. Extending the benchmark to 2-D/3-D geometries constitutes a substantial separate investigation that lies outside the present scope. revision: no

Circularity Check

0 steps flagged

Empirical benchmark with independent FVM validation; no derivation reduces to inputs

full rationale

The manuscript reports a numerical benchmark of eleven PINN loss-weighting and multi-scale configurations on a fixed 1D lithium-symmetric-cell PNP problem. All performance claims (RMSE rankings, wall-clock times, loss-landscape geometry) are obtained by direct comparison against an external finite-volume reference solution. No equation is derived from first principles, no parameter is fitted and then relabeled as a prediction, and no uniqueness theorem or ansatz is imported via self-citation. The work is therefore self-contained against external benchmarks and receives the default non-circularity score.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The work is a numerical benchmarking study and introduces no new physical parameters, axioms, or postulated entities.

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Works this paper leans on

54 extracted references · 37 canonical work pages

  1. [1]

    Automatic differentiation in machine learning: a survey.Journal of Machine Learning Research, 18(153):1–43, 2018

    Atılım Güne¸ s Baydin, Barak A Pearlmutter, Alexey Andreyevich Radul, and Jeffrey Mark Siskind. Automatic differentiation in machine learning: a survey.Journal of Machine Learning Research, 18(153):1–43, 2018. URL https://jmlr.org/papers/v18/17-468.html

    2018

  2. [2]

    Diffuse-charge dynamics in electrochemical systems

    Martin Z Bazant, Katsuyo Thornton, and Armand Ajdari. Diffuse-charge dynamics in electrochemical systems. Physical Review E, 70(2):021506, 2004. doi: 10.1103/PhysRevE.70.021506

    work page doi:10.1103/physreve.70.021506 2004

  3. [3]

    Monte carlo and quasi-monte carlo methods.Acta Numerica, 7:1–49, 1998

    Russel E Caflisch. Monte carlo and quasi-monte carlo methods.Acta Numerica, 7:1–49, 1998. doi: 10.1017/ S0962492900002804

    1998

  4. [4]

    Howard, and Panos Stinis

    Wenqian Chen, Amanda A. Howard, and Panos Stinis. Self-adaptive weights based on balanced residual decay rate for physics-informed neural networks and deep operator networks.Journal of Computational Physics, 542:114226,

  5. [5]

    doi: https://doi.org/10.1016/j.jcp.2025.114226

    ISSN 0021-9991. doi: https://doi.org/10.1016/j.jcp.2025.114226. URL https://www.sciencedirect. com/science/article/pii/S0021999125005091

    work page doi:10.1016/j.jcp.2025.114226 2025

  6. [6]

    Sepa- rable physics-informed neural networks

    Junwoo Cho, Seungtae Nam, Hyunmo Yang, Seok-Bae Yun, Youngjoon Hong, and Eunbyung Park. Sepa- rable physics-informed neural networks. In A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine, editors,Advances in Neural Information Processing Systems, volume 36, pages 23761–23788. Cur- ran Associates, Inc., 2023. URL https://proceedings.neuri...

    2023

  7. [7]

    Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next

    Salvatore Cuomo, Vincenzo Schiano Di Cola, Fabio Giampaolo, Gianluigi Rozza, Maziar Raissi, and Francesco Piccialli. Scientific machine learning through physics-informed neural networks: Where we are and what’s next. Journal of Scientific Computing, 92(3):88, 2022. doi: 10.1007/s10915-022-01939-z

    work page doi:10.1007/s10915-022-01939-z 2022

  8. [8]

    Approximation by superpositions of a sigmoidal function.Math

    George Cybenko. Approximation by superpositions of a sigmoidal function.Mathematics of Control, Signals and Systems, 2(4):303–314, 1989. doi: 10.1007/BF02551274

    work page doi:10.1007/bf02551274 1989

  9. [9]

    Ion transport in nanofluidic channels.Nano Letters, 4(1): 137–142, 2004

    Hirofumi Daiguji, Peidong Yang, and Arun Majumdar. Ion transport in nanofluidic channels.Nano Letters, 4(1): 137–142, 2004. doi: 10.1021/nl0348185

    work page doi:10.1021/nl0348185 2004

  10. [10]

    Error analysis for physics-informed neural networks (PINNs) approximating Kolmogorov PDEs.Advances in Computational Mathematics, 48(6):79, 2022

