Spectral Bias

Follow this topic Ask about this topic

Momentum

1 paper in the last four weeks, against 2 the four weeks before. 0.0% of all new papers.

Jul 13Week of Sep 28

Latest papers 25

Oct 1, 2026cs.AI

Contrastive Attention Mitigates Spectral Bias in Spiking Transformers

Spiking Transformers merge the energy-efficiency of spiking neural networks (SNNs) with the representational power of self-attention, creating a promising architecture for high-performance, energy-efficient computation. However, a performance gap persists versus its counterparts in artificial neural networks (ANNs). Unlike prior works attributing this to binary activations, we reveal that both spiking neurons and spiking self-attention (SSA) act as low-pass filters through multiscale spectral analysis. This characteristic leads to the dissipation of high-frequency components. To address this issue, we propose the Spiking Contrastive Attention (SCA) paradigm, which draw inspiration from the edge-detection and differential sensing properties of biological visual system. By extracting contrast prototypes via global contrastive aggregation and applying local differential refinement, SCA effectively enhances high-frequency information. Extensive experiments show that SCA is a general module that consistently boosts Spiking Transformers across image classification, semantic segmentation, and event-based tracking. Furthermore, it achieves lower complexity, offering superior efficiency over original SSA. These results establish its potential as a fundamental building block for energy-efficient Spiking Transformers.

Xiaoli Liu, Malu Zhang, Yang Yang

School of Computer Science and Engineering, University of Electronic Science and Technology of China

Spiking Transformers Self-Attention Spectral Bias Spiking Neural Networks

Sep 2, 2026cs.CV

Balancing Frequencies and Pixels in Flow Matching

Natural images follow a

1/f^2

spectral distribution: most signal energy lies in the low spatial frequencies, while the perceptually important structures such as textures and edges occupy sparse high-frequency bands. Pixel-space reconstruction objectives, however, treat all spatial errors uniformly, causing low frequencies to dominate the optimization signal and delaying the learning of fine-scale details. In this work, we identify this objective-level spectral imbalance as a key inefficiency in training pixel-space flow models. To address it, we propose a Focal Log-Frequency Loss (f-loss), a spectrally balanced objective that equalizes the learning signal across frequencies, emphasizing high-frequency components that are otherwise underrepresented in pixel-space objectives. Building on this, we introduce a simple training strategy that combines frequency and pixel supervision: we first emphasize frequency-domain learning early to capture all frequencies, and then transition to standard pixel-space v-loss for spatial refinement. This balancing mitigates the low-frequency bias of pixel losses and aligns the training signal with the evolving needs of the model. Our approach is conceptually simple, requires no architectural changes, and acts as a drop-in replacement for flow matching losses. Across multiple model scales, it accelerates convergence by up to 40% while consistently improving FID and perceptual fidelity. We will release code and models.

Lucas Degeorge, Paul Couairon, Arijit Ghosh +3

LIX, École Polytechnique, CNRS, IP Paris, France · LIGM, École Nationale des Ponts et Chaussées, IP Paris, UGE, CNRS, France · UC Berkeley

Image Generation Flow Matching Spectral Bias

Aug 13, 2026cs.LG

Sustaining Plasticity via Learnable Wavelet Activations in Continual Learning

Plasticity loss has emerged as a critical challenge in continual learning that significantly hinders the acquisition of sequential tasks. While optimizing activation designs offers a potential solution, current fixed-form functions suffer from an inherent spectral bias towards low-frequency variations, whereas learnable variants permit unconstrained updates that induce catastrophic forgetting. To address these limitations, we propose a novel learnable wavelet activation that decomposes the activation function into low-frequency and high-frequency components to explicitly counter spectral bias. Furthermore, we employ dynamic wavelet injection to adaptively enhance plasticity for new tasks, alongside a regularization strategy to ensure the stability of previous learned knowledge. Theoretically, we provide rigorous mathematical guarantees for the proposed framework, proving the structural necessity of the hybrid wavelet architecture for efficient

L^2

approximation and demonstrating that the decoupled learning rate mechanism successfully restores network plasticity for high-frequency information. Additionally, we provide a formal derivation of the loss-driven injection trigger mechanism to precisely guide the injection. Extensive empirical evaluations demonstrate that our approach maintains superior trainability and generalization throughout the learning process and achieves state-of-the-art performance across diverse continual learning benchmarks.

