REVIEW 4 major objections 5 minor 37 references
Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs
T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read A separated-variable spectral neural network with learnable Fourier frequencies solves high-frequency PDEs to orders-of-magnitude better accuracy than standard physics-informed networks, using 90% fewer parameters.
desk verdict SV-SNN is a competent Fourier-feature + separable-ansatz PINN whose benchmarks are all in-distribution; the general high-frequency claim is overstated but the architecture deserves refereeing. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the separated-variable spectral ansatz: $u(x,t)=\sum_{n=1}^{N} c_n \left(\prod_{j=1}^{d} \Phi_n^{(j)}(x_j)\right) T_n(t)$, where each $\Phi_n^{(j)}(x_j)=\sum_{k=1}^{K_j}[a_{n,k}^{(j)}\sin(w_{n,k}^{(j)}x_j)+b_{n,k}^{(j)}\cos(w_{n,k}^{(j)}x_j)]+\beta_n^{(j)}$ and the frequencies $w_{n,k}^{(j)}$ are learnable, initialized by a three-level sampler centered on the problem's characteristic frequency. This object carries the argument by making high-frequency spatial structure explicit in the basis and by letting spatial derivatives be computed analytically from the Fourier sum, so the PDE residual does not inherit automatic-differentiation noise. A second mechanism is the effective rank $r_{\mathrm{eff}}(\eta)$, defined by the cumulative singular-value energy of the Jacobian matrices; since neural-tangent-kernel eigenvalues are squares of Jacobian singular values, a decaying singular spectrum is quadratically amplified into slow high-frequency convergence, and the paper uses this to explain why the new architecture avoids the collapse seen in standard physics-informed networks.
What would settle it
Take the two-dimensional Helmholtz equation with $\kappa=48\pi$ but withhold the wavenumber, estimate the characteristic frequency solely from the Fourier transform of the source term, and train the network with an intentionally off-center initial frequency such as $w_{\mathrm{char}}=40\pi$; if the relative $L^2$ error rises to the same order as a standard physics-informed network, then the adaptive-frequency mechanism is not recovering the spectrum on its own.
Extended reading notes
Core claim
The central claim is that the separated-variable adaptive Fourier ansatz removes spectral bias rather than merely postponing it. Writing the trial solution as $u(x,t)=\sum_{n=1}^{N} c_n \prod_{j=1}^{d} \Phi_n^{(j)}(x_j)\, T_n(t)$, with each $\Phi_n^{(j)}$ a learnable-frequency sine/cosine series, gives the network a built-in Fourier vocabulary that can be initialized near the problem's characteristic frequency and refined during training. The paper pairs this architecture with an effective-rank analysis of the Jacobian matrices of the initial-condition, boundary-condition, and PDE-residual losses: because neural-tangent-kernel eigenvalues are the squares of Jacobian singular values, the rapid singular-value decay of a vanilla network is quadratically amplified into slow convergence of high-frequency error modes, while the proposed network's Jacobians retain a much heavier tail of active directions. Across heat, Helmholtz, Poisson, nonlinear elliptic, Taylor-Green, and steady Navier-Stokes benchmarks, the paper reports relative $L^2$ errors one to three orders of magnitude below standard physics-informed training, with parameter counts reduced by 90% or more.
Load-bearing premise
The entire method rests on the assumption that the true solution can be approximated by a handful of products of univariate sine and cosine sums with adjustable frequencies, and that the user can supply the problem's dominant frequency in advance; every benchmark sets that frequency from the known exact solution, so behavior when the frequency spectrum is unknown is not tested.
Editorial extensions
If this is right
- High-frequency wave-type PDEs, such as heat with $\kappa=100\pi$ and Helmholtz with $\kappa=48\pi$, become solvable by physics-informed training at accuracies that standard networks never reach.
- The 90%-plus reduction in parameter count means the method is plausible to run on a single GPU within a few thousand training epochs, rather than the tens of thousands needed by the compared physics-informed variants.
- The effective-rank criterion gives a concrete design principle: architectures should keep Jacobian singular-value spectra from collapsing if they are to avoid spectral bias.
- Because the same ansatz handles nonlinear elliptic equations and steady Navier-Stokes flows, the separation-of-variables structure is claimed to transfer beyond linear problems despite its classical limitations.