    Tim De Ryck and Siddhartha Mishra. Error analysis for physics-informed neural networks (PINNs) approximating Kolmogorov PDEs.Advances in Computational Mathematics, 48(6):79, 2022. doi: 10.1007/s10444-022-09985-9

    work page doi:10.1007/s10444-022-09985-9 2022

  11. [11]

    Multilevel domain decomposition- based architectures for physics-informed neural networks.Computer Methods in Applied Mechanics and Engineering, 429:117116, 2024

    Victorita Dolean, Alexander Heinlein, Siddhartha Mishra, and Ben Moseley. Multilevel domain decomposition- based architectures for physics-informed neural networks.Computer Methods in Applied Mechanics and Engineering, 429:117116, 2024. ISSN 0045-7825. doi: https://doi.org/10.1016/j.cma.2024.117116. URL https://www.sciencedirect.com/science/article/pii/S0...

    work page doi:10.1016/j.cma.2024.117116 2024

  12. [12]

    The deep ritz method: a deep learning-based numerical algorithm for solving variational problems

    Weinan E and Bing Yu. The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems.Communications in Mathematics and Statistics, 6(1):1–12, 2018. doi: 10.1007/s40304-018-0127-z

    work page doi:10.1007/s40304-018-0127-z 2018

  13. [13]

    Computing the field in proteins and channels.Journal of Membrane Biology, 150(1):1–25,

    Robert S Eisenberg. Computing the field in proteins and channels.Journal of Membrane Biology, 150(1):1–25,

  14. [14]

    doi: 10.1007/s002329900026

    work page doi:10.1007/s002329900026

  15. [15]

    A physics-informed deep learning framework for inversion and surrogate mod- eling in solid mechanics

    Ehsan Haghighat, Maziar Raissi, Adrian Moure, Hector Gomez, and Ruben Juanes. A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics.Computer Methods in Applied Mechanics and Engineering, 379:113741, 2021. doi: 10.1016/j.cma.2021.113741

    work page doi:10.1016/j.cma.2021.113741 2021

  16. [16]

    On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals.Numerische Mathematik, 2(1):84–90, 1960

    John H Halton. On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals.Numerische Mathematik, 2(1):84–90, 1960. doi: 10.1007/BF01386213

    work page doi:10.1007/bf01386213 1960

  17. [17]

    In: Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV)

    Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Delving deep into rectifiers: Surpassing human-level performance on ImageNet classification. InProceedings of the IEEE International Conference on Computer Vision (ICCV), pages 1026–1034, 2015. doi: 10.1109/ICCV .2015.123

    work page doi:10.1109/iccv 2015

  18. [18]

    5, 359–366,https: //doi.org/10.1016/0893-6080(89)90020-8

    Kurt Hornik, Maxwell Stinchcombe, and Halbert White. Multilayer feedforward networks are universal approxi- mators.Neural Networks, 2(5):359–366, 1989. doi: 10.1016/0893-6080(89)90020-8

    work page doi:10.1016/0893-6080(89)90020-8 1989

  19. [19]

    Enriched physics-informed neural networks for dynamic poisson-nernst-planck systems.Mathematics and Computers in Simulation, 237:231–246, 2025

    Xujia Huang, Fajie Wang, Benrong Zhang, and Hanqing Liu. Enriched physics-informed neural networks for dynamic poisson-nernst-planck systems.Mathematics and Computers in Simulation, 237:231–246, 2025. ISSN 0378-4754. doi: https://doi.org/10.1016/j.matcom.2025.04.037. URL https://www.sciencedirect.com/ science/article/pii/S0378475425001752

    work page doi:10.1016/j.matcom.2025.04.037 2025

  20. [20]

    M. F. Hutchinson. A stochastic estimator of the trace of the influence matrix for Laplacian smooth- ing splines.Communications in Statistics – Simulation and Computation, 18(3):1059–1076, 1989. doi: 10.1080/03610918908812806

    work page doi:10.1080/03610918908812806 1989

  21. [21]

    Neural tangent kernel: Convergence and generalization in neural networks

    Arthur Jacot, Franck Gabriel, and Clément Hongler. Neural tangent kernel: Convergence and generalization in neural networks. InAdvances in Neural Information Processing Systems, vol- ume 31, pages 8571–8580, 2018. URL https://proceedings.neurips.cc/paper/2018/hash/ 5a4be1fa34e62bb8a6ec6b91d2462f5a-Abstract.html