Zeyang Zhang, Tieliang Gong, Junyan Lu +1

School of Computer Science and Technology, Xi’an Jiaotong University and Institute of Multimedia Knowledge Fusion and Engineering, Xi’an 710049, China

Loss of Plasticity Continual Learning Spectral Bias Neural Network Activation Functions

Aug 6, 2026cs.LG

Alternating Levenberg-Marquardt Training of Physics-Informed Neural Networks with Fourier-Enhanced Features

Physics-informed neural networks (PINNs) often fail to accurately resolve partial differential equations (PDEs) with high-frequency or multi-scale solutions, as well as strongly nonlinear problems. Two factors underlie this difficulty: spectral bias, the tendency of neural networks to underfit high-frequency features; and representation-coefficient coupling, the entanglement of representation learning and coefficient fitting within a single nonconvex optimization objective. In this work, we propose the Fourier-enhanced alternating Levenberg--Marquardt PINN (FALM-PINN), an optimization framework that decouples representation learning from coefficient fitting. The upper-level problem learns a Fourier-enhanced basis that enriches the latent space with high-frequency components, while the lower-level problem resolves the coupling by fitting the projection coefficients on this basis, solving a nonlinear least-squares problem with the Levenberg--Marquardt algorithm. The framework applies to general nonlinear and coupled PDE systems, and reduces to a single-step convex optimization problem for linear PDEs. We prove global convergence of the alternating training scheme in both cases. Numerical examples on multiple challenging high-frequency and nonlinear PDEs show that FALM-PINN achieves relative

L^2

errors up to two orders of magnitude lower than state-of-the-art baselines.

Yulun Wu, Matthieu Barreau, Miguel Aguiar +1

Division of Decision and Control Systems, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, Stockholm, 100 44, Sweden

Neural PDE Solvers Spectral Bias Physics-Informed ML

Jul 13, 2026cs.LG

SPARC-Net: A Spectral, Causality-Aware, and Hard-Constrained Physics-Informed Architecture for Stiff and Shock-Dominated Partial Differential Equations

Physics-Informed Neural Networks (PINNs) provide a meshless approach for solving partial differential equations (PDEs), but suffer severe degradation in stiff and shock-dominated problems, where small PDE residuals can correspond to globally inaccurate solutions. We show these failures are multi-causal, arising from the concurrent interplay of (i) spectral bias against sharp features, (ii) imbalanced multi-term optimization and loss-weight collapse, (iii) violation of temporal causality, and (iv) under-resolved collocation. We present SPARC-Net, a unified architecture and training framework that jointly addresses all four pathologies. SPARC-Net leverages an adaptive multi-scale spectral encoder with a learnable spectral gate, a gated residual backbone, adaptive activations, and a hard-constraint output ansatz that exactly enforces initial and boundary conditions, structurally eliminating loss-weight collapse. Training employs stabilized gradient-norm loss balancing, floored causality-respecting residual weighting, and residual-based adaptive collocation (RAD). Validated against exact analytic and high-order spectral reference solutions across four canonical benchmarks -- viscous Burgers', Allen-Cahn, convection (beta=30), and reaction -- SPARC-Net yields substantial improvements over vanilla PINNs: relative L2 error drops from 1.47e-1 to 1.14e-1 on Burgers' (22% reduction), 9.93e-1 to 5.78e-2 on Allen-Cahn (94% reduction), and 9.82e-1 to 3.54e-3 on reaction (100% reduction). A characteristic-coordinate encoder for hyperbolic transport further reduces convection error from 5.14e-1 to 9.88e-5 (100% reduction). We report five-seed mean +/- standard deviation errors, Wilcoxon significance tests, full ablation studies, hyperparameter sensitivities, an extension to the 2D heat equation, and comparisons against parameter-matched baselines.

Divyavardhan Singh, Dimple Sonone, Hammad Mohammad +1

Dept. of Electronics Engg. SVNIT Surat, India

PDE Solving Spectral Bias Physics-Informed ML

Jul 2, 2026cs.LG

Frequency Shift Physics-Informed Extreme Learning Machine for Solving High-Frequency Partial Differential Equations

Solving partial differential equations (PDEs) with high-frequency solutions remains a central challenge in physics-informed machine learning due to spectral bias -- the tendency of neural networks to learn low-frequency components preferentially. This paper proposes a Frequency Shift Physics-Informed Extreme Learning Machine (FS-PIELM) framework that addresses this limitation through an additive mechanism for weight initialization. Rather than multiplying random weights by a scaling factor, the method translates the mean of the Gaussian weight distribution while keeping the variance fixed at unity, thereby avoiding the variance amplification inherent in scaling-based methods. Two variants are developed: FS-PIELM-L assigns independent frequency magnitudes to individual neurons, while FS-PIELM-G groups neurons for improved robustness. Theoretical analysis shows that the frequency variance under the proposed framework remains bounded and approaches unity regardless of target frequency, in contrast to the quadratic growth of conventional approaches. The method preserves the computational efficiency of extreme learning machines, requiring only a single linear solve. Experiments on seven benchmark problems spanning six equation types -- Helmholtz, wave, Poisson, Klein-Gordon, heat, and advection-diffusion -- on both regular and complex geometries show that the linear variant achieves the best accuracy in six of seven cases, with improvements of one to nearly five orders of magnitude over existing PIELM variants. The code and data accompanying this manuscript will be made publicly available at https://github.com/xgxgnpu/Physics-informed-vibe-coding/tree/main/FS-PIELM.

