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Physics-Informed Neural Networks for Complex Eigenfrequency Identification and Mode Structure Reconstruction of the Ground-State ITG Branch

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read CEI-PINN jointly recovers the complex eigenfrequency and 2D complex mode field of the ground-state ITG branch from sparse observations, reaching a relative eigenfrequency error of 0.029 at 6% supervision.

desk verdict A real engineering contribution—Fourier features, complex layers, and staged training for complex eigenfrequency inversion—but the headline accuracy rests on a regularization term that is never defined and is shown to be load-bearing. read the letter →

arxiv 2608.01850 v1 pith:VOE6XYS2 submitted 2026-08-03 cs.AI

Dengdi Sun , Bingbing Zhang , Xiao Wang , Zikang Yan , Yuqiang Tao , Qingquan Yang , Guosheng Xu , Jin Tang This is my paper
classification cs.AI
keywords physics-informedneuralnetworkscomplexeigenfrequencyidentificationion-temperature-gradientmodecomplex-valuedfieldreconstructionsparseobservationsthree-stagetrainingtokamakpedestalFourierfeatureencoding
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that a physics-informed neural network can solve a coupled inverse problem that standard PINNs cannot: from only about 6% sparse observations of the complex-valued mode field of a steep-gradient ion-temperature-gradient (ITG) drift wave, it identifies the complex eigenfrequency (oscillation frequency and growth rate) and reconstructs the full 2D complex mode field on an annular tokamak cross-section. The key claim is that the failure of standard PINNs is caused by 'parameter compensation': the network can lower the equation residual by adjusting the high-dimensional field to hide an incorrect eigenfrequency. The paper's framework, CEI-PINN, addresses this with Fourier feature encoding, complex-valued feature propagation, and a three-stage training schedule that freezes the field while identifying the eigenfrequency and then freezes the eigenfrequency while refining the field. Experiments report a relative eigenfrequency error of 0.029 and an aligned field error of 0.18 at 6% supervision, beating four representative PINN baselines. If correct, this offers a path to infer drift-wave stability from sparse measurements in regimes where traditional eigensolvers are expensive or inapplicable.

What carries the argument

Three mechanisms carry the argument. (1) Fourier feature encoding maps normalized spatial coordinates into multiscale sine/cosine pairs, giving the network direct access to the mode's rapid poloidal oscillations (dominant angular index 63) without learning them from scratch, while preserving automatic differentiation with respect to the original coordinates. (2) Complex-valued feature propagation uses a block-structured linear transformation that couples the real and imaginary representations at every hidden layer, so the two components of the field maintain a consistent phase relationship rather than interacting only through the final loss. (3) Three-stage training freezes the field network

What would settle it

Train CEI-PINN on a dataset generated with slightly perturbed physical parameters (e.g., shift s or epsilon_n by 1–5%) while keeping the assumed operator fixed, or add Gaussian noise to the 6% supervised field values; if E_omega grows above 0.1 or the converged branch changes, the inversion is not robust to model mismatch and noise.

Watch

Extended reading notes

Core claim

The central discovery is the identification of a failure mechanism and a remedy. The authors show that in the ITG inverse eigenproblem, the mode field and the complex eigenfrequency enter the same complex-valued PDE residual, so joint optimization lets errors in one be absorbed by the other—an inaccurate provisional field can bias the eigenfrequency, and an inaccurate eigenfrequency can reshape the unsupervised field. CEI-PINN breaks this loop by treating the eigenfrequency as two trainable scalar parameters, not network outputs; by propagating features through a complex-valued (block-structured) network so the real and imaginary field components interact during forward passes; and by decomp

Load-bearing premise

The load-bearing premise is that the physical coefficients in the governing operator (rho_i, k_theta, q, s, omega_*e, epsilon_n, eta_i, tau, delta_e) are exactly known and that the 6% observations are noiseless; under realistic uncertainty or noise, the reported accuracy may degrade or branch selection may fail.