- The ablation study shows that mode count is a practical tuning knob: a single mode fails on the double-cylinder Navier-Stokes case, while four modes achieve small errors with near-linear growth in training time.
Reading between the lines
- A natural next test is to remove the known-frequency assumption: initialize the frequency sampling from a Fourier analysis of the initial or source data, and check whether the accuracy advantage survives when the estimated characteristic frequency is deliberately wrong.
- The same effective-rank argument predicts that any architecture that flattens the Jacobian singular spectrum, such as orthogonal-polynomial spectral features using Chebyshev or Legendre bases, would show analogous high-frequency gains, generalizing the mechanism beyond Fourier dictionaries.
- The low-rank separated form suggests an adaptive variant that grows the number of modes or frequencies during training when the effective rank saturates, which the paper does not implement.
- A useful follow-up would compare time-to-accuracy at matched error levels, since the headline speedup combines a tenfold epoch reduction with the architecture change.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces Separated-Variable Spectral Neural Networks (SV-SNN), in which the PDE solution is represented as a finite sum of products of univariate Fourier-feature networks for spatial coordinates and shallow fully-connected networks for time. The method uses learnable frequencies, a three-level frequency sampling strategy seeded by a user-specified characteristic frequency, and analytical spatial differentiation with automatic temporal differentiation. The authors report large reductions in relative L2 error and parameter count versus a vanilla PINN on heat, Helmholtz, Poisson, nonlinear elliptic, Taylor-Green vortex, and steady Navier-Stokes benchmarks, and they propose an effective-rank analysis of Jacobian matrices as a diagnostic for spectral bias and parameter-space collapse.
Significance. If the claimed capability held broadly, SV-SNN would be a valuable contribution to physics-informed learning for high-frequency PDEs: the separated-variable form reduces the parameter count from O(K^d) to O(dK), the hybrid differentiation strategy is a practical way to avoid automatic-differentiation errors for oscillatory spatial modes, and the effective-rank diagnostic is inexpensive to compute. The paper is also explicitly machine-checkable in its experimental setup and discloses the absence of code. However, the central claim is not yet supported: every benchmark exact solution lies exactly inside the ansatz, and the characteristic frequency is taken from the known solution, initial condition, or wavenumber. The theoretical analysis is descriptive rather than a proof of approximation power. The significance is therefore conditional on additional out-of-class tests and a more careful baseline comparison.
major comments (4)
- [Section 4, Tables 3-11, Eq. (2.9)] All benchmark exact solutions in Section 4 are contained in the SV-SNN ansatz of Eq. (2.9) (sums of products of univariate Fourier sums), and the characteristic frequency in Eq. (2.12)-(2.13) is set from the known solution, initial condition, source, or wavenumber in every case. For instance, the heat solutions are e^{-t}\sin(\kappa x), the Helmholtz solutions are \sin(\kappa x)\sin(\kappa y), the nonlinear elliptic solution is (x+y)\cos(10x)\sin(10y), the complex-source Poisson solution is \sin(15x^2)+\sin(15y^2), and the Taylor-Green and double-cylinder Navier-Stokes cases use manufactured separated products. The reported 1-3 orders-of-magnitude improvements therefore demonstrate fitting of functions in the model class rather than general high-frequency PDE solving. I request at least one nontrivial benchmark whose exact solution is non-separable (e.g., containing mixed terms such as \sin(10x+3y) or a solution not decomposable into axis-aligned products) and for which w_char is estimated from the problem data (initial condition/forcing) rather than read off the exact answer. Alternatively, the abstract and introduction should be reframed to claim only separable or approximately separable high-frequency problems.
- [Section 3.2, Eqs. (3.8)-(3.19)] The 'theoretical framework' is presented as a contribution, but it is a post-hoc diagnostic, not a proof. The effective-rank definition in Eq. (3.10) is descriptive, the relationship in Eq. (3.15) is standard Jacobian-NTK algebra, and the sentence 'we speculate that the physical mechanism...' makes the speculative character explicit. No approximation theorem or convergence rate is provided for the finite-rank separated ansatz with the small N and K used in the experiments. Since the central claim depends on the ansatz being able to represent the target solutions, the paper should either state an approximation bound for a well-defined function class (e.g., functions with finite separation rank and bandlimited Fourier content) or explicitly label the SVD analysis as an empirical diagnostic rather than a theoretical foundation.