Xiong Xiong, Ruonan Zhai, Zheng Zeng +3

School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710072, China · MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi’an, 710072, China · College of Science, Shijiazhuang University, Shijiazhuang, 050035, China +2

PDE Solving Frequency-Domain Feature Learning Spectral Bias Physics-Informed ML

Jun 26, 2026cs.CV

ScaLe-INR: Scale and Learn Implicit Neural Representations

Implicit Neural Representations (INRs) parameterized by multilayer perceptrons excel at modeling continuous signals. However, a key challenge persists as INRs fundamentally suffer from spectral bias and information cross-talk. When a single network attempts to capture multi-scale phenomena, high-frequency weight updates destructively interfere with the underlying low-frequency structural approximation. We introduce Scale and Learn INR (ScaLe-INR), a novel multi-branch architecture that resolves these limitations by explicitly matching the signal's frequency spectrum with the optimal operating region of the INR. Drawing upon the Fourier inverse scaling theorem we demonstrate that applying directional coordinate scaling expands a network's representational bandwidth along specific spatial axes. To mathematically enforce functional disentanglement and minimize task-specific information leakage between branches, we propose a Directional Edge Guidance Loss, a spatially-conditioned sparsity prior derived from ground-truth gradients. By constraining the high-frequency branches to act as strict, localized edge-filters, ScaLe-INR eliminates spectral cross-talk, accelerates convergence, and achieves high-fidelity signal reconstruction on complex multi-scale topologies. We evaluate ScaLe-INR across diverse reconstruction and inverse tasks, demonstrating substantial performance gains over existing state-of-the-art (SOTA) methods. The proposed architecture improves upon the nearest baselines by +5.16 dB in image reconstruction and +0.65 dB in image denoising. Furthermore, it achieve an impressive figure of 50.02 dB on audio reconstruction and 0.999 IOU(Intersection Over Union) on 3D reconstruction which beats the all SOTA models.

Buwaneka Epakanda, Athulya Ratnayake, Pandula Thennakoon +4

Implicit Neural Representations Frequency-Domain Feature Learning Spectral Bias

Jun 25, 2026cs.CV

Calibrated Harmonic Overlaid Implicit Neural Representations for Multi-Dimensional Data

Implicit neural representation (INR) has emerged as a powerful prior for multi-dimensional data (e.g., multispectral images and videos). However, most INR methods employing periodic activation functions (e.g., Sine) predominantly rely on function composition. This mechanism introduces optimization instability as network depth increases, thereby limiting their performance. Meanwhile, these methods fail to incorporate proper physical priors to effectively alleviate spectrum bias. To address these issues, inspired by the commonalities between deep periodic networks and generalized Fourier series, we propose a novel Calibrated Harmonic Overlaid Implicit Neural Representation (CHOIR). Specifically, we utilize Coordinated Harmonic Superposition (CHS) to replace the conventional function composition used in most INRs, thereby ensuring optimization stability when scaling network depth. Furthermore, we introduce a Perceptual Spectrum Calibration (PSC) to mitigate spectrum bias. This calibration embeds the ubiquitous power-law spectrum prior of natural images and adjusts the globally fixed spectrum towards a physically plausible log-uniform distribution. Extensive experiments on various multidimensional data recovery problems demonstrate that our method achieves superior performance over state-of-the-art approaches. Code is available at https://github.com/chorl0229/CHOIR.

Honghang Chen, Xiujun Zhang, Xiaoli Sun +1

Shenzhen University, Shenzhen, China · Shenzhen Polytechnic University, Shenzhen, China · Southern Illinois University Carbondale, USA

Implicit Neural Representations Spectral Bias

Jun 22, 2026cs.CV

Spectral Gating via Damped Oscillations for Adaptive Implicit Neural Representations

Implicit Neural Representations (INRs) have been proven successful in encoding continuous signals through coordinate-based networks, yet facing a spectral dilemma: periodic activations capture fine details but act as all-pass filters that memorise noise, while spatially compact activations regularise effectively but suffer from low-frequency bias. Existing attempts to resolve this trade-off introduce computational overhead or tuning frailty. We propose to model each neuron's activation as the steady-state response of a sinusoidally-forced damped harmonic oscillator, whose amplitude naturally governs the network's spectral selectivity during training. By jointly optimising the oscillator parameters alongside the network weights, our method adapts to the target signal's spectral content without explicit regularisation. Initialised in the stopband, the network exhibits a coarse-to-fine learning curriculum that progressively expands its spectral gate, capturing low-frequency structures first and high-frequency details only when justified by the reconstruction objective. Comprehensive experiments show that our approach consistently achieves state-of-the-art or competitive results against established INRs, while requiring no task-specific tuning of any hyperparameters.