    2018

  22. [22]

    Ameya D Jagtap and George Em Karniadakis. Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems.Computer Methods in Applied Mechanics and Engineering, 365:113028, 2020. doi: 10.1016/j.cma.2020.113028

    work page doi:10.1016/j.cma.2020.113028 2020

  23. [23]

    Springer, Berlin, Heidelberg, 1996

    Joseph W Jerome.Analysis of Charge Transport: A Mathematical Study of Semiconductor Devices. Springer, Berlin, Heidelberg, 1996. doi: 10.1007/978-3-642-79987-7

    work page doi:10.1007/978-3-642-79987-7 1996

  24. [24]

    Physics-informed machine learning

    George Em Karniadakis, Ioannis G Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang. Physics- informed machine learning.Nature Reviews Physics, 3(6):422–440, 2021. doi: 10.1038/s42254-021-00314-5

    work page doi:10.1038/s42254-021-00314-5 2021

  25. [25]

    Multi-task learning using uncertainty to weigh losses for scene geometry and semantics

    Alex Kendall, Yarin Gal, and Roberto Cipolla. Multi-task learning using uncertainty to weigh losses for scene geometry and semantics. InProceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 7482–7491, 2018. doi: 10.1109/CVPR.2018.00781

    work page doi:10.1109/cvpr.2018.00781 2018

  26. [26]

    Adam: A method for stochastic optimization, 2014

    Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization, 2014. URL https://arxiv. org/abs/1412.6980

    Pith/arXiv arXiv 2014

  27. [27]

    Characterizing possible failure modes in physics-informed neural networks

    Aditi Krishnapriyan, Amir Gholami, Shandian Zhe, Robert Kirby, and Michael W Mahoney. Characterizing possible failure modes in physics-informed neural networks. InAdvances in Neural Information Processing Systems, volume 34, pages 26548–26560, 2021. URL https://proceedings.neurips.cc/paper/2021/ hash/df438e5206f31600e6ae4af72f2725f1-Abstract.html

    2021

  28. [28]

    Artificial neural networks for solving ordinary and partial differential equations

    Isaac E Lagaris, Aristidis Likas, and Dimitrios I Fotiadis. Artificial neural networks for solving ordinary and partial differential equations.IEEE Transactions on Neural Networks, 9(5):987–1000, 1998. doi: 10.1109/72.712178

    work page doi:10.1109/72.712178 1998

  29. [29]

    Visualizing the loss landscape of neural nets

    Hao Li, Zheng Xu, Gavin Taylor, Christoph Studer, and Tom Goldstein. Visualizing the loss landscape of neural nets. InAdvances in Neural Information Processing Systems, vol- ume 31, pages 6389–6399, 2018. URL https://proceedings.neurips.cc/paper/2018/hash/ a41b3bb3e6b050b6c9067c67f663b915-Abstract.html

    2018

  30. [30]

    Hou, and Max Tegmark

    Ziming Liu, Yixuan Wang, Sachin Vaidya, Fabian Ruehle, James Halverson, Marin Soljaˇci´c, Thomas Y . Hou, and Max Tegmark. Kan: Kolmogorov-arnold networks, 2025. URLhttps://arxiv.org/abs/2404.19756

    Pith/arXiv arXiv 2025

  31. [31]

    DeepXDE: a deep learning library for solving differential equations

    Lu Lu, Xuhui Meng, Zhiping Mao, and George Em Karniadakis. DeepXDE: A deep learning library for solving differential equations.SIAM Review, 63(1):208–228, 2021. doi: 10.1137/19M1274067

    work page doi:10.1137/19m1274067 2021

  32. [32]

    The old and the new: Can physics-informed deep-learning replace traditional linear solvers? Frontiers in Big Data, 4:669097, 2021

    Stefano Markidis. The old and the new: Can physics-informed deep-learning replace traditional linear solvers? Frontiers in Big Data, 4:669097, 2021. doi: 10.3389/fdata.2021.669097. 19 A Systematic Benchmark of Physics-Informed Neural Network Architectures for the Stiff Poisson–Nernst–Planck System: Adaptive Loss Weighting and Multi-Scale ResolutionA PREPRINT

    work page doi:10.3389/fdata.2021.669097 2021

  33. [33]