- [Tables 3-11 and Table 2] The PINN baseline is not described sufficiently: no architecture (depth, width, activation), no learning-rate schedule, no number of collocation points, no training epochs are reported for the vanilla PINN used in Tables 3-11. The text in Section 4 states that 10 random-seed experiments were run and Table 2 reports averages, but no error bars or standard deviations are given in Tables 3-11. Without these details, the comparison is not reproducible and the magnitude of the claimed improvement cannot be assessed. I request full baseline hyperparameters and error statistics (mean and standard deviation over seeds) for all tables.
- [Tables 15-16 and Abstract] The training-time claim in the abstract ('60×') is derived from a comparison in which SV-SNN runs for 5,000 epochs while the competing PINN-based methods run for 50,000 epochs (Table 15 caption). This conflates convergence speed with per-epoch cost and is not a fair 'training time' comparison unless time-to-accuracy is measured. I request a matched-accuracy comparison (e.g., wall-clock time to reach a fixed relative L2 error) or at minimum a per-epoch cost comparison with the same number of optimization steps. As written, the 60x speedup claim is misleading.
minor comments (5)
- [Appendix] There is an unresolved cross-reference: 'We present the specific training algorithm for SV-SNN in Appendix ??' appears at the end of Section 3.1; the appendix is not included in the manuscript.
- [Sections 4.1-4.2] Several sentences are truncated mid-number, e.g., 'reducing 91', 'improving 97', 'improving 94', 'improving 98', 'approximately 96', and 'improving 94.9'. These appear to be intended percentage reductions but the percent signs and rest of the sentences are missing. Please proofread the final text.
- [Section 4.1.1, Eq. (4.7) and surrounding text] The notation 'N (20π, 202)' should be written as 'N(20π, 20^2)' to denote a Gaussian with variance 20^2; the same issue appears in the frequency sampling description in Section 2.3.3 and in the ablation tables.
- [Tables 2 and 15] There is an inconsistency for the Helmholtz κ=24π problem: Table 2 reports ReL2E=1.62e-2 with training time 205.76s, while Table 15 reports AvgReL2E=1.27e-2 with training time 132.54s for the same problem. The difference may be due to different network settings or random seeds; please clarify which configuration is used in each table.
- [Section 4.1, Eq. (4.1)] The relative L2 error is defined as an integral over the domain, but the actual computation appears to be on discrete test points. Please specify the quadrature/discretization used in the reported ReL2E values.
Circularity Check
No circular derivation; in-class benchmark design limits generalization but does not make the solver's outputs true by construction.
full rationale
The paper's derivation chain is self-contained: the ansatz (Eq. 2.9) is a separated-variable Fourier spectral representation, the loss (Eqs. 3.4-3.7) is the PDE residual plus initial/boundary mismatch, and training minimizes that loss. The only hyperparameter that could be suspected of being fitted to the answer is the characteristic frequency wchar in Eqs. (2.12)-(2.13), but in every experiment it is read off from known problem data, not from the target solution values: e.g., 'Since initial condition frequency is 20π, we define characteristic frequency wchar = 20π' (Sec. 4.1.1), and for Helmholtz it is taken from the given wavenumber. Moreover, the ablation in Table 13 shows that for the κ=24π Helmholtz problem the best wchar is 20π, not the true 24π, so the reported accuracy is not forced by using the exact solution frequency. No fitted parameter is later relabeled as a prediction: the effective-rank values reff in Tables 3-11 are computed post hoc from the trained Jacobians and serve as descriptive diagnostics, not as inputs used to set the architecture. It is true that every manufactured exact solution in Section 4 lies inside the ansatz class (e.g., u_exact = e^{-t} sin(20πx) has N=1, Φ=sin(20πx), T=e^{-t}), but that is an in-class test design typical of method-of-manufactured-solutions benchmarks; the network still has to optimize unknown coefficients from the initial, boundary, and residual constraints, so the results are not true by construction. No load-bearing self-citations are present; the references are to external work on spectral bias, Fourier features, and PINNs. The central claim of general high-frequency PDE capability for non-separable solutions is untested, which is a scope/correctness limitation, not a circularity.