Alex Costanzino, Pierluigi Zama Ramirez, Giuseppe Lisanti +1

CVLab, University of Bologna · Ca’ Foscari University of Venice

Implicit Neural Representations Frequency-Domain Feature Learning Spectral Bias

Jun 22, 2026cs.LG

The Fractal Neural Operator: Overcoming Spectral Bias in Chaotic Attractors via Prime-Harmonic Weierstrass Encodings

Deep learning models, particularly Transformers and Neural Operators, exhibit a well-documented "spectral bias," effectively acting as low-pass filters that smooth out high-frequency information. While benign in fluid dynamics, this bias is catastrophic for Chaotic Dynamical Systems, where the underlying strange attractor is characterized by fractal geometry and infinite spectral density. We introduce the Fractal Neural Operator (FNO), a novel architecture that utilizes a non-resonant prime number basis to approximate continuous dynamical systems. Unlike geometric encodings (

2^k

), which suffer from spectral gaps and resonance, our Harmonic Weierstrass Encoder injects infinite spectral resolution into the latent space. We demonstrate that FNO extends the valid prediction horizon of the Lorenz-63 system to 347 Lyapunov times, exceeding state-of-the-art Reservoir Computing baselines by a factor of 2.3x. These results suggest that "chaos" is not inherently unpredictable to neural networks, but rather requires non-differentiable, fractal embedding manifolds.

Kanishk Awadhiya

Independent Researcher

Spectral Bias Chaotic Dynamical Systems Neural Operators Chaotic Time-Series Prediction

Jun 15, 2026cs.LG

RepNN: Tackling spectral bias in deep neural networks for regression and PDE problems via parameter reparameterization

Deep neural networks (DNNs) have achieved remarkable success in scientific computing, yet they often suffer from spectral bias in capturing oscillatory and multiscale behaviors. In this study, we investigate this limitation by examining the failure of shallow ReLU neural networks in fitting high-frequency functions. This observation identifies two important factors in resolving rapid oscillations: the initial slope scale and the distribution of partition points induced by the networks. Motivated by this analysis, we propose RepNN, a reparameterized neural network model with ReLU or tanh activations designed for high-frequency and multiscale problems. The key idea is to reparameterize the weights and biases in the first hidden layer, which enables effective control of the initial slope scale and provides an appropriate distribution of the initial partition points. Furthermore, treating the reparameterized weights and biases as trainable parameters allows the DNN to achieve adaptive frequency scaling during training. In addition, we derive quantitative estimates for the output and slope magnitudes of the reparameterized DNN to guide the initialization of the proposed method. Numerical experiments, including multiscale one-, two-, and four-dimensional function approximations, forward and inverse PDE problems in combination with physics-informed neural networks (PINNs), and operator learning for an earthquake problem using real data, demonstrate that RepNN improves the predicted accuracy of vanilla DNNs in capturing highly oscillatory features. These results indicate that RepNN provides an effective and flexible approach for overcoming spectral bias and applying DNNs to multiscale problems.

Yong Wang, Tao Zhou, Xuhui Meng

Institute of Interdisciplinary Research for Mathematics and Applied Science, School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China · Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

PDE Solving Spectral Bias Neural Network Approximation Theory

Jun 9, 2026cs.LG

SirenFNO: Efficient and Full Frequency Learning of Fourier Neural Operators

Fourier neural operators (FNOs) are effective and efficient surrogates for approximating solutions of PDEs and generalize across discretizations. However, owing to the reliance on frequency truncation to maintain learning efficiency of FNOs, empirical studies suggest that FNOs exhibit spectral bias toward low-frequency information, which may hinder the learning capability especially for certain PDEs with strong high-frequency oscillations. To address this limitation, we propose SirenFNO, a novel framework that leverages sinusoidal representation networks (SIRENs) to learn implicit neural representations and performs mode-wise kernel parameterization. Our SIREN parameterization learns a full-grid spectrum with a constant and discretization-independent parameter count, thereby eliminating the need for frequency truncation. We further extend SirenFNO with functional tensor decompositions to enhance parameter and learning efficiency. Empirical results show that our SirenFNO consistently outperforms FNO with approximately

4

to

15

times parameter reductions with preserved discretization invariance, and our functional decomposition variants obtain performance improvements with a maximum of

73

times fewer parameters across multiple PDE benchmarks.

