arxiv: 2605.12997 · v1 · submitted 2026-05-13 · 💻 cs.LG

Recognition: unknown

Frequency Bias and OOD Generalization in Neural Operators under a Variable-Coefficient Wave Equation

An Luo, Runlong Xie

Authors on Pith no claims yet

Pith reviewed 2026-05-14 19:43 UTC · model grok-4.3

classification 💻 cs.LG

keywords neural operatorsFNODeepONetOOD generalizationfrequency biaswave equationPDE surrogatedistribution shift

0 comments

The pith

FNO shows sharp error jumps on unseen high frequencies in wave equations while DeepONet degrades more gradually.

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

The paper tests two neural operator architectures on a one-dimensional variable-coefficient wave equation to measure how they generalize when input frequency or coefficient smoothness changes from the training distribution. FNO maintains low error under smoothness shifts but suffers a sharp rise in error on high-frequency inputs outside its training range. DeepONet starts with higher overall error yet shows only mild additional degradation under the same frequency shift. These patterns are traced to differences in how each model encodes and processes frequency content. The work matters because neural operators are intended as fast surrogates for repeated PDE solves across new inputs, yet frequency shifts are common in physical applications.

Core claim

Under structured out-of-distribution settings that vary input frequency independently of coefficient smoothness, the Fourier Neural Operator exhibits a sharp increase in error on unseen high-frequency inputs, while the Deep Operator Network shows milder degradation despite its higher baseline error. Both architectures remain stable under shifts in coefficient smoothness, with FNO achieving lower error in that regime. The performance differences arise from each architecture's distinct representation of and response to variations in frequency structure.

What carries the argument

Structured OOD test settings on the variable-coefficient wave equation that separately control input frequency content and coefficient smoothness to isolate architectural biases in FNO versus DeepONet.

If this is right

FNO requires training data or architectural modifications that cover broader frequency ranges to avoid abrupt failure modes.
DeepONet may be more suitable than FNO for wave problems where input frequencies can vary widely after deployment.
Operator learning pipelines must explicitly account for frequency content when constructing training distributions to achieve reliable generalization.
The choice between spectral and branch-trunk architectures directly influences robustness to frequency shifts in physics-informed operator models.

Where Pith is reading between the lines

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

Similar frequency-shift tests on other PDE families such as advection or elasticity could reveal whether the observed bias pattern generalizes beyond waves.
Explicit frequency augmentation during training might reduce the gap between in-distribution accuracy and OOD performance for spectral operators.
The results suggest that representation bias in neural operators can be diagnosed by measuring error sensitivity to isolated input features like frequency.
Extending the analysis to multi-dimensional or nonlinear wave equations would test whether the one-dimensional findings scale to more realistic settings.

Load-bearing premise

That independently varying input frequency and coefficient smoothness produces distribution shifts representative of those encountered in practical PDE applications.

What would settle it

A controlled experiment that trains FNO on a dataset spanning the full target frequency range and then measures whether the sharp error increase on the highest frequencies disappears.

Figures

Figures reproduced from arXiv: 2605.12997 by An Luo, Runlong Xie.

**Figure 1.** Figure 1: An overview of our experimental pipeline for operator learning under the variable-coefficient wave equation. Input functions consist of an initial displacement field u0(x) and a spatially varying wave-speed coefficient c(x). A finite-difference solver generates the terminal solution u(x, T) at a fixed time horizon, forming supervised training data. FNO and DeepONet are then trained to approximate the mappi… view at source ↗

**Figure 2.** Figure 2: Comparison of relative L2 errors for FNO and DeepONet across Validation, in-distribution (ID), high-frequency OOD, and wave-speed smoothness OOD settings. FNO achieves lower error on Validation and ID datasets, but exhibits substantially larger degradation under high-frequency OOD, whereas DeepONet shows comparatively smoother degradation behavior [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: OOD-frequency degradation curves for FNO and DeepONet across all test samples. X-axis shows mode index k, Y-axis shows mean squared modal error. Left: ID spectral error; Right: OOD-frequency spectral error. FNO errors accumulate in multiple mid-to-high frequency modes under OOD-frequency conditions, while DeepONet exhibits smoother error growth. This figure provides a detailed view of how prediction erro… view at source ↗