Assumptions & free parameters
free parameters (6)
- Characteristic frequency w_char =
varies: 20π,100π,500π for heat; κ=24π,48π for Helmholtz; 7π,15 for Poisson
- Gaussian variance σ2 for frequency sampling =
202 for most cases, 182 optimal in Table 14
- Number of modes N =
4, 6, 8, or 10 per test case
- Number of Fourier features K per direction =
16, 32, 40, 50, or 64
- Frequency sampling range [w_min, w_max] =
e.g., [1,20π] and [20π,40π] for heat; similar ranges for others
- Effective-rank threshold η =
0.99
assumptions (4)
- domain assumption Finite-rank separated-variable ansatz u = Σ c_n Φ_n(x)T_n(t) (Eqs. 2.6, 2.9)
- domain assumption Characteristic frequency w_char is known or computable from problem data (Eq. 2.12)
- standard math NTK linearization of residual dynamics (Eqs. 3.17-3.19)
- standard math Singular values of Jacobian blocks determine NTK eigenvalues as λ_i = σ_i² (Eq. 3.15)
Cite this review
Pith. "Pith review of Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs." pith.science (2026). https://pith.science/paper/MQ7O7KZ3
@misc{pith2026250800628,
author = {Pith},
title = {Pith review of: Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs},
year = {2026},
howpublished = {\url{https://pith.science/paper/MQ7O7KZ3}},
note = {Machine review of arXiv:2508.00628}
}
read the original abstract
Solving high-frequency oscillatory partial differential equations (PDEs) is a critical challenge in scientific computing, with applications in fluid mechanics, quantum mechanics, and electromagnetic wave propagation. Traditional physics-informed neural networks (PINNs) suffer from spectral bias, limiting their ability to capture high-frequency solution components. We introduce Separated-Variable Spectral Neural Networks (SV-SNN), a novel framework that addresses these limitations by integrating separation of variables with adaptive spectral methods. Our approach features three key innovations: (1) decomposition of multivariate functions into univariate function products, enabling independent spatial and temporal networks; (2) adaptive Fourier spectral features with learnable frequency parameters for high-frequency capture; and (3) theoretical framework based on singular value decomposition to quantify spectral bias. Comprehensive evaluation on benchmark problems including Heat equation, Helmholtz equation, Poisson equations and Navier-Stokes equations demonstrates that SV-SNN achieves 1-3 orders of magnitude improvement in accuracy while reducing parameter count by over 90\% and training time by 60\%. These results establish SV-SNN as an effective solution to the spectral bias problem in neural PDE solving. The implementation will be made publicly available upon acceptance at https://github.com/xgxgnpu/SV-SNN.
Figures
Figures from the paper (11 more)
Reference graph
Works this paper leans on
-
[1]
Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
Maziar Raissi, Paris Perdikaris, and George Em 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
2019
-
[2]
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
2021
-
[3]
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
2022
-
[4]
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
2021
-
[5]
Ameya D Jagtap and George Em Karniadakis. Extended physics-informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations. Communications in Computational Physics, 28(5):2002–2041, 2020
work page 2002
-
[6]
Physics-informed neural networks (pinns) for fluid mechanics: A review
Shengze Cai, Zhiping Mao, Zhicheng Wang, Minglang Yin, and George Em Karniadakis. Physics-informed neural networks (pinns) for fluid mechanics: A review. Acta Mechanica Sinica, 37(12):1727–1738, 2021
work page 2021
-
[7]
Jaideep Pathak, Shashank Subramanian, Peter Harrington, Sanjeev Raja, Ashesh Chattopadhyay, Morteza Mardani, Thorsten Kurth, David Hall, Zongyi Li, Kamyar Azizzadenesheli, et al. Fourcastnet: A global data-driven high- resolution weather model using adaptive fourier neural operators. arXiv preprint arXiv:2202.11214, 2022
arXiv 2022
-
[8]
Weiwei He, Qingshan Jia, Siyuan Wang, and Hanbo Zhang. Multi-level physics informed deep learning for solving partial differential equations in computational structural mechanics. Communications Engineering, 5(1):303, 2024
work page 2024
Show all 37 references
-
[9]
Fe-pirbn: Feature-enhanced physics- informed radial basis neural networks for solving high-frequency electromagnetic scattering problems
Yizheng Chen, Lu Lu, George Em Karniadakis, and Luca Dal Negro. Fe-pirbn: Feature-enhanced physics- informed radial basis neural networks for solving high-frequency electromagnetic scattering problems. Journal of Computational Physics, 527:113772, 2025