Editorial extensions

If this is right

  • Sparse observations (about 6% of the grid) are sufficient to identify the complex eigenfrequency and mode field of the ground-state ITG branch when the operator coefficients are known.
  • The three-stage schedule, rather than any single architectural component, is what suppresses parameter compensation: joint training yields relative error 0.139, alternating 0.038, three-stage 0.029.
  • The model is robust across a range of initial eigenfrequencies, converging near the reference from 30 different starting values.
  • At deployment, each forward evaluation takes 0.0043 s versus 3.678 s for the physical eigensolver—an 855-fold inference speedup after training.
  • Performance does not improve monotonically with supervision; the interaction between data quantity, coupling, and optimization is such that more observations can hurt.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the trained network's k_theta conditioning input generalizes across parameter values, the approach could become a fast surrogate for scanning equilibrium parameters without re-solving the eigenvalue problem; the paper does not test extrapolation.
  • For real experimental data, the assumed-exact coefficients and noiseless observations are likely violated; a natural next test is injecting noise or mismatched parameters into the loss and measuring degradation of E_omega.
  • The aligned L2 field metric allows a global complex rescaling, so the reported 0.18 does not by itself certify pointwise phase accuracy; reporting local phase error in the high-energy radial band would sharpen the claim.
  • The same staged-decoupling idea could be applied to identify higher-order branches or multiple coexisting branches, but branch selection would require an additional mechanism the paper leaves open.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes CEI-PINN, a physics-informed neural framework for jointly identifying the complex eigenfrequency and reconstructing the two-dimensional complex-valued mode field of the ground-state ion-temperature-gradient (ITG) branch from sparse observations. The method combines Fourier feature encoding, complex-valued feature propagation, and a three-stage training schedule. On a benchmark generated with a custom eigensolver, the authors report a relative eigenfrequency error E_omega=0.029 and an aligned field error E_phi^align=0.18 under 6% supervision, with ablation studies and an initialization scan supporting the design choices. The paper positions this as a step toward physics-informed inversion of drift-wave eigenmodes under sparse observations.

Significance. If the reported accuracy is reproducible, the paper addresses a genuinely relevant inverse problem in plasma physics: recovering a complex eigenvalue and its 2D complex eigenfunction from sparse field measurements. The strengths of the paper are its clear problem formulation, the quantified characterization of the reference mode (localization, poloidal oscillation index, complex coupling) in Appendix C, the systematic ablations of representation components and training strategy, and the initialization robustness study in Table VI. These are valuable contributions to the PINN-for-eigenvalue literature. However, the central quantitative claim cannot yet be fully assessed because a load-bearing loss term, the eigenfrequency regularizer R_omega in Eq. (15), is never defined, and the loss weights mentioned in Eq. (16) are absent.

