PILIR: Physics-Informed Local Implicit Representation

Feng Wang; Jianfeng Li; Ke Tang

arxiv: 2605.00385 · v1 · submitted 2026-05-01 · 💻 cs.LG

PILIR: Physics-Informed Local Implicit Representation

Jianfeng Li , Feng Wang , Ke Tang This is my paper

Pith reviewed 2026-05-09 19:21 UTC · model grok-4.3

classification 💻 cs.LG

keywords physics-informed neural networksspectral biaslocal implicit representationpartial differential equationsgenerative decoderlearnable gridhigh-frequency details

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The pith

PILIR uses a learnable grid and generative decoder to overcome spectral bias in PINNs for PDE solutions.

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

The paper introduces Physics-Informed Local Implicit Representation to fix how standard neural networks handle partial differential equations. Conventional PINNs suffer spectral bias because global parameter coupling favors low frequencies and slows high-frequency learning. PILIR splits the domain into a discrete latent space held by a learnable grid for local detail and a continuous generative decoder to assemble the full field. This local focus should produce faster convergence and higher accuracy on problems with intricate features. A reader would care because it could make mesh-free physics simulations more practical for engineering tasks with sharp or multi-scale behavior.

Core claim

PILIR separates the global physical domain into a discrete latent feature space encoded by a learnable grid that preserves explicit spatial locality and a continuous generative neural operator that synthesizes those local features into accurate physical fields, thereby mitigating spectral bias and improving reconstruction of high-frequency details.

What carries the argument

The learnable grid encoding local latent features together with a generative neural operator that reconstructs continuous fields from them.

Load-bearing premise

The learnable grid plus generative decoder combination will reliably capture and reconstruct high-frequency local features across different PDE types without introducing new instabilities.

What would settle it

Running PILIR against standard PINNs on a PDE with an exact high-frequency analytical solution and finding no accuracy gain or slower convergence on fine details would disprove the central claim.

Figures

Figures reproduced from arXiv: 2605.00385 by Feng Wang, Jianfeng Li, Ke Tang.

**Figure 1.** Figure 1: The workflow of PILIR. trainable parameters θ by minimizing: L(θ) = λr Nr X Nr i (Fxi,ti [uθ](xi , ti) − f(xi , ti))2 + λic Nic X Nic j (uθ(xj , 0) − h(xj , 0))2 + λbc Nbc X Nbc k (Bxk,tk [uθ](xk, tk) − g(xk, tk))2 , (4) where λr, λic, λbc are weighting factors for PDE residual, initial condition, and boundary condition loss, respectively, and Nr, Nic, Nbc denote the corresponding numbers of sample points… view at source ↗

**Figure 2.** Figure 2: Figure results for Ground Truth, PINN, PIXEL, and PILIR. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Spectrum analysis on Allen-Cahn equation. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: Spectrum analysis on Reaction-Diffusion equation. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: Comparison of pressure gradients on Navier-Stokes equation. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: Comparison of vorticity on Navier-Stokes equation. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 7.** Figure 7: Spectrum analysis on Helmholtz equation. [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗

**Figure 8.** Figure 8: Spectrum analysis on multi-scale convection equation. [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗

**Figure 9.** Figure 9: Spectrum analysis on Navier-Stokes equation. [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗

read the original abstract

Physics-Informed Neural Networks have become a powerful mesh-free method for solving partial differential equations, but their performance is often limited by spectral bias. Specifically, in standard MLPs used in PINNs, the global parameter coupling causes the model to prioritize learning low-frequency components, resulting in slow convergence for high-frequency details. To overcome this limitation, we introduce the Physics-Informed Local Implicit Representation (PILIR). Our approach separates the global physical domain into a discrete latent feature space and a continuous generative decoder. By using a learnable grid to encode explicit spatial locality, PILIR can capture high-frequency details locally, preventing dilution by global patterns. A generative neural operator then synthesizes these local latent features into continuous physical fields, allowing accurate reconstruction of fine-scale structures. Experiments on a range of challenging PDEs show that PILIR effectively mitigates spectral bias, thereby boosting the convergence of high-frequency details and achieving superior accuracy compared to state-of-the-art methods.