**Figure 4.** Figure 4: Representative prediction examples comparing FNO and DeepONet under different OOD conditions. Top row shows predicted terminal waveforms versus ground truth for ID, OOD-frequency, and OOD-smoothness inputs. Bottom row shows the corresponding pointwise prediction errors. The figure illustrates that FNO suffers larger local distortions and phase errors under high-frequency OOD, whereas DeepONet maintains smo… view at source ↗

**Figure 6.** Figure 6: Ablation study on the number of retained Fourier modes in FNO. X-axis indicates retained mode count (8, 16, 32), and Y-axis shows relative L2 error for ID (dashed line) and OODfrequency (bars). While ID performance remains stable and optimal at intermediate modes, OOD-frequency error increases with more retained modes, demonstrating that simply increasing spectral capacity does not prevent OOD degradati… view at source ↗

**Figure 7.** Figure 7: Additional OOD-frequency prediction examples illustrating representative failure patterns under high-frequency initial conditions. Each sample includes the initial condition, spatial coefficient field, terminal prediction, and residual distribution. D.2. Additional Smoothness OOD Examples [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗

**Figure 8.** Figure 8: Additional OOD-smoothness prediction examples under shifted coefficient smoothness. Each panel follows the same convention as [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗

**Figure 9.** Figure 9: Additional spectral error curves across validation, ID, OOD-frequency, and OOD-smoothness evaluation splits. The curves provide a frequency-domain view of how prediction error varies across sine modes for FNO and DeepONet. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗

read the original abstract

Neural operators learn to map initial conditions to the terminal solution of partial differential equations (PDEs), providing a surrogate for the full operator mapping. This enables rapid prediction across different input configurations. While recent neural operator architectures have demonstrated strong performance on diverse PDE tasks, their behavior under structured distribution shifts remains insufficiently understood. To investigate this, we study operator learning in a wave propagation setting governed by a one-dimensional variable-coefficient wave equation, using two representative architectures, the Fourier Neural Operator (FNO) and the Deep Operator Network (DeepONet). To examine their generalization under distribution shifts, we consider structured out-of-distribution (OOD) settings that independently vary input frequency and coefficient smoothness. The results show that under smoothness shifts, both models maintain stable performance, with FNO achieving lower error. In contrast, under frequency shifts, FNO exhibits a sharp increase in error under unseen high-frequency inputs, whereas DeepONet shows milder degradation despite higher overall error. Our analysis reveals that these differences arise from how each architecture represents and responds to variations in frequency structure. Together, these findings highlight a fundamental gap between strong in-distribution performance and generalization under distribution shifts in operator learning, underscoring the role of architectural representation bias in developing more reliable neural operators for physics-based PDE simulations beyond the training distribution.

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.

Desk Editor's Note private letter to a colleague

FNO degrades more sharply than DeepONet under high-frequency OOD inputs on the variable-coefficient wave equation, but the comparison lacks the details needed to rule out confounds.

read the letter

The main thing to know is that the authors report FNO error rising sharply on unseen high-frequency initial conditions while DeepONet degrades more mildly, even though its baseline error is higher; both models stay relatively stable when only coefficient smoothness changes, with FNO still ahead in that case. They isolate the two shifts in a controlled way on the 1D variable-coefficient wave equation, which gives a cleaner look at architectural differences than most prior operator-learning papers provide. That structured empirical comparison is the real contribution here and it is worth noting for anyone working on generalization in neural operators. The setup itself is reasonable for probing frequency bias in PDE surrogates. The soft spots are more substantial. The abstract supplies no error magnitudes, training hyperparameters, number of runs, or statistical checks, so the claimed “sharp increase” cannot be sized or verified. The stress-test concern also holds on the evidence given: high-frequency inputs can interact with spatially varying coefficients through local wavelength changes, and nothing in the reported results (no coefficient-conditioned curves or residual spectra) separates that interaction from pure frequency representation differences. Without those checks the attribution to architectural frequency bias stays provisional. This paper is aimed at the neural-operator subfield rather than broader PDE or physics audiences. Readers already following FNO versus DeepONet comparisons will get the most out of it. It deserves peer review because the question is relevant and the experimental design is deliberate, but the authors will need to add full methods, quantitative results, and a direct response to the coefficient-frequency coupling issue before the central claim can be treated as solid.