major comments (4)
  1. [Section IV.C, Eq. (15); Appendix E, Table VIII] The regularizer L_reg = R_omega(omega) is never given a functional form. Table VIII shows that removing this term collapses the solution to omega = 1.021 + 0.045i, a degenerate nonphysical eigenfrequency. Since Eq. (2) is homogeneous and admits phi=0 for any omega, and since the eigenfunction has a complex scaling freedom, this term is load-bearing for selecting the target branch. Please specify R_omega explicitly: its functional form (e.g., quadratic, Huber, box constraint), its center and width, and the value of its coefficient in Eq. (16). If R_omega is centered near the reference eigenfrequency, the headline E_omega=0.029 reflects in part an external prior, and the claim that the framework 'identifies' the eigenfrequency from sparse data plus PDE residual needs to be qualified.
  2. [Section IV.C, Eq. (16)] The sentence 'the four coefficients balance the constraints' implies that Eq. (16) should contain four weights, but the displayed objective L = L_data + L_bc + L_pde + L_reg has no coefficients and no numerical values are reported. Reproducing the method requires knowing how data, boundary, PDE residual, and regularization are balanced, especially because the ablations in Table VIII remove entire terms. Please add the explicit weighted objective and report the coefficient values used in the experiments.
  3. [Section V.C, Table II; Section V.B] The claim 'under 6% supervision' is supported by a single random seed and a supervision ratio that appears to be chosen post hoc. The results are nonmonotonic: 7%, 8%, and 10% supervision land in the wrong basin with E_omega around 0.13-0.14, while 3%-6% and 9% succeed. Please report multiple seeds or repeated runs, and provide a principled rule for selecting the supervision ratio. Without this, it is unclear whether the 6% result is a stable property of CEI-PINN or an artifact of a particular random draw.
  4. [Appendix C, Appendix D, Section III.B] The benchmark uses noiseless observations generated by the same physical operator that is used as the PDE constraint, and all physical coefficients in L0 and L1 are fixed. This is a reasonable benchmark, but the paper does not analyze sensitivity to measurement noise or to uncertainty in rho_i, k_theta, q, s, omega_*e, epsilon_n, eta_i, tau, delta_e. Given the branch-selection ambiguity acknowledged in Appendix A, such sensitivity is directly relevant to the claim of identifying eigenfrequencies 'from sparse observations' in practice. Please add a noise/parameter-perturbation study, or explicitly scope the claim to the noiseless, perfectly known-coefficient setting.
minor comments (4)
  1. [Eq. (13)] The residual is printed as '(L0 + L1) phi_theta' with a mismatched parenthesis. Should be '(L0 + L1) phi_theta' or the equivalent with proper parentheses.
  2. [Section III.B] The sentence 'nonzero observations select the target branch and exclude this trivial solution' is too strong: nonzero observations exclude the exactly-zero field at the sampled points, but they do not by themselves exclude all other eigenbranches. The later caveat in Appendix A is more accurate.
  3. [Figure 6] The plot shows a 'Regularization loss' curve, but the regularizer itself is not defined in the main text or appendix. Please either define it or remove the curve.
  4. [Table IX] The '855.3-fold speedup after training' compares a one-shot eigensolver run with a single PINN forward pass, excluding the 47-minute training time. The sentence 'providing an approximately 855.3-fold speedup after training' should be clarified so it is not read as an end-to-end speedup.

Circularity Check

0 steps flagged · score 2.0 of 10

No demonstrated circularity: the eigenfrequency is genuinely inferred from sparse data plus PDE residual; the only concern is an essential but underspecified regularizer that could in principle encode the target.

full rationale

The central inversion is not circular by construction. The reference eigenfrequency and mode field come from an external eigensolver (Appendix C), not from the PINN or from the predicted ω; the sparse observations, boundary conditions, and PDE residual are independent constraints, and ω is a trainable parameter optimized against those constraints (Eqs. (10)-(16), (18)-(19)). Ablations show the PDE loss is necessary (No-PDE stalls at the initial value, Table VIII) and the result is robust across initializations (Table VI), so the recovery is not simply copied from the initial guess or from a self-cited uniqueness theorem. The paper does not invoke any load-bearing self-citation, and no equation defines the target in terms of the fitted parameters or vice versa. The one caveat is the eigenfrequency regularizer Lreg=Rω(ω) in Eq. (15): its functional form is never given, and Table VIII shows that removing it collapses to a degenerate nonphysical solution (ω=1.021+0.045i). If Rω were centered on or near the known reference value, the reported Eω=0.029 could be partly supplied by the regularizer rather than inferred from data and physics. However, the paper provides no equation or text showing that Rω is defined from the target, so this remains an omitted-detail / verifiability concern, not a demonstrated circular step. Overall, the derivation is self-contained apart from this unspecified regularizer, and no circular reduction can be exhibited from the paper's own equations.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests mainly on the assumed physical model, the unverifiable reference solver, and hand-chosen hyperparameters (especially the undefined regularizer). The eigenfrequency itself is the target of inference, not an input parameter, so it is not listed as a free parameter.