2025
-
[10]
Physics-informed neural wavefields with gabor basis functions.Neural Networks, 177:106380, 2024
Jian Huang and Tariq Alkhalifah. Physics-informed neural wavefields with gabor basis functions.Neural Networks, 177:106380, 2024
2024
-
[11]
A physics-informed deep learning framework for inversion and surrogate modeling in solid geophysics
Ehsan Haghighat, Maziar Raissi, Adrian Moure, Hector Gomez, and Ruben Juanes. A physics-informed deep learning framework for inversion and surrogate modeling in solid geophysics. Computer Methods in Applied Mechanics and Engineering, 379:113741, 2021
2021
-
[12]
Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning
Siddhartha Mishra and Roberto Molinaro. Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning. Acta Numerica, 33:87–113, 2024
2024
-
[13]
A versatile framework to solve the helmholtz equation using physics-informed neural networks
Chao Song, Tariq Alkhalifah, and Umair Bin Waheed. A versatile framework to solve the helmholtz equation using physics-informed neural networks. Geophysical Journal International, 228(3):1750–1762, 2022
2022
-
[14]
Deep neural helmholtz operators for 3-d elastic wave propagation and inversion
James Smith, Siamak Azizzadenesheli, and Zachary Ross. Deep neural helmholtz operators for 3-d elastic wave propagation and inversion. Geophysical Journal International, 239(3):1469–1492, 2024. 32
2024
-
[15]
Physics-informed neural networks for modal wave field predictions in 3d room acoustics
Alexander Riedlinger, Gonzalo Ramos, Jesus Carbajo, Steffen Marburg, and Gerhard Müller. Physics-informed neural networks for modal wave field predictions in 3d room acoustics. Applied Sciences, 15(2):939, 2025
2025
-
[16]
A neural multigrid solver for helmholtz equations with high wavenumber and heteroge- neous media
Chen Cui and Kai Jiang. A neural multigrid solver for helmholtz equations with high wavenumber and heteroge- neous media. arXiv preprint arXiv:2404.02493, 2024
2024 arXiv
-
[17]
Nsno: Neumann series neural operator for solving helmholtz equations in inhomogeneous medium
Tong Wang, Zongyi Li, Nikola Kovachki, Jazmin Azizzadenesheli, Anima Anandkumar, and Stanley Osher. Nsno: Neumann series neural operator for solving helmholtz equations in inhomogeneous medium. Journal of Systems Science and Complexity, 37(4):1294–1320, 2024
2024
-
[18]
The helmholtz equation with uncertainties in the wavenumber
Olivier Sété. The helmholtz equation with uncertainties in the wavenumber. arXiv preprint arXiv:2209.14740, 2022
2022 arXiv
-
[19]
Training-image based geostatistical inversion using a spatial generative adversarial neural network
Eric Laloy, Romain Hérault, Diederik Jacques, and Niklas Linde. Training-image based geostatistical inversion using a spatial generative adversarial neural network. Water Resources Research, 54(1):381–406, 2018
2018
-
[20]
On the spectral bias of neural networks
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. In International Conference on Machine Learning, pages 5301–5310. PMLR, 2019
-
[21]
On the eigenvector bias of fourier feature networks: From regression to solving multi-scale pdes with physics-informed neural networks
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
2021
-
[22]
Frequency principle: Fourier analysis sheds light on deep neural networks
Zhi-Qin John Xu, Yaoyu Zhang, Tao Luo, Yanyang Xiao, and Zheng Ma. Frequency principle: Fourier analysis sheds light on deep neural networks. Communications in Computational Physics, 28(5):1746–1767, 2020
2020
-
[23]
On understanding and overcoming spectral biases of deep neural network learning methods for solving pdes
Zhi-Qin John Xu, Lulu Zhang, and Wei Cai. On understanding and overcoming spectral biases of deep neural network learning methods for solving pdes. Journal of Computational Physics, 530:113905, 2025
2025
-
[24]
Understanding and mitigating gradient flow pathologies in physics-informed neural networks
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
2021
-
[25]
Characterizing possible failure modes in physics-informed neural networks
Aditi S Krishnapriyan, Amir Gholami, Shandian Zhe, Robert M Kirby, and Michael W Mahoney. Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34:26548–26560, 2021
2021
-
[26]
Fourier features let networks learn high frequency functions in low dimensional domains
Matthew Tancik, Pratul P Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan T Barron, and Ren Ng. Fourier features let networks learn high frequency functions in low dimensional domains. InAdvances in Neural Informatio...