Referee Report

2 major / 2 minor

Summary. The paper examines out-of-distribution generalization of neural operators (FNO and DeepONet) for mapping initial conditions to terminal solutions of a 1D variable-coefficient wave equation. It considers structured shifts that independently vary input frequency and coefficient smoothness, reporting that both architectures remain stable under smoothness shifts (with FNO lower error), while FNO shows a sharp error increase on unseen high-frequency inputs and DeepONet degrades more mildly despite higher baseline error; the differences are attributed to architectural differences in frequency representation.

Significance. If the central empirical distinction holds after addressing potential confounds, the work would usefully illustrate how operator architectures encode frequency structure and why in-distribution performance does not guarantee robustness under realistic PDE distribution shifts. This is relevant for surrogate modeling in physics applications where input spectra can vary.

major comments (2)

[Results / Experimental Setup] The claim that observed error differences arise from intrinsic frequency-representation bias (abstract and results) is load-bearing for the central conclusion, yet the experimental design does not isolate frequency shifts from coefficient interactions. In the variable-coefficient wave equation, high-frequency initial conditions can generate solution components whose local wavelengths couple to spatial coefficient variations; keeping the marginal coefficient distribution fixed does not automatically remove this interaction. No coefficient-conditioned error curves, residual spectral analysis, or ablation that holds coefficient smoothness fixed while varying frequency content are reported to rule out the alternative explanation that FNO's Fourier modes are more sensitive to imperfect coefficient handling under high-frequency excitation.
[Abstract and Experiments] The abstract and experimental sections supply no training details (optimizer, learning-rate schedule, number of epochs, batch size), error metrics (L2, relative L2, pointwise), statistical tests, or data-exclusion rules. This leaves the reported sharp error increase for FNO and the milder degradation for DeepONet only partially supported and difficult to reproduce or compare with prior operator-learning benchmarks.

minor comments (2)

[Methods] Notation for the variable-coefficient wave equation (e.g., the precise form of the coefficient function and boundary conditions) should be stated explicitly in the methods section rather than assumed from the abstract.
[Figures] Figure captions and axis labels for error-vs-frequency plots should include the exact definition of the error norm and the range of frequencies used in training versus OOD test sets.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address each major comment below and describe the revisions that will be incorporated.

read point-by-point responses

Referee: [Results / Experimental Setup] The claim that observed error differences arise from intrinsic frequency-representation bias (abstract and results) is load-bearing for the central conclusion, yet the experimental design does not isolate frequency shifts from coefficient interactions. In the variable-coefficient wave equation, high-frequency initial conditions can generate solution components whose local wavelengths couple to spatial coefficient variations; keeping the marginal coefficient distribution fixed does not automatically remove this interaction. No coefficient-conditioned error curves, residual spectral analysis, or ablation that holds coefficient smoothness fixed while varying frequency content are reported to rule out the alternative explanation that FNO's Fourier modes are more sensitive to imperfect coefficient handling under high-frequency excitation.

Authors: We appreciate the referee's observation on possible residual interactions. Our design independently varies frequency content and coefficient smoothness while holding marginal coefficient distributions fixed, and the resulting error patterns are consistent with known architectural properties (FNO's global Fourier basis versus DeepONet's local branch-trunk structure). Nevertheless, we agree that explicit isolation would strengthen the attribution. In the revision we will add coefficient-conditioned error curves, residual spectral analysis of the solutions, and an ablation that holds coefficient smoothness fixed while sweeping frequency content. These additions will directly address the alternative explanation and better support the architectural-bias interpretation. revision: partial
Referee: [Abstract and Experiments] The abstract and experimental sections supply no training details (optimizer, learning-rate schedule, number of epochs, batch size), error metrics (L2, relative L2, pointwise), statistical tests, or data-exclusion rules. This leaves the reported sharp error increase for FNO and the milder degradation for DeepONet only partially supported and difficult to reproduce or compare with prior operator-learning benchmarks.