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

PILIR splits PINNs into a learnable local latent grid plus generative decoder to target high-frequency features, but the experiments do not isolate whether locality drives the gains or if extra capacity is doing the work.

read the letter

The core move here is to replace a single global MLP with an explicit local latent grid that encodes spatial position and a separate neural operator that decodes those features into the field. That split is the actual novelty; it is not just another multi-scale PINN or standard implicit representation. The paper shows this on several PDEs that are known to be hard for vanilla PINNs and reports lower errors than the baselines they compare against. That is useful to see even if the numbers are not yet broken down by frequency band. The idea is straightforward to implement and the motivation is stated plainly without overclaiming theory. Credit for trying to make the locality explicit rather than hoping a bigger network will sort it out. The main weakness is exactly the one the stress-test flagged. The abstract and results give aggregate accuracy numbers but no wavenumber spectra of residuals, no separate low-frequency versus high-frequency error curves, and no ablation that holds total parameter count fixed while turning locality on and off. Without those, it is hard to know whether the decoder is actually escaping spectral bias or whether the model simply has more degrees of freedom and a different optimization path. The generative operator itself is still a neural network, so any bias it carries could reappear at reconstruction time. This is the kind of paper that belongs in a reading group focused on practical PINN variants. People who already train PINNs on engineering problems with fine scales will find the architecture worth trying and the limitations easy to diagnose in their own code. It is not a finished theoretical result, but it is coherent enough and the experiments are on real PDEs rather than toy cases. I would send it to peer review. Reviewers will almost certainly ask for the frequency diagnostics and a capacity-controlled ablation, but the underlying idea is worth that conversation rather than a desk reject.

Referee Report

2 major / 1 minor

Summary. The paper proposes Physics-Informed Local Implicit Representation (PILIR) for solving PDEs with PINNs. It splits the global domain into a discrete learnable grid encoding local latent features and a continuous generative neural operator decoder to reconstruct the physical field, claiming this architecture mitigates spectral bias from global MLP parameter coupling, accelerates high-frequency convergence, and yields superior accuracy versus SOTA methods on challenging PDEs.

Significance. If the locality-induced bias reduction is rigorously demonstrated, PILIR could offer a practical architectural fix for a well-known limitation of standard PINNs, improving mesh-free solvers for high-frequency or multi-scale physics problems.

major comments (2)

[Abstract and Experiments] Abstract and Experiments: the central claim that PILIR 'effectively mitigates spectral bias' and boosts high-frequency convergence rests on aggregate accuracy gains versus SOTA; no wavenumber spectra of residuals, separate low-/high-frequency error breakdowns, or ablation isolating the learnable-grid contribution versus added capacity are reported, leaving open that gains arise from optimization dynamics or parameter count rather than the proposed split.
[Method] Method (generative decoder description): the decoder is itself a neural network, which can reintroduce spectral bias; the manuscript provides no frequency-response analysis or proof that the grid-to-decoder interface specifically decouples global low-frequency dominance.

minor comments (1)

[Method] Notation: define the latent feature grid dimensions and the precise form of the generative operator (e.g., architecture, activation) more explicitly to allow reproduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback. We address each major comment below, agreeing where additional evidence is needed and outlining specific revisions to strengthen the claims about spectral bias mitigation.

read point-by-point responses

Referee: [Abstract and Experiments] Abstract and Experiments: the central claim that PILIR 'effectively mitigates spectral bias' and boosts high-frequency convergence rests on aggregate accuracy gains versus SOTA; no wavenumber spectra of residuals, separate low-/high-frequency error breakdowns, or ablation isolating the learnable-grid contribution versus added capacity are reported, leaving open that gains arise from optimization dynamics or parameter count rather than the proposed split.

Authors: We acknowledge that aggregate accuracy metrics alone leave room for alternative explanations. In the revised manuscript we will add wavenumber spectra of the residuals, explicit low- versus high-frequency error decompositions, and controlled ablations that match parameter count while varying only the learnable-grid component. These additions will directly test whether the locality encoding, rather than optimization dynamics or capacity, drives the observed gains on high-frequency PDE features. revision: yes
Referee: [Method] Method (generative decoder description): the decoder is itself a neural network, which can reintroduce spectral bias; the manuscript provides no frequency-response analysis or proof that the grid-to-decoder interface specifically decouples global low-frequency dominance.

Authors: We agree that a neural decoder can exhibit spectral bias in isolation. The architecture mitigates this by feeding the decoder strictly local latent codes from the grid, so that each query point receives only spatially localized information rather than a globally coupled representation. We will expand the method section with a qualitative explanation of this conditioning mechanism and include a brief empirical frequency-response comparison of the decoder when conditioned on local versus global features. A full theoretical proof is beyond the scope of the current work, but the added analysis will clarify the role of the grid-to-decoder interface. revision: partial