Pengqing Shi, Jie Yin, Stephen Tierney +1

The University of Sydney, Australia

Fourier Neural Operator Spectral Bias PDE Operator Learning

Jun 2, 2026cs.LG

Correcting Neural Operator Spectral Bias via Diffusion Posterior Sampling with Sparse Observations

Neural operator surrogates (NO) approximate PDE solutions orders of magnitude faster than numerical solvers, but suffer from spectral bias: high-frequency content is systematically attenuated, limiting reliability where fine-scale structure matters. Sparse sensor measurements of the field are often available too, offering pointwise accuracy without spectral distortion but covering only a small fraction of the domain. We address this by treating NO predictions as auxiliary observations in a diffusion posterior sampling framework. Our method, FreqNO-DPS (https://github.com/niccoloperrone/FreqNO-DPS), combines an unconditional score-based diffusion prior, trained on high-fidelity simulations, with diffusion posterior sampling (DPS) conditioned on sparse observations and guided by a frozen neural operator. Naive integration reintroduces the surrogate's spectral bias; we resolve this with a closed-form, spectrally shaped guidance score that weights the surrogate by its frequency-dependent accuracy and needs no denoiser backpropagation. A distribution-free analysis bounds the approximation error across the frequency-diffusion-time plane and shows the guidance's frequency dependence is preserved regardless of distributional assumptions. On 3D elastic wavefield prediction at 5% and 2% sensor coverage, the method reaches near-zero spectral bias across all bands, where both the surrogate and sensor-only DPS show systematic high-frequency attenuation. Isotropic guidance, the natural baseline, improves pointwise accuracy but carries the bias into the posterior nearly intact, confirming that frequency-dependent calibration is essential, not merely beneficial. The framework needs only paired surrogate/reference data and exploits no problem-specific structure beyond the residual's approximate spectral diagonality, verifiable for new surrogates via the coherence diagnostic we provide.

Niccolò Perrone, Fanny Lehmann, Stefania Fresca +1

Université Paris-Saclay, CentraleSupélec, CNRS, ENS Paris-Saclay · Laboratoire de Mécanique Paris-Saclay UMR 9026 · 8-10 rue Joliot Curie, 91190 Gif-sur-Yvette, France +5

PDE Surrogate Modeling Spectral Bias Diffusion-Based Inverse Problems Diffusion Posterior Sampling Neural Operators

May 28, 2026cs.CV

Colored Noise Diffusion Sampling

Diffusion models achieve state-of-the-art image synthesis, with their generative trajectories fundamentally exhibiting a spectral bias, resolving low-frequency global structures early and high-frequency fine details later. Conventional stochastic differential equation (SDE) solvers fail to account for this dynamic, naively injecting uniform white noise throughout the entire process and misusing the finite energy budget. In this work, we establish a mathematical framework that reconsiders SDE inference as a targeted, frequency-decoupled energy transfer. Leveraging this framework, we introduce Colored Noise Sampling (CNS), a novel, training-free stochastic solver. Rather than injecting uniform white noise, CNS utilizes a dynamic, timestep- and frequency-dependent schedule that more efficiently allocates injected energy toward structurally unresolved frequency bands. By actively exploiting the model's inherent spectral bias, CNS systematically steers the generated distribution toward the true data manifold. Extensive experiments demonstrate that CNS significantly outperforms standard ODE and SDE baselines as a strictly plug-and-play, inference-time sampler substitution across diverse architectures (SiT, JiT, FLUX). Compared to standard sampling on ImageNet-256, CNS achieves substantial unguided FID reductions, improving from 8.26 to 6.27 on SiT-XL/2, 32.39 to 26.69 on JiT-B/16, and 11.88 to 8.31 on JiT-H/16, while yielding consistent relative FID improvements with Classifier-Free Guidance. Project page is available at https://hadardavidson.github.io/CNS/.

Hadar Davidson, Noam Issachar, Sagie Benaim

The Hebrew University of Jerusalem

Diffusion Sampling Spectral Bias Diffusion-Based Image Generation

May 26, 2026cs.LG

Aperiodic and Low-Frequency Spectral Bias in Reconstruction based EEG Foundation Models

EEG foundation models, pre-trained on large-scale unlabelled EEG data, have emerged as a promising direction towards learning generalizable EEG representations. Despite showing positive results in data-rich regimes, they often fail to outperform significantly smaller supervised models in low-resource settings compared to fully supervised models. We provide a mechanistic account of this shortcoming, attributing it to a fundamental mismatch between reconstruction-based pretext tasks and the idiosyncratic spectral structure of EEG signals, which decompose into distinct high-power aperiodic and low-power oscillatory components. Using controlled, synthetically-generated EEG inputs, we demonstrate that EEG foundation model embeddings are biased to capture the aperiodic components of the EEG signal while under-representing oscillatory components, particularly at higher frequencies. Additionally, linear probe evaluations on real-world BCI datasets further reveal that embeddings encode subject identity more strongly than task-relevant information, thereby reinforcing the low-frequency and aperiodic component bias in foundation model embeddings trained primarily on reconstruction based objectives. Together, these findings elucidate a failure mode in reconstruction based EEG foundation models and motivate future work to incorporate auxiliary losses explicitly targeting high-frequency oscillatory structure as a path toward more capable and generalizable EEG representations.