2020
-
[27]
Diminishing spectral bias in physics-informed neural networks using spatially-adaptive fourier feature encoding
Yizheng Wang, Shibo Han, Pradeep Chintalapudi, Guang Geng, Riccardo Conti, Matteo Ripepi, Ercan M Dede, and J Andrzej Domaradzki. Diminishing spectral bias in physics-informed neural networks using spatially-adaptive fourier feature encoding. Neural Networks, 182:106826, 2024
2024
-
[28]
Spectrum-informed multistage neural networks: Multiscale function approximators of machine precision
Jakin Ng, Yongji Wang, and Ching-Yao Lai. Spectrum-informed multistage neural networks: Multiscale function approximators of machine precision. In International Conference on Machine Learning, pages 37842–37865. PMLR, 2024
2024
-
[29]
Binary structured physics-informed neural networks for solving equations with rapidly changing solutions
Yanzhi Liu, Ruifan Wu, and Ying Jiang. Binary structured physics-informed neural networks for solving equations with rapidly changing solutions. Journal of Computational Physics, 515:113341, 2024
2024
-
[30]
Frequency-adaptive multi-scale deep neural networks
Jizu Huang, Rukang You, and Tao Zhou. Frequency-adaptive multi-scale deep neural networks. Computer Methods in Applied Mechanics and Engineering, 437:117751, 2025
2025
-
[31]
Fourier neural operator for parametric partial differential equations
Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895, 2020
2010 arXiv
-
[32]
Spectral bias and task-model alignment explain generalization in kernel regression and infinitely wide neural networks
Abdulkadir Canatar, Blake Bordelon, and Cengiz Pehlevan. Spectral bias and task-model alignment explain generalization in kernel regression and infinitely wide neural networks. Nature Communications, 12(1):2914, 2021
2021
-
[33]
Spectral bias in practice: The role of function frequency in generalization
Sara Fridovich-Keil, Raphael Gontijo-Lopes, and Rebecca Roelofs. Spectral bias in practice: The role of function frequency in generalization. arXiv preprint arXiv:2110.02424, 2021
2021 arXiv
-
[34]
On the activation function dependence of the spectral bias of neural networks
Qingguo Hong, Dan Lelescu, and Leo G Rebholz. On the activation function dependence of the spectral bias of neural networks. arXiv preprint arXiv:2208.04924, 2022
2022 arXiv
-
[35]
Spectral and finite difference solutions of the burgers equation
Claude Basdevant, Michel Deville, Pierre Haldenwang, Jean Michel Lacroix, Jalil Ouazzani, Roger Peyret, Paolo Orlandi, and Anthony T Patera. Spectral and finite difference solutions of the burgers equation. Computers & Fluids, 14(1):23–41, 1986. 33
1986
-
[36]
Message passing neural pde solvers
Johannes Brandstetter, Daniel Worrall, and Max Welling. Message passing neural pde solvers. arXiv preprint arXiv:2202.03376, 2023
2023 arXiv
-
[37]
Spectral methods in MATLAB
Lloyd N Trefethen. Spectral methods in MATLAB. SIAM, 2000. 34 Algorithm 1 Separated-Variable Spectral Neural Network (SV-SNN) Training Algorithm Require: Partial differential equation F [u] = 0 , initial condition I[u](x, 0) = g0(x), boundary condition B[u] = gB(x, t) Require:...
2000
Reviewed August 6, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.