free parameters (4)
  • Fourier frequency count n_f = 8
    Selected by ablation (Table III) as best trade-off; n_f=2 and n_f=4 fail to recover the imaginary part (E_i ~ 0.50), n_f=16 increases error.
  • Eigenfrequency regularization R_omega and its weight = unspecified
    A soft regularizer restricting omega to a physically admissible range is introduced in Eq. (15) but its functional form and coefficient are never given; the loss-ablation (Table VIII) shows that removing it collapses the solution to omega=1.021+0.045i.
  • 3-stage epoch schedule and learning rates = 2000/15000/30000/40000 epochs; lr 5e-3/1e-3/1e-5
    Hand-chosen hyperparameters controlling when the PDE residual activates and which parameters are frozen; central to the claimed stability but no sensitivity analysis beyond comparing joint/alternating/three-stage.
  • Random seed = 42
    A single seed is used for sampling observations and initialization; no repeated runs or variance are reported, so the headline numbers could depend on this seed.
assumptions (4)
  • domain assumption The long-wavelength fluid operator (L0+L1) with fixed coefficients accurately describes the ground-state ITG branch.
    Eqs. (2)-(4) are taken as the governing physics; all conclusions are relative to this model. If the model is wrong for real pedestal conditions, the identified eigenfrequency is not the true ITG mode.
  • domain assumption The reference eigenpair generated by the custom eigensolver is the correct ground-truth for evaluation.
    Appendix C states the reference was generated with a custom eigensolver using EAST-provided parameters; the solver and parameters are not publicly available, so ground truth is not independently verifiable.
  • domain assumption Dirichlet boundary conditions phi=0 on inner and outer boundaries of the annulus.
    Eq. (11) imposes g=0; the reference data presumably satisfies them. If the physical boundary conditions differ, the reconstructed mode will be biased.
  • standard math Automatic differentiation and Adam optimization converge to a good minimum of the staged loss.
    The paper relies on AD for derivatives and Adam for training (Section V.B); no formal convergence guarantees are given, and the non-monotonic supervision results (Table II) indicate the optimizer can land in different basins.

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Cite this review

Pith. "Pith review of Physics-Informed Neural Networks for Complex Eigenfrequency Identification and Mode Structure Reconstruction of the Ground-State ITG Branch." pith.science (2026). https://pith.science/paper/VOE6XYS2

@misc{pith2026260801850,
  author       = {Pith},
  title        = {Pith review of: Physics-Informed Neural Networks for Complex Eigenfrequency Identification and Mode Structure Reconstruction of the Ground-State ITG Branch},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VOE6XYS2}},
  note         = {Machine review of arXiv:2608.01850}
}
read the original abstract

Physics-informed neural networks (PINNs) combine sparse observations with physical equations, providing an important approach for modeling complex plasma processes and inferring unknown physical quantities. The steep-gradient pedestal of high-confinement-mode tokamaks is closely linked to plasma confinement and edge transport. Analyzing ion-temperature-gradient (ITG) drift waves in this region requires jointly identifying complex eigenfrequencies and reconstructing two-dimensional complex-valued mode fields. Localized high-frequency oscillations, strong real-imaginary coupling, and nonlinear coupling between the mode field and eigenfrequency challenge PINN representation and joint optimization. To address these challenges, we propose a physics-informed neural framework combining Fourier feature encoding, complex-valued feature propagation, and three-stage training. Under sparse observations and physical constraints, it jointly solves for the complex eigenfrequency and mode field of a representative ground-state ITG branch. Experiments show that the framework accurately recovers the target complex eigenfrequency and two-dimensional complex-valued mode field and outperforms representative PINN baselines. It also provides a basis for analyzing higher-order and multiple-branch drift-wave modes.

Figures

Figures reproduced from arXiv: 2608.01850 by the authors.