Aditya Kommineni, Emily Zhou, Kleanthis Avramidis +7

Ming Hsieh Department of Electrical and Computer Engineering1 University of Southern California · Thomas Lord Department of Computer Science2[cs.LG] University of Southern California · Department of Mathematical Sciences3 Aalborg University

Self-Supervised Time Series Representation Learning EEG Decoding Frequency-Domain Feature Learning Spectral Bias EEG Foundation Models

May 21, 2026cs.LG

Iterative Refinement Neural Operators are Learned Fixed-Point Solvers: A Principled Approach to Spectral Bias Mitigation

Neural operators serve as fast, data-driven surrogates for scientific modeling but typically rely on a monolithic, single-pass inference procedure that struggles to resolve high-frequency details, a limitation known as spectral bias. We introduce the Iterative Refinement Neural Operator (IRNO), which augments pre-trained operators with a learned refinement module iteratively applied via fixed-point iteration. IRNO decomposes the prediction into a coarse initialization followed by successive residual corrections, paralleling classical numerical solvers. Under local assumptions, we establish contraction of the induced operator, ensuring convergence to a unique fixed point. To explicitly target high-frequency errors, we propose a progressive spectral loss that adaptively increases penalty on high-frequency components over refinement steps during training. Across physical systems, IRNO consistently lowers error, with up to 56.05% improvement on turbulent flow. On Active Matter, spectral analysis reveals that, relative to base operator, the normalized error ratios decrease to 27.72-36.10% in low-, 5.07-6.68% in mid-, and 1.48-2.04% in high-frequencies, remaining stable beyond the trained iteration count. Code is available at https://github.com/xiaotianliu-dartmouth/Iterative_Refinement_Neural_Operator

Xiaotian Liu, Shuyuan Shang, Xiaopeng Wang +2

1Dartmouth College · 2CUHK Shenzhen · 3Lawrence Berkeley National Lab

Neural PDE Solvers Spectral Bias Iterative Refinement Neural Operators

May 16, 2026stat.ML

A Fourier perspective on the learning dynamics of neural networks: from sample complexities to mechanistic insights

Neural networks trained with gradient-based methods exhibit a strong simplicity bias: they learn simpler statistical features of their data before moving to more complex features. Previous analyses of this phenomenon have largely focused on settings with (quasi-)isotropic inputs. In this work, we study the simplicity bias from a Fourier perspective, which allows us to include two key features of natural images in the analysis: approximate translation-invariance and power-law spectra. We first show experimentally that simple neural networks trained on image classification tasks first rely on amplitude information -- related to pair-wise correlations between pixels -- before exploiting phase information, which encodes edges and higher-order correlations. In view of this, we introduce a synthetic data model for translation-invariant inputs that allows precise control over amplitudes and phases while remaining tractable. We rigorously establish that for isotropic and high-dimensional inputs, classification based on phase information alone is a genuinely hard task: online stochastic gradient descent (SGD) cannot distinguish the structured inputs from noise within

n \ll N^3

steps, but needs at least

n \gg N^3 \log^2{N}

steps. In contrast, we show both experimentally and theoretically that power-law spectra can dramatically accelerate the speed of learning phase information, even if the spectra do not help with classification. Simulations with two-layer networks trained on textures and with deep convolutional networks on ImageNet and CIFAR100 confirm this non-trivial interaction between amplitudes and phases, providing mechanistic insights into how deep neural networks can learn natural image distributions efficiently.

Fabiola Ricci, Claudia Merger, Sebastian Goldt

International School of Advanced Studies (SISSA), Trieste, Italy

Sample Complexity Frequency-Domain Feature Learning Spectral Bias Neural Network Training Dynamics

May 13, 2026cs.LG

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

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.

Runlong Xie, An Luo

Independent Researcher · School of Statistics, University of Minnesota, MN, USA.