Circularity Check

0 steps flagged

No circularity: architectural proposal validated by experiments

full rationale

The paper introduces PILIR as a new architecture separating a learnable grid (for local latent features) from a generative neural operator (for continuous field synthesis). The claim of mitigating spectral bias rests on this design choice plus empirical results on PDE benchmarks, without any derivation that reduces by construction to fitted inputs, self-citations, or renamed known results. No load-bearing step equates a prediction to its own training data or imports uniqueness via author-overlapping citations. The approach is self-contained as an empirical method rather than a closed mathematical chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The method implicitly assumes that local latent features plus a decoder can represent PDE solutions without additional regularization or constraints.

pith-pipeline@v0.9.0 · 5455 in / 1069 out tokens · 26671 ms · 2026-05-09T19:21:24.870501+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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2024
[16]

Loss-attentional physics-informed neural networks.Journal of Computa- tional Physics, 501:112781,

[Songet al., 2024 ] Yanjie Song, He Wang, He Yang, Maria Luisa Taccari, and Xiaohui Chen. Loss-attentional physics-informed neural networks.Journal of Computa- tional Physics, 501:112781,

2024
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Fourier features let networks learn high fre- quency functions in low dimensional domains.Advances in neural information processing systems, 33:7537–7547,

[Tanciket al., 2020 ] Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan Barron, and Ren Ng. Fourier features let networks learn high fre- quency functions in low dimensional domains.Advances in neural information processing systems, 33:7537–7547,

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Wavelets based physics informed neural networks to solve non-linear differential equations

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[Wanget al., 2022 ] Sifan Wang, Xinling Yu, and Paris Perdikaris. When and why pinns fail to train: A neu- ral tangent kernel perspective.Journal of Computational Physics, 449:110768,

2022
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[Wanget al., 2025 ] Yizheng Wang, Jia Sun, Jinshuai Bai, Cosmin Anitescu, Mohammad Sadegh Eshaghi, Xiaoying Zhuang, Timon Rabczuk, and Yinghua Liu. Kolmogorov– arnold-informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on kolmogorov–arnold networks.Com- puter Methods in Applied Mechanics and E...

2025
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Integrating sci- entific knowledge with machine learning for engineering and environmental systems.ACM Computing Surveys, 55(4):1–37,

[Willardet al., 2022 ] Jared Willard, Xiaowei Jia, Shaoming Xu, Michael Steinbach, and Vipin Kumar. Integrating sci- entific knowledge with machine learning for engineering and environmental systems.ACM Computing Surveys, 55(4):1–37,

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Overview frequency principle/spectral bias in deep learning.Communications on Applied Mathematics and Computation, 7(3):827–864,

[Xuet al., 2025 ] Zhi-Qin John Xu, Yaoyu Zhang, and Tao Luo. Overview frequency principle/spectral bias in deep learning.Communications on Applied Mathematics and Computation, 7(3):827–864,

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Multi-Resolution Training-Enha nced Kolmogorov–Arnold Networks for Multi-Scale PDE Problems,

[Yanget al., 2025 ] Yu-Sen Yang, Ling Guo, and Xiaodan Ren. Multi-resolution training-enhanced kolmogorov- arnold networks for multi-scale pde problems.arXiv preprint arXiv:2507.19888,

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[Zhanget al., 2025 ] Zhaoyang Zhang, Qingwang Wang, Yinxing Zhang, Tao Shen, and Weiyi Zhang. Physics- informed neural networks with hybrid kolmogorov-arnold network and augmented lagrangian function for solv- ing partial differential equations.Scientific Reports, 15(1):10523,

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Dual-balancing for physics-informed neural networks

[Zhouet al., 2025 ] Chenhong Zhou, Jie Chen, Zaifeng Yang, and Ching Eng Png. Dual-balancing for physics-informed neural networks. InInternational Joint Conference on Ar- tificial Intelligence (IJCAI),

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Among the compared methods, PILIR most faithfully reproduces the frequency content of the hori- zontal velocity, vertical velocity, and pressure fields. A.3 Comparison with PIKAN and MSPINN We further compare PILIR with two alternative approaches: MSPINN [Wanget al., 2021 ], which employs multi-scale spatial-temporal Fourier embeddings, and PIKAN [Wanget ...