Authors: We agree that these implementation details are necessary for reproducibility and fair comparison. The revised manuscript will expand the experimental section to report the optimizer, learning-rate schedule, number of epochs, batch size, the precise error metric (relative L2 norm), results of statistical tests across multiple random seeds, and the data-exclusion criteria. These additions will make the reported OOD trends fully supported and directly comparable to existing neural-operator benchmarks. revision: yes

Circularity Check

0 steps flagged

No circularity: purely empirical comparison with no derivations or self-referential predictions

full rationale

The paper conducts an empirical study training FNO and DeepONet on solutions to the 1D variable-coefficient wave equation and measuring test error under controlled shifts in input frequency and coefficient smoothness. No mathematical derivations, uniqueness theorems, ansatzes, or predictions are claimed; results are reported directly from numerical experiments. The central observations (FNO error spike under high-frequency OOD inputs, milder DeepONet degradation) are data-driven comparisons, not quantities that reduce by construction to fitted parameters or self-citations. No load-bearing steps match any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on empirical observations from experiments on the wave equation; no free parameters or invented entities are introduced.

axioms (1)

domain assumption The one-dimensional variable-coefficient wave equation serves as a representative testbed for studying operator-learning generalization under structured distribution shifts.
Invoked to justify the choice of PDE and the two shift types examined.

pith-pipeline@v0.9.0 · 5529 in / 1163 out tokens · 61118 ms · 2026-05-14T19:43:48.297455+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

71 extracted references · 71 canonical work pages

[1]

Langley , title =

P. Langley , title =. Proceedings of the 17th International Conference on Machine Learning (ICML 2000) , address =. 2000 , pages =

work page 2000
[2]

T. M. Mitchell. The Need for Biases in Learning Generalizations. 1980

work page 1980
[3]

M. J. Kearns , title =

work page
[4]

Machine Learning: An Artificial Intelligence Approach, Vol. I. 1983

work page 1983
[5]

R. O. Duda and P. E. Hart and D. G. Stork. Pattern Classification. 2000

work page 2000
[6]

Suppressed for Anonymity , author=

work page
[7]

Newell and P

A. Newell and P. S. Rosenbloom. Mechanisms of Skill Acquisition and the Law of Practice. Cognitive Skills and Their Acquisition. 1981

work page 1981
[8]

A. L. Samuel. Some Studies in Machine Learning Using the Game of Checkers. IBM Journal of Research and Development. 1959

work page 1959
[9]

2010 , publisher =

Partial Differential Equations , author =. 2010 , publisher =

work page 2010
[10]

2002 , publisher =

Finite Volume Methods for Hyperbolic Problems , author =. 2002 , publisher =

work page 2002
[11]

2004 , publisher =

Finite Difference Schemes and Partial Differential Equations , author =. 2004 , publisher =

work page 2004
[12]

2015 , publisher =

Applied Partial Differential Equations , author =. 2015 , publisher =

work page 2015
[13]

2007 , publisher =

Partial Differential Equations: An Introduction , author =. 2007 , publisher =

work page 2007
[14]

2013 , publisher =

Uncertainty Quantification: Theory, Implementation, and Applications , author =. 2013 , publisher =

work page 2013
[15]

, journal =

Stuart, Andrew M. , journal =. Inverse problems: A

work page
[16]

2015 , publisher =

Reduced Basis Methods for Partial Differential Equations: An Introduction , author =. 2015 , publisher =

work page 2015
[17]

SIAM Review , volume =

A survey of projection-based model reduction methods for parametric dynamical systems , author =. SIAM Review , volume =

work page
[18]

2014 , publisher =

Reduced Order Methods for Modeling and Computational Reduction , editor =. 2014 , publisher =

work page 2014
[19]

2019 , publisher =

Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control , author =. 2019 , publisher =

work page 2019
[20]

Journal of Computational Physics , volume =

Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations , author =. Journal of Computational Physics , volume =

work page
[21]

Nature Reviews Physics , volume =

Physics-informed machine learning , author =. Nature Reviews Physics , volume =

work page
[22]

ACM Computing Surveys , volume =

Integrating scientific knowledge with machine learning for engineering and environmental systems , author =. ACM Computing Surveys , volume =

work page
[23]

Journal of Scientific Computing , volume =

Scientific machine learning through physics-informed neural networks: Where we are and what's next , author =. Journal of Scientific Computing , volume =

work page
[24]

United States Department of Energy Report , year =

Workshop report on basic research needs for scientific machine learning: Core technologies for artificial intelligence , author =. United States Department of Energy Report , year =

work page
[25]

Learning nonlinear operators via

Lu, Lu and Jin, Pengzhan and Pang, Guofei and Zhang, Zhongqiang and Karniadakis, George Em , journal =. Learning nonlinear operators via

work page
[26]

International Conference on Learning Representations , year =

Fourier Neural Operator for Parametric Partial Differential Equations , author =. International Conference on Learning Representations , year =

work page
[27]