Fourier Neural Operator PDE Solving OOD Generalization Spectral Bias PDE Operator Learning Deep Operator Networks

May 12, 2026cs.LG

Spectral Energy Centroid: a Metric for Improving Performance and Analyzing Spectral Bias in Implicit Neural Representations

Implicit Neural Representations (INRs) model continuous signals using multilayer perceptrons (MLPs), enabling compact, differentiable, and high-fidelity representations of data across diverse domains. However, due to the low-frequency bias of MLPs that prevents effective learning of small details, the model's frequency must be carefully tuned through the embedding layer. Prior work established that this tuning can be performed before training based on the target signal, but it did not account for the significant effect of model depth, indicating that our understanding of the relationship between frequency and INR performance remains limited. To gain insights into this relationship, we utilize the Spectral Energy Centroid (SEC) metric that quantifies the frequency of target images and the spectral bias of INR models. We show that SEC is a versatile tool for INR analysis, demonstrating its utility across three tasks: (1) a data-driven strategy (SEC-Conf) for hyperparameter selection that outperforms existing heuristics and is robust to model depth, (2) a reliable proxy for signal complexity, and (3) effective alignment of spectral biases across diverse INR architectures.

Tomasz Dądela, Adam Kania, Maciej Rut +1

Jagiellonian University IDEAS Kraków, Poland

Implicit Neural Representations Spectral Bias Hyperparameter Optimization

May 8, 2026cs.LG

Convergence Analysis of Newton's Method for Neural Networks in the Overparameterized Limit

A convergence analysis is developed for the regularized Newton method for training neural networks (NNs) in the overparameterized limit. As the number of hidden units tends to infinity, the NN training dynamics converge in probability to the solution of a deterministic limit equation involving a ``Newton neural tangent kernel'' (NNTK). Explicit rates characterizing this convergence are provided and, in the infinite-width limit, we prove that the NN converges exponentially fast to the target data (i.e., a global minimizer with zero loss). We show that this convergence is uniform across the frequency spectrum, addressing the spectral bias inherent in gradient descent. The eigenvalues of the NTK for gradient descent accumulate at zero, leading to slow convergence for target data with high-frequency components. In contrast, the NNTK has uniformly lower bounded eigenvalues if the regularization parameter is selected appropriately, allowing Newton's method to converge more quickly for data with high-frequency components. Mathematical challenges that need to be addressed in our analysis include the implicit parameter update of the Newton method with a potentially indefinite Hessian matrix and the fact that the dimension of this linear system of equations tends to infinity as the NN width grows. This complicates deriving the training dynamics in the overparameterized limit as well as proving the convergence of the finite-width dynamics thereto. The analysis identifies a scaling formula for selecting the regularization parameter, which we show can vanish at a suitable rate as the number of hidden units becomes larger. We prove that, for sufficiently large numbers of hidden units, the regularized Hessian remains positive definite during training and the Newton updates for individual NN parameters converge to zero, showing that the model behaves as a linearization around the initialization.

Konstantin Riedl, Konstantinos Spiliopoulos, Justin Sirignano

Mathematical Institute University of Oxford · Department of Mathematics & Statistics Boston University

Neural Network Optimization Spectral Bias Second-Order Optimization Neural Network Training Dynamics Neural Tangent Kernel

May 8, 2026cs.LG

Diagnosing Spectral Ceilings in Equivariant Neural Force Fields

We introduce a spectral-injection diagnostic for measuring which angular frequencies a trained equivariant force-field backbone preserves: inject a controlled angular-frequency perturbation into a molecular force field, attach a lightweight Spectral Prediction Network (SPN) to the frozen backbone, and read off which frequencies are recoverable. On aspirin, a quadratic SPN attached to an L = 2 NequIP backbone recovers the boundary signal at l = 4 but collapses at l = 5: a 11.7x cliff at the predicted drL boundary, with p dropping from 0.913 to 0.078. The same boundary-vs-above contrast persists across n = 4 independently trained backbones (raw-gain delta contrast, hierarchical cluster bootstrap) and is corroborated by a denominator-free injected-residual metric (R2_inj(4) = 0.374 versus R2_inj(5) = 0.006). A finite-degree span theorem calibrates the diagnostic: for a single marked direction, degree-d polynomials of degree-L spherical-harmonic features span exactly H less than or equal to dL with multiplicity-one saturation at the boundary (scoped to single-direction degree-bounded probes, not a function-class upper bound on multi-atom MPNNs). A synthetic C5 calibration plus capacity, activation, and cross-architecture controls rule out parameter count alone as the explanation.

Hyunmog Kim

Independent Researcher.