Figure 1
Figure 1. Illustration of ITG instability feedback mechanism inverse problem [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Schematic diagram of a PINN framework for solving a simplified 2D [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Overview of CEI-PINN for complex field reconstruction and eigenfrequency identification. Top: model architecture and three-stage training workflow; [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Complex-valued mode-field reconstruction under 6% supervision. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Schematic comparison of the radial plasma profiles in L-mode and [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Weighted-loss convergence in three-stage training. Dashed lines at [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Distribution of converged eigenfrequencies across initializations. The [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

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Reference graph

Works this paper leans on

45 extracted references · 1 linked inside Pith

  1. [1]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,

    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, vol. 378, pp. 686–707, 2019

    2019

  2. [2]

    Physics-informed machine learning,

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, “Physics-informed machine learning,”Nature Reviews Physics, vol. 3, no. 6, pp. 422–440, 2021

    2021

  3. [3]

    Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations,

    M. Raissi, A. Yazdani, and G. E. Karniadakis, “Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations,”Science, vol. 367, no. 6481, pp. 1026–1030, 2020

    2020

  4. [4]

    B-PINNs: Bayesian physics- informed neural networks for forward and inverse PDE problems with noisy data,

    L. Yang, X. Meng, and G. E. Karniadakis, “B-PINNs: Bayesian physics- informed neural networks for forward and inverse PDE problems with noisy data,”Journal of Computational Physics, vol. 425, p. 109913, 2021

    2021

  5. [5]

    Multi-output physics-informed neural networks for forward and inverse pde problems with uncertainties,

    M. Yang and J. T. Foster, “Multi-output physics-informed neural networks for forward and inverse pde problems with uncertainties,” Computer Methods in Applied Mechanics and Engineering, vol. 402, p. 115041, 2022

    2022

  6. [6]

    Physics-informed neural networks for quantum eigenvalue problems,

    H. Jin, M. Mattheakis, and P. Protopapas, “Physics-informed neural networks for quantum eigenvalue problems,” in2022 International Joint Conference on Neural Networks (IJCNN), 2022, pp. 1–8. 8

    2022

  7. [7]

    Solving two-dimensional quantum eigenvalue problems using physics-informed machine learn- ing,

    E. G. Holliday, J. F. Lindner, and W. L. Ditto, “Solving two-dimensional quantum eigenvalue problems using physics-informed machine learn- ing,”arXiv preprint arXiv:2302.01413, 2023

    arXiv 2023

  8. [8]

    Local gyrokinetic collisional theory of the ion-temperature gradient mode,

    B. J. Frei, A. C. D. Hoffmann, and P. Ricci, “Local gyrokinetic collisional theory of the ion-temperature gradient mode,”Journal of Plasma Physics, vol. 88, no. 3, p. 905880304, 2022

    2022

Show all 45 references
  1. [9]

    Self-organized evolution of the internal transport barrier in ion-temperature-gradient driven gyrokinetic turbu- lence,

    S. Wang, Z. Wang, and T. Wu, “Self-organized evolution of the internal transport barrier in ion-temperature-gradient driven gyrokinetic turbu- lence,”Physical Review Letters, vol. 132, no. 6, p. 065106, 2024

    2024

  2. [10]

    Complex structure of turbulence across the ASDEX upgrade pedestal,

    L. A. Leppin, T. Görler, M. Cavedon, M. G. Dunne, E. Wolfrum, F. Jenko, and the ASDEX Upgrade Team, “Complex structure of turbulence across the ASDEX upgrade pedestal,”Journal of Plasma Physics, vol. 89, no. 6, p. 905890605, 2023

    2023

  3. [11]

    Using a local gyrokinetic code to study global ion temperature gradient modes in tokamaks,

    P. A. Abdoul, D. Dickinson, C. M. Roach, and H. R. Wilson, “Using a local gyrokinetic code to study global ion temperature gradient modes in tokamaks,”Plasma Physics and Controlled Fusion, vol. 57, no. 6, p. 065004, 2015

    2015

  4. [12]