Neural operator: Learning maps between function spaces with applications to

Kovachki, Nikola and Li, Zongyi and Liu, Burigede and Azizzadenesheli, Kamyar and Bhattacharya, Kaushik and Stuart, Andrew and Anandkumar, Anima , journal =. Neural operator: Learning maps between function spaces with applications to

work page
[28]

Pathak, Jaideep and Subramanian, Shashank and Harrington, Peter and Raja, Sanjeev and Chattopadhyay, Ashesh and Mardani, Morteza and Kurth, Thorsten and Hall, David and Li, Zongyi and Azizzadenesheli, Kamyar and Hassanzadeh, Pedram and Kashinath, Karthik and Anandkumar, Anima , journal =

work page
[29]

and Detre, John A

Kissas, Georgios and Yang, Yaohua and Hwuang, En-Jui and Witschey, Walter R. and Detre, John A. and Perdikaris, Paris , journal =. Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow

work page
[30]

Deep learning methods for Reynolds-averaged

Thuerey, Nils and Wei. Deep learning methods for Reynolds-averaged. AIAA Journal , volume =

work page
[31]

Universal differential equations for scientific machine learning.arXiv preprint arXiv:2001.04385, 2020

Universal differential equations for scientific machine learning , author =. arXiv preprint arXiv:2001.04385 , year =

work page arXiv 2001
[32]

ICLR Workshop on Integration of Deep Neural Models and Differential Equations , year =

Neural Operator: Graph Kernel Network for Partial Differential Equations , author =. ICLR Workshop on Integration of Deep Neural Models and Differential Equations , year =

work page
[33]

Learning the solution operator of parametric partial differential equations with physics-informed

Wang, Sifan and Wang, Hanwen and Perdikaris, Paris , journal =. Learning the solution operator of parametric partial differential equations with physics-informed

work page
[34]

arXiv preprint arXiv:2111.03794 , year =

Physics-informed neural operator for learning partial differential equations , author =. arXiv preprint arXiv:2111.03794 , year =

work page arXiv
[35]

Advances in Neural Information Processing Systems , volume =

Choose a Transformer: Fourier or Galerkin , author =. Advances in Neural Information Processing Systems , volume =

work page
[36]

and Azizzadenesheli, Kamyar , journal =

Rahman, Md Ashiqur and Ross, Zachary E. and Azizzadenesheli, Kamyar , journal =

work page
[37]

Computer Methods in Applied Mechanics and Engineering , volume =

A comprehensive and fair comparison of two neural operators for data-driven PDE solving , author =. Computer Methods in Applied Mechanics and Engineering , volume =

work page
[38]

Computer Methods in Applied Mechanics and Engineering , volume =

Wavelet neural operator for solving parametric partial differential equations in computational mechanics problems , author =. Computer Methods in Applied Mechanics and Engineering , volume =

work page
[39]

Journal of Computational Physics , volume =

Koopman neural operator as a mesh-free solver of non-linear partial differential equations , author =. Journal of Computational Physics , volume =

work page
[40]

Improved generalization with deep neural operators for engineering systems:

Kobayashi, Kazuma and Daniell, James and Alam, Syed Bahauddin , journal =. Improved generalization with deep neural operators for engineering systems:

work page
[41]

Advances in Neural Information Processing Systems Datasets and Benchmarks Track , year=

AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier-Stokes Solutions , author=. Advances in Neural Information Processing Systems Datasets and Benchmarks Track , year=

work page
[42]

ICLR Workshop on Geometrical and Topological Representation Learning , year=

An Extensible Benchmarking Graph-Mesh Dataset for Studying Steady-State Incompressible Navier-Stokes Equations , author=. ICLR Workshop on Geometrical and Topological Representation Learning , year=

work page
[43]

and Welling, Max , booktitle =

Brandstetter, Johannes and Worrall, Daniel E. and Welling, Max , booktitle =. Message Passing Neural

work page
[44]

Numerische Mathematik , volume =

Error estimates for deep learning methods in fluid dynamics , author =. Numerische Mathematik , volume =

work page
[45]

Advances in Neural Information Processing Systems , volume=

Learning Dissipative Dynamics in Chaotic Systems , author=. Advances in Neural Information Processing Systems , volume=

work page
[46]

1986 , publisher =

Theoretical Acoustics , author =. 1986 , publisher =

work page 1986
[47]

