FS-PIELM shifts the mean of Gaussian weights (variance fixed at 1) in PIELM to bound frequency variance and achieve 1-5 orders of magnitude better accuracy on high-frequency PDE benchmarks while retaining single linear solve efficiency.
arXiv preprint arXiv:1912.01198 , year=
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abstract
An intriguing phenomenon observed during training neural networks is the spectral bias, which states that neural networks are biased towards learning less complex functions. The priority of learning functions with low complexity might be at the core of explaining generalization ability of neural network, and certain efforts have been made to provide theoretical explanation for spectral bias. However, there is still no satisfying theoretical result justifying the underlying mechanism of spectral bias. In this paper, we give a comprehensive and rigorous explanation for spectral bias and relate it with the neural tangent kernel function proposed in recent work. We prove that the training process of neural networks can be decomposed along different directions defined by the eigenfunctions of the neural tangent kernel, where each direction has its own convergence rate and the rate is determined by the corresponding eigenvalue. We then provide a case study when the input data is uniformly distributed over the unit sphere, and show that lower degree spherical harmonics are easier to be learned by over-parameterized neural networks. Finally, we provide numerical experiments to demonstrate the correctness of our theory. Our experimental results also show that our theory can tolerate certain model misspecification in terms of the input data distribution.
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Neural tangent kernel from neural reparameterization modulates sensitivity and wave tangent kernels to produce spectral filtering, wavenumber modulation, and frequency bias that improve NeurFWI convergence.
The global empirical NTK for finite-width networks has a universal Kronecker-core form that makes it structurally low-rank and biases gradient descent toward dominant modes of joint input-hidden activity.
Establishes convergence guarantees for overparameterized 2-layer ReLU networks in flow matching, generalization bounds for the velocity-field objective, and Wasserstein guarantees for generated samples, using multi-task representation learning bounds.
beignet replaces random Fourier feature embeddings in PINNs with a trainable multi-resolution Fourier feature pyramid, achieving higher accuracy on PDE benchmarks with fewer parameters and near machine precision residuals on the inviscid Burgers blowup using Adam.
Simple feed-forward neural networks trained on crossing symmetry plus a single anchor value reproduce CFT correlators to percent-level accuracy, and the authors conjecture this works because physical correlators are the smoothest allowed functions.
SPIN lets weak LLMs become strong by self-generating training data from previous model versions and training to prefer human-annotated responses over its own outputs, outperforming DPO even with extra GPT-4 data on benchmarks.
A GAN with elliptical-spiral feature mixing, star-operation blocks, and Sobel edge loss produces more realistic synthetic handwriting and lowers HTR error rates on English and Vietnamese datasets.
Quantum computers may enable more natural manipulation of Fourier spectra in ML models via the Quantum Fourier Transform, potentially leading to resource-efficient spectral methods.
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Frequency Shift Physics-Informed Extreme Learning Machine for Solving High-Frequency Partial Differential Equations
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Deciphering Neural Reparameterized Full-Waveform Inversion with Neural Sensitivity Kernel and Wave Tangent Kernel
Neural tangent kernel from neural reparameterization modulates sensitivity and wave tangent kernels to produce spectral filtering, wavenumber modulation, and frequency bias that improve NeurFWI convergence.
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The Global Empirical NTK: Self-Referential Bias and Dimensionality of Gradient Descent Learning
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A Theory on Flow Matching with Neural Networks
Establishes convergence guarantees for overparameterized 2-layer ReLU networks in flow matching, generalization bounds for the velocity-field objective, and Wasserstein guarantees for generated samples, using multi-task representation learning bounds.
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Fourier Feature Pyramids for Physics-Informed Neural Networks
beignet replaces random Fourier feature embeddings in PINNs with a trainable multi-resolution Fourier feature pyramid, achieving higher accuracy on PDE benchmarks with fewer parameters and near machine precision residuals on the inviscid Burgers blowup using Adam.
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Neural Spectral Bias and Conformal Correlators I: Introduction and Applications
Simple feed-forward neural networks trained on crossing symmetry plus a single anchor value reproduce CFT correlators to percent-level accuracy, and the authors conjecture this works because physical correlators are the smoothest allowed functions.
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