    Generalised ballooning theory of two-dimensional tokamak modes,

    ——, “Generalised ballooning theory of two-dimensional tokamak modes,”Plasma Physics and Controlled Fusion, vol. 60, no. 2, p. 025011, 2017

    2017

  5. [13]

    Linear gyrokinetic theory of two- dimensional ion-temperature-gradient mode in tokamaks,

    Y . Qiu, J. Wang, and S. Wang, “Linear gyrokinetic theory of two- dimensional ion-temperature-gradient mode in tokamaks,”Physics of Plasmas, vol. 31, no. 3, p. 032107, 2024

    2024

  6. [14]

    Global theory to understand toroidal drift waves in steep gradient,

    H.-S. Xie and B. Li, “Global theory to understand toroidal drift waves in steep gradient,”Physics of Plasmas, vol. 23, no. 8, p. 082513, 2016

    2016

  7. [15]

    Numerical study of long-wavelength drift- wave instabilities in steep gradients,

    Y . Q. Tao, L. Wang, P. J. Tang, G. S. Xu, N. Yan, P. J. Sun, Q. Q. Yang, X. Lin, and Y . Ye, “Numerical study of long-wavelength drift- wave instabilities in steep gradients,”Plasma Physics and Controlled Fusion, vol. 67, no. 8, p. 085033, 2025

    2025

  8. [16]

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

    S. Wang, X. Yu, and P. Perdikaris, “When and why pinns fail to train: A neural tangent kernel perspective,”Journal of Computational Physics, vol. 449, p. 110768, 2022

    2022

  9. [17]

    Characterizing possible failure modes in physics-informed neural networks,

    A. S. Krishnapriyan, A. Gholami, S. Zhe, R. M. Kirby, and M. W. Mahoney, “Characterizing possible failure modes in physics-informed neural networks,” inAdvances in Neural Information Processing Sys- tems, vol. 34, 2021, pp. 26 548–26 560

    2021

  10. [18]

    Challenges in training PINNs: A loss landscape perspective,

    P. Rathore, W. Lei, Z. Frangella, L. Lu, and M. Udell, “Challenges in training PINNs: A loss landscape perspective,” inProceedings of the 41st International Conference on Machine Learning, ser. Proceedings of Machine Learning Research, vol. 235. PMLR, 2024, pp. 42 159– 42 191

    2024

  11. [19]

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

    M. Tancik, P. P. Srinivasan, B. Mildenhall, S. Fridovich-Keil, N. Ragha- van, U. Singhal, R. Ramamoorthi, J. T. Barron, and R. Ng, “Fourier features let networks learn high frequency functions in low dimensional domains,” inAdvances in Neural Information Processing Systems, vo...

    2020

  12. [20]

    Deep complex networks,

    C. Trabelsi, O. Bilaniuk, Y . Zhang, D. Serdyuk, S. Subramanian, J. F. Santos, S. Mehri, N. Rostamzadeh, Y . Bengio, and C. J. Pal, “Deep complex networks,” inInternational Conference on Learning Representations, 2018

    2018

  13. [21]

    Properties of ion temperature gradient and trapped electron modes in tokamak plasmas with inverted density profiles,

    H. Du, H. Jhang, T. S. Hahm, J. Q. Dong, and Z. X. Wang, “Properties of ion temperature gradient and trapped electron modes in tokamak plasmas with inverted density profiles,”Physics of Plasmas, vol. 24, no. 12, p. 122501, 2017

    2017

  14. [22]

    On drift wave instabilities excited by strong plasma gradients in toroidal plasmas,

    H. Chen and L. Chen, “On drift wave instabilities excited by strong plasma gradients in toroidal plasmas,”Physics of Plasmas, vol. 25, no. 1, p. 014502, 2018

    2018

  15. [23]

    Shear, periodicity, and plasma ballooning modes,

    J. W. Connor, R. J. Hastie, and J. B. Taylor, “Shear, periodicity, and plasma ballooning modes,”Physical Review Letters, vol. 40, no. 6, pp. 396–399, 1978