Machine Learning Interatomic Potentials Spectral Bias Equivariant Neural Networks Spectral Methods Neural Network Expressivity

May 1, 2026cs.LG

Topological Neural Tangent Kernel

Graph neural tangent kernels give a principled infinite-width theory for graph neural networks, but inherit a basic limitation of graph models: they see only pairwise structure. Many relational systems contain higher-order interactions that are more naturally represented by simplicial complexes. We introduce the Topological Neural Tangent Kernel (TopoNTK), an infinite-width kernel for simplicial message passing on edge features. TopoNTK combines lower Hodge interactions, capturing graph-like coupling through shared vertices, with upper Hodge interactions, capturing coupling through filled simplices. This makes the kernel sensitive to topology invisible to graph kernels, allowing complexes with the same graph but different filled simplices to induce different kernels. Beyond expressivity, the Hodge structure gives the kernel an interpretable learning geometry. Edge signals decompose into gradient-like, harmonic, and local circulation components, and the spectrum of the TopoNTK determines how quickly each component is learned. This yields a topological form of spectral bias: components aligned with large-eigenvalue modes are learned quickly, while global harmonic modes, retained through the residual channel, often lie at smaller eigenvalues and are learned more slowly. We prove expressivity, Hodge-alignment, spectral learning, and stability properties, and validate them on synthetic simplicial tasks and DBLP higher-order link prediction. The results show that topology is not merely extra structure; it can provide coordinates that make relational learning more faithful, interpretable, and effective.

Sanjukta Krishnagopal

Department of Computer Science, University of California Santa Barbara

Spectral Bias Neural Tangent Kernel Topological Representation Learning

May 1, 2026cs.LG

PILIR: Physics-Informed Local Implicit Representation

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.

Jianfeng Li, Feng Wang, Ke Tang

School of Computer Science, Wuhan University · School of Computer Science and Technology, Southern University of Science and Technology

Neural PDE Solvers Implicit Neural Representations Spectral Bias PDE Operator Learning Physics-Informed ML Physics-Informed Neural Operators

Apr 16, 2026cs.CV

Implicit Neural Representations: A Signal Processing Perspective

Implicit neural representations (INRs) mark a fundamental shift in signal modeling, moving from discrete sampled data to continuous functional representations. By parameterizing signals as neural networks, INRs provide a unified framework for representing images, audio, video, 3D geometry, and beyond as continuous functions of their coordinates. This functional viewpoint enables signal operations such as differentiation to be carried out analytically through automatic differentiation rather than through discrete approximations. In this article, we examine the evolution of INRs from a signal processing perspective, emphasizing spectral behavior, sampling theory, and multiscale representation. We trace the progression from standard coordinate based networks, which exhibit a spectral bias toward low frequency components, to more advanced designs that reshape the approximation space through specialized activations, including periodic, localized, and adaptive functions. We also discuss structured representations, such as hierarchical decompositions and hash grid encodings, that improve spatial adaptivity and computational efficiency. We further highlight the utility of INRs across a broad range of applications, including inverse problems in medical and radar imaging, compression, and 3D scene representation. By interpreting INRs as learned signal models whose approximation spaces adapt to the underlying data, this article clarifies the field's core conceptual developments and outlines open challenges in theoretical stability, weight space interpretability, and large scale generalization.

Dhananjaya Jayasundara, Vishal M. Patel

Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD 21218 USA

Implicit Neural Representations Spectral Bias Neural Network Approximation Theory

Feb 16, 2026cs.LG

Pseudo-differential-enhanced physics-informed neural networks

We present pseudo-differential enhanced physics-informed neural networks (PINNs), an extension of gradient enhancement but in Fourier space. Gradient enhancement of PINNs dictates that the PDE residual is taken to a higher differential order than prescribed by the PDE, added to the objective as an augmented term in order to improve training and overall learning fidelity. We propose the same procedure after application via Fourier transforms, since differentiating in Fourier space is multiplication with the Fourier wavenumber under suitable decay. Our methods are fast and efficient. Our methods oftentimes achieve superior PINN versus numerical error in fewer training iterations, potentially pair well with few/fixed samples in collocation, and can achieve breakthrough rather than gradual effects on the error. Moreover, our methods are suitable for fractional derivatives. We establish that our methods, due to the dynamical effects, improve spectral eigenvalue decay of the neural tangent kernel (NTK), and so our methods contribute towards the learning of high frequencies in early training, mitigating the effects of frequency bias up to the polynomial order and possibly greater with smooth activations. In particular, our primary contribution can be characterized as gradient enhancement affects spectral bias, and we specialize to Fourier space, although our theory does not prove the ambient case exactly due to a result with the Plancherel theorem on the descent paths. Our methods accommodate advanced techniques in PINNs, such as Fourier feature embeddings. A pitfall of discrete Fourier transforms via the Fast Fourier Transform (FFT) is mesh subjugation, and so we demonstrate compatibility of our methods for greater mesh flexibility and invariance on alternative Euclidean and non-Euclidean domains via Monte Carlo methods and otherwise, although possessing trade-offs of their own.

Andrew Gracyk

Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States

PDE Solving Spectral Bias Physics-Informed ML