2012 , publisher =

Wave Motion in Elastic Solids , author =. 2012 , publisher =

work page 2012
[48]

2002 , publisher =

Quantitative Seismology , author =. 2002 , publisher =

work page 2002
[49]

Geophysics , volume =

An overview of full-waveform inversion in exploration geophysics , author =. Geophysics , volume =

work page
[50]

Virieux, Jean , journal =

work page
[51]

Geophysics , volume =

Inversion of seismic reflection data in the acoustic approximation , author =. Geophysics , volume =

work page
[52]

1995 , publisher =

Waves and Fields in Inhomogeneous Media , author =. 1995 , publisher =

work page 1995
[53]

2007 , publisher =

Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems , author =. 2007 , publisher =

work page 2007
[54]

Mathematical Geosciences , volume =

Stochastic seismic waveform inversion using generative adversarial networks as a geological prior , author =. Mathematical Geosciences , volume =

work page
[55]

Proceedings of the 40th International Conference on Machine Learning , pages =

Group Equivariant Fourier Neural Operators for Partial Differential Equations , author =. Proceedings of the 40th International Conference on Machine Learning , pages =

work page
[56]

Hao, Zhongkai and Ying, Chengyang and Wang, Zhengyi and Su, Hang and Dong, Yinpeng and Liu, Songming and Cheng, Ze and Song, Jian and Zhu, Jun , booktitle =

work page
[57]

Machine Learning: Science and Technology , volume =

Applications of physics informed neural operators , author =. Machine Learning: Science and Technology , volume =

work page
[58]

, journal =

Khan, Asad and Huerta, Eliu A. , journal =

work page
[59]

Communications on Pure and Applied Mathematics , volume =

Trainability and accuracy of artificial neural networks: An interacting particle system approach , author =. Communications on Pure and Applied Mathematics , volume =

work page
[60]

Proceedings of the 41st International Conference on Machine Learning , year =

Transolver: A Fast Transformer Solver for PDEs on General Geometries , author =. Proceedings of the 41st International Conference on Machine Learning , year =

work page
[61]

Advances in Neural Information Processing Systems , year =

Herde, Maximilian and Raoni. Advances in Neural Information Processing Systems , year =

work page
[62]

Advances in Neural Information Processing Systems , year =

Universal Physics Transformers: A Framework for Efficiently Scaling Neural Operators , author =. Advances in Neural Information Processing Systems , year =

work page
[63]

Scientific Reports , year =

Temporal Neural Operator for Modeling Time-Dependent Physical Phenomena , author =. Scientific Reports , year =

work page
[64]

An Accurate Neural Solver for

Luo, Hongkai and others , year =. An Accurate Neural Solver for

work page
[65]

Methods of Mathematical Physics,

Courant, Richard and Hilbert, David , year =. Methods of Mathematical Physics,

work page
[66]

2005 , publisher=

The Finite Element Method: Its Basis and Fundamentals , author=. 2005 , publisher=

work page 2005
[67]

Shields and George Em Karniadakis , title =

Somdatta Goswami and Katiana Kontolati and Michael D. Shields and George Em Karniadakis , title =. Nature Machine Intelligence , year =

work page
[68]

Convolutional Neural Operators for Robust and Accurate Learning of

Bogdan Raoni. Convolutional Neural Operators for Robust and Accurate Learning of. Advances in Neural Information Processing Systems , volume =. 2023 , url =

work page 2023
[69]

Advances in Neural Information Processing Systems , volume =

Zongyi Li and Nikola Kovachki and Chris Choy and Boyi Li and Jean Kautz and Anima Anandkumar , title =. Advances in Neural Information Processing Systems , volume =. 2023 , url =

work page 2023
[70]

Communications on Applied Mathematics and Computation , year =

Zhi-Qin John Xu and Yaoyu Zhang and Tao Luo , title =. Communications on Applied Mathematics and Computation , year =. doi:10.48550/arXiv.2201.07395 , url =

work page doi:10.48550/arxiv.2201.07395
[71]

Benjamin Erichson and Kush Bhatia and Michael W

Jerry Weihong Liu and N. Benjamin Erichson and Kush Bhatia and Michael W. Mahoney and Christopher R. Does In-Context Operator Learning Generalize to Domain-Shifted Settings? , booktitle =. 2023 , url =

work page 2023