    1978

  16. [24]

    Finite ballooning angle effects on ion temperature gradient driven mode in gyrokinetic flux tube simulations,

    R. Singh, S. Brunner, R. Ganesh, and F. Jenko, “Finite ballooning angle effects on ion temperature gradient driven mode in gyrokinetic flux tube simulations,”Physics of Plasmas, vol. 21, no. 3, p. 032115, 2014

    2014

  17. [25]

    Saad,Numerical Methods for Large Eigenvalue Problems, revised edition ed

    Y . Saad,Numerical Methods for Large Eigenvalue Problems, revised edition ed. Philadelphia, PA, USA: SIAM, 2011

    2011

  18. [26]

    R. B. Lehoucq, D. C. Sorensen, and C. Yang,ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. Philadelphia, PA, USA: SIAM, 1998

    1998

  19. [27]

    Unconventional ballooning structures for toroidal drift waves,

    H.-S. Xie and Y . Xiao, “Unconventional ballooning structures for toroidal drift waves,”Physics of Plasmas, vol. 22, no. 9, p. 090703, 2015

    2015

  20. [28]

    Multiple ion tem- perature gradient driven modes in transport barriers,

    M. K. Han, Z.-X. Wang, J. Q. Dong, and H. Du, “Multiple ion tem- perature gradient driven modes in transport barriers,”Nuclear Fusion, vol. 57, no. 4, p. 046019, 2017

    2017

  21. [29]

    Threshold for the destabilisation of the ion-temperature-gradient mode in magnetically confined toroidal plasmas,

    A. Zocco, P. Xanthopoulos, H. Doerk, J. W. Connor, and P. Helander, “Threshold for the destabilisation of the ion-temperature-gradient mode in magnetically confined toroidal plasmas,”Journal of Plasma Physics, vol. 84, no. 1, p. 715840101, 2018

    2018

  22. [30]

    Scientific machine learning through physics-informed neu- ral networks: Where we are and what’s next,

    S. Cuomo, V . S. Di Cola, F. Giampaolo, G. Rozza, M. Raissi, and F. Piccialli, “Scientific machine learning through physics-informed neu- ral networks: Where we are and what’s next,”Journal of Scientific Computing, vol. 92, no. 3, p. 88, 2022

    2022

  23. [31]

    Understanding and mitigating gradient flow pathologies in physics-informed neural networks,

    S. Wang, Y . Teng, and P. Perdikaris, “Understanding and mitigating gradient flow pathologies in physics-informed neural networks,”SIAM Journal on Scientific Computing, vol. 43, no. 5, pp. A3055–A3081, 2021

    2021

  24. [32]

    Leveraging physics-informed neural computing for transport simulations of nuclear fusion plasmas,

    J. Seo, I. H. Kim, and H. Nam, “Leveraging physics-informed neural computing for transport simulations of nuclear fusion plasmas,”Nuclear Engineering and Technology, vol. 56, no. 12, pp. 5396–5404, 2024

    2024

  25. [33]

    Reconstruction of plasma equilibrium and separatrix using convolutional physics-informed neural operator,

    M. Bonotto, D. Abate, and L. Pigatto, “Reconstruction of plasma equilibrium and separatrix using convolutional physics-informed neural operator,”Fusion Engineering and Design, vol. 200, p. 114193, 2024

    2024

  26. [34]

    Grad–shafranov equilibria via data-free physics informed neural networks,

    B. Jang, A. A. Kaptanoglu, R. Gaur, S. Pan, M. Landreman, and W. Dorland, “Grad–shafranov equilibria via data-free physics informed neural networks,”Physics of Plasmas, vol. 31, no. 3, p. 032510, 2024

    2024

  27. [35]

    On the spectral bias of neural networks,

    N. Rahaman, A. Baratin, D. Arpit, F. Draxler, M. Lin, F. Hamprecht, Y . Bengio, and A. Courville, “On the spectral bias of neural networks,” in Proceedings of the 36th International Conference on Machine Learning, ser. Proceedings of Machine Learning Research, vol. 97. PMLR, 2...