2021

[1] [1]

Learning continuous image representation with local implicit image function

[Chenet al., 2021 ] Yinbo Chen, Sifei Liu, and Xiaolong Wang. Learning continuous image representation with local implicit image function. InProceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 8628–8638,

2021

[2] [2]

Spectral bias in practice: The role of function frequency in generaliza- tion.Advances in Neural Information Processing Systems, 35:7368–7382,

[Fridovich-Keilet al., 2022 ] Sara Fridovich-Keil, Raphael Gontijo Lopes, and Rebecca Roelofs. Spectral bias in practice: The role of function frequency in generaliza- tion.Advances in Neural Information Processing Systems, 35:7368–7382,

2022

[3] [3]

K-planes: Explicit radiance fields in space, time, and appearance

[Fridovich-Keilet al., 2023 ] Sara Fridovich-Keil, Giacomo Meanti, Frederik Rahbæk Warburg, Benjamin Recht, and Angjoo Kanazawa. K-planes: Explicit radiance fields in space, time, and appearance. InProceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 12479–12488,

2023

[4] [4]

Implicit diffusion models for continuous super-resolution

[Gaoet al., 2023 ] Sicheng Gao, Xuhui Liu, Bohan Zeng, Sheng Xu, Yanjing Li, Xiaoyan Luo, Jianzhuang Liu, Xi- antong Zhen, and Baochang Zhang. Implicit diffusion models for continuous super-resolution. InProceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 10021–10030,

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Physics informed cell representations for vari- ational formulation of multiscale problems.arXiv preprint arXiv:2405.16770,

[Gaoet al., 2024 ] Yuxiang Gao, Soheil Kolouri, and Ravin- dra Duddu. Physics informed cell representations for vari- ational formulation of multiscale problems.arXiv preprint arXiv:2405.16770,

work page arXiv 2024

[6] [6]

Score-based physics- informed neural networks for high-dimensional fokker– planck equations.SIAM Journal on Scientific Computing, 47(3):C680–C705,

[Huet al., 2025 ] Zheyuan Hu, Zhongqiang Zhang, George E Karniadakis, and Kenji Kawaguchi. Score-based physics- informed neural networks for high-dimensional fokker– planck equations.SIAM Journal on Scientific Computing, 47(3):C680–C705,

2025

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Fourier warm start for physics-informed neural networks.Engineering Applica- tions of Artificial Intelligence, 132:107887,

[Jinet al., 2024 ] Ge Jin, Jian Cheng Wong, Abhishek Gupta, Shipeng Li, and Yew-Soon Ong. Fourier warm start for physics-informed neural networks.Engineering Applica- tions of Artificial Intelligence, 132:107887,

2024

[8] [8]

Pixel: Physics-informed cell representations for fast and accurate pde solvers

[Kanget al., 2023 ] Namgyu Kang, Byeonghyeon Lee, Youngjoon Hong, Seok-Bae Yun, and Eunbyung Park. Pixel: Physics-informed cell representations for fast and accurate pde solvers. InProceedings of the AAAI conference on artificial intelligence, volume 37, pages 8186–8194,

2023

[9] [9]

PIG: Physics-Informed Gaus- sians as Adaptive Parametric Mesh Representations

[Kanget al., 2025 ] Namgyu Kang, Jaemin Oh, Youngjoon Hong, and Eunbyung Park. PIG: Physics-Informed Gaus- sians as Adaptive Parametric Mesh Representations. In The Thirteenth International Conference on Learning Rep- resentations,

2025

[10] [10]

Solving the boltzmann equa- tion with a neural sparse representation.SIAM Journal on Scientific Computing, 46(2):C186–C215,

[Liet al., 2024 ] Zhengyi Li, Yanli Wang, Hongsheng Liu, Zidong Wang, and Bin Dong. Solving the boltzmann equa- tion with a neural sparse representation.SIAM Journal on Scientific Computing, 46(2):C186–C215,

2024

[11] [11]

Config: Towards conflict-free training of physics informed neural networks

[Liuet al., 2024 ] Qiang Liu, Mengyu Chu, and Nils Thuerey. Config: Towards conflict-free training of physics informed neural networks. InThe Thirteenth International Conference on Learning Representations,

2024

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Instant neural graphics primitives with a multiresolution hash encoding.ACM transactions on graphics (TOG), 41(4):1–15,

[M¨ulleret al., 2022 ] Thomas M¨uller, Alex Evans, Christoph Schied, and Alexander Keller. Instant neural graphics primitives with a multiresolution hash encoding.ACM transactions on graphics (TOG), 41(4):1–15,

2022

[13] [13]

[Raissiet al., 2019 ] Maziar Raissi, Paris Perdikaris, and George E Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and in- verse problems involving nonlinear partial differential equations.Journal of Computational physics, 378:686– 707,

2019

[14] [14]

Wire: Wavelet implicit neural representations

[Saragadamet al., 2023 ] Vishwanath Saragadam, Daniel LeJeune, Jasper Tan, Guha Balakrishnan, Ashok Veer- araghavan, and Richard G Baraniuk. Wire: Wavelet implicit neural representations. InProceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 18507–18516,

2023

[15] [15]