    Journal of Systems and Soft- ware202, 111722 (2023).https://doi.org/10.1016/j.jss.2023.111722

    Levi D McClenny and Ulisses M Braga-Neto. Self-adaptive physics-informed neural networks.Journal of Computational Physics, 474:111722, 2023. doi: 10.1016/j.jcp.2023.111722

    work page doi:10.1016/j.jcp.2023.111722 2023

  34. [34]

    Siddhartha Mishra and Roberto Molinaro. Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs.IMA Journal of Numerical Analysis, 42(2): 981–1022, 2022. doi: 10.1093/imanum/drab032

    work page doi:10.1093/imanum/drab032 2022

  35. [35]

    Variationally correct neural residual regression for parametric pdes: On the viability of controlled ac- curacy

    Siddhartha Mishra and Roberto Molinaro. Estimates on the generalization error of physics-informed neural networks for approximating PDEs.IMA Journal of Numerical Analysis, 43(1):1–43, 2023. doi: 10.1093/imanum/ drac032

    work page doi:10.1093/imanum/ 2023

  36. [36]

    Ben Moseley, Andrew Markham, and Tarje Nissen-Meyer. Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations.Advances in Computa- tional Mathematics, 49(4):62, 2023. doi: 10.1007/s10444-023-10065-9

    work page doi:10.1007/s10444-023-10065-9 2023

  37. [37]

    John Wiley & Sons, Hoboken, NJ, 3rd edition, 2004

    John Newman and Karen E Thomas-Alyea.Electrochemical Systems. John Wiley & Sons, Hoboken, NJ, 3rd edition, 2004

    2004

  38. [38]

    PhysicsNeMo Sym: An open-source framework for physics-informed machine learning

    NVIDIA Corporation. PhysicsNeMo Sym: An open-source framework for physics-informed machine learning. https://github.com/NVIDIA/physicsnemo, 2024. Accessed: 2025

    2024

  39. [39]

    Hamprecht, Yoshua Bengio, and Aaron Courville

    Nasim Rahaman, Aristide Baratin, Devansh Arpit, Felix Draxler, Min Lin, Fred A. Hamprecht, Yoshua Bengio, and Aaron Courville. On the spectral bias of neural networks, 2019. URL https://arxiv.org/abs/1806.08734

    Pith/arXiv arXiv 2019

  40. [40]

    Raissi, P

    M. Raissi, P. Perdikaris, and G.E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.Journal of Computational Physics, 378:686–707, 2019. ISSN 0021-9991. doi: https://doi.org/10.1016/j.jcp.2018.10.045. URL https: //www.sciencedirect.com/scien...

    work page doi:10.1016/j.jcp.2018.10.045 2019

  41. [41]

    On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs.Communications in Computational Physics, 28 (5):2042–2074, 2020

    Yeonjong Shin, Jérôme Darbon, and George Em Karniadakis. On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs.Communications in Computational Physics, 28 (5):2042–2074, 2020. doi: 10.4208/cicp.OA-2020-0193

    work page doi:10.4208/cicp.oa-2020-0193 2042

  42. [42]

    Analysis and simulation of one-dimensional transport models for lithium symmetric cells

    Anudeep Subramaniam, Jixian Chen, Taekyu Jang, Nicholas R Geise, Ryan M Kasse, Michael F Toney, and Venkat R Subramanian. Analysis and simulation of one-dimensional transport models for lithium symmetric cells. Journal of The Electrochemical Society, 166(15):A3806, 2019. doi: 10.1149/2.0261915jes

    work page doi:10.1149/2.0261915jes 2019

  43. [43]

    Fourier features let networks learn high frequency functions in low dimensional domains

    Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh Sing- hal, Ravi Ramamoorthi, Jonathan Barron, and Ren Ng. Fourier features let networks learn high frequency functions in low dimensional domains. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors,Advances in Neural Information Process...