    2019

  28. [36]

    Frequency principle: Fourier analysis sheds light on deep neural networks,

    Z.-Q. J. Xu, Y . Zhang, T. Luo, Y . Xiao, and Z. Ma, “Frequency principle: Fourier analysis sheds light on deep neural networks,”Communications in Computational Physics, vol. 28, no. 5, pp. 1746–1767, 2020

    2020

  29. [37]

    Extended physics-informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equa- tions,

    A. D. Jagtap and G. E. Karniadakis, “Extended physics-informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equa- tions,”Communications in Computational Physics, vol. 28, no. 5, pp. 200...

    2002

  30. [38]

    A comprehensive study of non-adaptive and residual-based adaptive sampling for physics- informed neural networks,

    C. Wu, M. Zhu, Q. Tan, Y . Kartha, and L. Lu, “A comprehensive study of non-adaptive and residual-based adaptive sampling for physics- informed neural networks,”Computer Methods in Applied Mechanics and Engineering, vol. 403, p. 115671, 2023

    2023

  31. [39]

    Complex-valued physics-informed machine learning for efficient solving of quintic non- linear schrödinger equations,

    L. Zhang, M. Du, X. Bai, Y . Chen, and D. Zhang, “Complex-valued physics-informed machine learning for efficient solving of quintic non- linear schrödinger equations,”Physical Review Research, vol. 7, no. 1, p. 013164, 2025

    2025

  32. [40]

    The two-dimensional kinetic ballooning theory for ion temperature gradient mode in tokamak,

    T. Xie, Y . Z. Zhang, S. M. Mahajan, S. L. Hu, H. He, and Z. Y . Liu, “The two-dimensional kinetic ballooning theory for ion temperature gradient mode in tokamak,”Physics of Plasmas, vol. 24, no. 10, p. 102506, 2017

    2017

  33. [41]

    Symmetry breaking of ion temperature gradient mode structure: From local to global analysis,

    Z. X. Lu, E. Fable, W. A. Hornsby, C. Angioni, A. Bottino, P. Lauber, and F. Zonca, “Symmetry breaking of ion temperature gradient mode structure: From local to global analysis,”Physics of Plasmas, vol. 24, no. 4, p. 042502, 2017

    2017

  34. [42]

    Global itg eigenmodes: From ballooning angle and radial shift to reynolds stress and nonlinear saturation,

    J. Zielinski, M. Becoulet, A. I. Smolyakov, X. Garbet, G. T. A. Huijsmans, P. Beyer, and S. Benkadda, “Global itg eigenmodes: From ballooning angle and radial shift to reynolds stress and nonlinear saturation,”Physics of Plasmas, vol. 27, no. 7, p. 072507, 2020

    2020

  35. [43]

    A dive into spectral inference net- works: Improved algorithms for self-supervised learning of continuous spectral representations,

    J. Wu, S. F. Wang, and P. Perdikaris, “A dive into spectral inference net- works: Improved algorithms for self-supervised learning of continuous spectral representations,”Applied Mathematics and Mechanics, vol. 44, no. 7, pp. 1199–1224, 2023

    2023

  36. [44]

    On the eigenvector bias of fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks,

    S. Wang, H. Wang, and P. 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, vol. 384, p. 113938, 2021

    2021

  37. [45]

    Complex-valued neural networks: A comprehensive survey,

    C. Y . Lee, H. Hasegawa, and S. C. Gao, “Complex-valued neural networks: A comprehensive survey,”IEEE/CAA Journal of Automatica Sinica, vol. 9, no. 8, pp. 1406–1426, 2022. 9 APPENDIXA PHYSICS OF THEITG INVERSEPROBLEM At the heart of magnetic-confinement fusion is the ability t...

    2022

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.