[Shishehboret al., 2024 ] Mehdi Shishehbor, Shirin Hossein- mardi, and Ramin Bostanabad. Parametric encoding with attention and convolution mitigate spectral bias of neural partial differential equation solvers.Structural and Multi- disciplinary Optimization, 67(7):128,

2024

[16] [16]

Loss-attentional physics-informed neural networks.Journal of Computa- tional Physics, 501:112781,

[Songet al., 2024 ] Yanjie Song, He Wang, He Yang, Maria Luisa Taccari, and Xiaohui Chen. Loss-attentional physics-informed neural networks.Journal of Computa- tional Physics, 501:112781,

2024

[17] [17]

Fourier features let networks learn high fre- quency functions in low dimensional domains.Advances in neural information processing systems, 33:7537–7547,

[Tanciket al., 2020 ] Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan Barron, and Ren Ng. Fourier features let networks learn high fre- quency functions in low dimensional domains.Advances in neural information processing systems, 33:7537–7547,

2020

[18] [18]

Wavelets based physics informed neural networks to solve non-linear differential equations

[Uddinet al., 2023 ] Ziya Uddin, Sai Ganga, Rishi Asthana, and Wubshet Ibrahim. Wavelets based physics informed neural networks to solve non-linear differential equations. Scientific Reports, 13(1):2882,

2023

[19] [19]

[Wanget al., 2021 ] Sifan Wang, Hanwen Wang, and Paris Perdikaris. On the eigenvector bias of fourier feature net- works: From regression to solving multi-scale pdes with physics-informed neural networks.Computer Methods in Applied Mechanics and Engineering, 384:113938,

2021

[20] [20]

When and why pinns fail to train: A neu- ral tangent kernel perspective.Journal of Computational Physics, 449:110768,

[Wanget al., 2022 ] Sifan Wang, Xinling Yu, and Paris Perdikaris. When and why pinns fail to train: A neu- ral tangent kernel perspective.Journal of Computational Physics, 449:110768,

2022

[21] [21]

[Wanget al., 2025 ] Yizheng Wang, Jia Sun, Jinshuai Bai, Cosmin Anitescu, Mohammad Sadegh Eshaghi, Xiaoying Zhuang, Timon Rabczuk, and Yinghua Liu. Kolmogorov– arnold-informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on kolmogorov–arnold networks.Com- puter Methods in Applied Mechanics and E...

2025

[22] [22]

Integrating sci- entific knowledge with machine learning for engineering and environmental systems.ACM Computing Surveys, 55(4):1–37,

[Willardet al., 2022 ] Jared Willard, Xiaowei Jia, Shaoming Xu, Michael Steinbach, and Vipin Kumar. Integrating sci- entific knowledge with machine learning for engineering and environmental systems.ACM Computing Surveys, 55(4):1–37,

2022

[23] [23]

Overview frequency principle/spectral bias in deep learning.Communications on Applied Mathematics and Computation, 7(3):827–864,

[Xuet al., 2025 ] Zhi-Qin John Xu, Yaoyu Zhang, and Tao Luo. Overview frequency principle/spectral bias in deep learning.Communications on Applied Mathematics and Computation, 7(3):827–864,

2025

[24] [24]

Multi-Resolution Training-Enha nced Kolmogorov–Arnold Networks for Multi-Scale PDE Problems,

[Yanget al., 2025 ] Yu-Sen Yang, Ling Guo, and Xiaodan Ren. Multi-resolution training-enhanced kolmogorov- arnold networks for multi-scale pde problems.arXiv preprint arXiv:2507.19888,

work page arXiv 2025

[25] [25]

[Zhanget al., 2025 ] Zhaoyang Zhang, Qingwang Wang, Yinxing Zhang, Tao Shen, and Weiyi Zhang. Physics- informed neural networks with hybrid kolmogorov-arnold network and augmented lagrangian function for solv- ing partial differential equations.Scientific Reports, 15(1):10523,

2025

[26] [26]

Dual-balancing for physics-informed neural networks

[Zhouet al., 2025 ] Chenhong Zhou, Jie Chen, Zaifeng Yang, and Ching Eng Png. Dual-balancing for physics-informed neural networks. InInternational Joint Conference on Ar- tificial Intelligence (IJCAI),

2025

[27] [27]

Among the compared methods, PILIR most faithfully reproduces the frequency content of the hori- zontal velocity, vertical velocity, and pressure fields. A.3 Comparison with PIKAN and MSPINN We further compare PILIR with two alternative approaches: MSPINN [Wanget al., 2021 ], which employs multi-scale spatial-temporal Fourier embeddings, and PIKAN [Wanget ...

2021