    2020

  44. [44]

    Understanding and mitigating gradient flow pathologies in physics-informed neural networks.SIAM Journal on Scientific Computing, 43 (5):A3055–A3081, 2021

    Sifan Wang, Yujun Teng, and Paris Perdikaris. Understanding and mitigating gradient flow pathologies in physics-informed neural networks.SIAM Journal on Scientific Computing, 43(5):A3055–A3081, 2021. doi: 10.1137/20M1318043

    work page doi:10.1137/20m1318043 2021

  45. [45]

    Learning the solution operator of parametric partial differential equations with physics-informed DeepONets.Science Advances, 7(40): eabi8605, 2021

    Sifan Wang, Hanwen Wang, and Paris Perdikaris. Learning the solution operator of parametric partial differential equations with physics-informed deeponets.Science Advances, 7(40):eabi8605, 2021. doi: 10.1126/sciadv.abi8605. URLhttps://www.science.org/doi/abs/10.1126/sciadv.abi8605

    work page doi:10.1126/sciadv.abi8605 2021

  46. [46]

    Sifan Wang, Hanwen Wang, and Paris Perdikaris. On the eigenvector bias of fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks.Computer Methods in Applied Mechanics and Engineering, 384:113938, 2021. doi: 10.1016/j.cma.2021.113938

    work page doi:10.1016/j.cma.2021.113938 2021

  47. [47]

    When and why pinns fail to train: A neural tangent kernel perspective

    Sifan Wang, Xinling Yu, and Paris Perdikaris. When and why pinns fail to train: A neural tangent kernel perspective. Journal of Computational Physics, 449:110768, 2022. ISSN 0021-9991. doi: https://doi.org/10.1016/j.jcp.2021. 110768. URLhttps://www.sciencedirect.com/science/article/pii/S002199912100663X

    work page doi:10.1016/j.jcp.2021 2022

  48. [49]

    Yizheng Wang, Jia Sun, Jinshuai Bai, Cosmin Anitescu, Mohammad Sadegh Eshaghi, Xiaoying Zhuang, Timon Rabczuk, and Yinghua Liu. Kolmogorov–arnold-informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on kolmogorov–arnold networks.Computer Methods in Applied Mechanics and Engineering, 433:117518,...

    work page doi:10.1016/j.cma 2025

  49. [50]

    Dendrites and pits: Untangling the complex behavior of lithium metal anodes through operando video microscopy.ACS Cent Sci, 2(11):790–801, October 2016

    Kevin N Wood, Eric Kazyak, Alexander F Chadwick, Kuan-Hung Chen, Ji-Guang Zhang, Katsuyo Thornton, and Neil P Dasgupta. Dendrites and pits: Untangling the complex behavior of lithium metal anodes through operando video microscopy.ACS Cent Sci, 2(11):790–801, October 2016

    2016

  50. [51]

    Chenxi Wu, Min Zhu, Qinyang Tan, Yen Kartha, and Lu Lu. A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks.Computer Methods in Applied Mechanics and Engineering, 403:115671, 2023. doi: 10.1016/j.cma.2022.115671

    work page doi:10.1016/j.cma.2022.115671 2023

  51. [52]

    A finite element iterative solver for a pnp ion channel model with neumann boundary condition and membrane surface charge.Journal of Computational Physics, 423:109915, 2020

    Dexuan Xie and Zhen Chao. A finite element iterative solver for a pnp ion channel model with neumann boundary condition and membrane surface charge.Journal of Computational Physics, 423:109915, 2020. ISSN 0021-

    2020

  52. [53]

    URL https://www.sciencedirect.com/science/ article/pii/S0021999120306896

    doi: https://doi.org/10.1016/j.jcp.2020.109915. URL https://www.sciencedirect.com/science/ article/pii/S0021999120306896

    work page doi:10.1016/j.jcp.2020.109915 2020

  53. [54]

    Training behavior of deep neural network in frequency domain, 2019

    Zhi-Qin John Xu, Yaoyu Zhang, and Yanyang Xiao. Training behavior of deep neural network in frequency domain, 2019. URLhttps://arxiv.org/abs/1807.01251

    arXiv 2019

  54. [55]

    Adahessian: An adaptive second order optimizer for machine learning.Proceedings of the AAAI Conference on Artificial Intelligence, 35(12):10665–10673, May 2021

    Zhewei Yao, Amir Gholami, Sheng Shen, Mustafa Mustafa, Kurt Keutzer, and Michael Mahoney. Adahessian: An adaptive second order optimizer for machine learning.Proceedings of the AAAI Conference on Artificial Intelligence, 35(12):10665–10673, May 2021. doi: 10.1609/aaai.v35i12.17275. URL https://ojs.aaai.org/ index.php/AAAI/article/view/17275. 21

    work page doi:10.1609/aaai.v35i12.17275 2021