Equivariant neural networks use symmetry to interpret complex data.
problem Interpreting and understanding the behavior of equivariant neural networks.
method Decompose layers into simple representations and analyze nonlinear activation functions.
result Equivariant neural networks can be interpreted using a filtration generalizing Fourier series.
A new method integrates Fourier basis expansion and mapping for improved time series forecasting.
problem Inconsistent starting cycles and series length issues in Fourier-based methods.
method Fourier Basis Mapping (FBM) method that integrates time-frequency features through Fourier basis expansion and mapping.
result FBM addresses inconsistencies and preserves temporal characteristics, achieving SOTA performance.
FNN approximates functions and solves PDEs with periodic BCs.
problem Approximating and solving periodic functions and PDEs.
method Fourier neural network architecture with activation and loss functions.
result FNN can solve PDEs with periodic BCs and is interpretable.
Paper introduces FNM framework for learning finite-dimensional parametrized models.
problem Efficiently learning finite-dimensional parametrized models from limited data.
method Fourier Neural Mappings (FNMs) framework for operator learning.
result End-to-end learning of PtO maps can be less data-efficient than learning the solution operator first.
DAFNO learns surrogates for complex systems on irregular geometries.
problem Learning accurate surrogates for complex physical systems on irregular geometries.
method DAFNO incorporates a smoothed characteristic function in the integral layer architecture of FNOs, leveraging FFT for rapid computations.
result DAFNO achieves state-of-the-art accuracy on material modeling and airfoil simulation datasets.
Study reveals how neural network biases align with adversarial attack frequencies.
problem Correlation between neural network biases and adversarial attacks.
method Fourier transform analysis of network implicit bias and adversarial perturbations.
result Network bias and adversarial attack frequencies are highly correlated.
Random Fourier features improve tabular deep learning convergence.
problem Tabular deep learning convergence issues.
method Random Fourier projections as a pre-processing step, projecting inputs into a fixed feature space.
result Random Fourier pre-processing accelerates tabular deep learning convergence.
This paper analyzes SHAP values using Fourier expansions for model interpretability.
problem Understanding and interpreting SHAP values in complex models.
method Developed a spectral framework using Fourier expansions for SHAP values in various model regimes.
result SHAP values are Lipschitz continuous in the deterministic regime and converge to Gaussian process values in the probabilistic regime.
HFNO enhances interpretability of turbulent flows through parallel wavenumber bin processing.
problem Opaque inner workings of Fourier Neural Operators (FNOs) hinder physical interpretability.
method Introduces HFNO, a novel FNO-based architecture that processes wavenumber bins in parallel, enhancing interpretability.
result HFNO decomposes turbulent flows across various scales, enabling increased interpretability and multiscale modeling.
Derives representations invariant under crystallographic groups for functions.
problem Representing and learning functions invariant under crystallographic groups.
method Derives linear and nonlinear representations of functions invariant under crystallographic groups.
result Derives orthonormal crystallographically invariant basis functions and embedding maps.
Unified method for deriving ridgelet transforms for various neural network architectures.
problem Deriving closed-form expressions for ridgelet transforms in modern neural network architectures.
method Unified Fourier slice method to derive ridgelet transforms for diverse neural network types.
result Systematic method to derive ridgelet transforms for various neural network architectures.
Study approximates operator learning for PDEs using Fourier multipliers.
problem Approximating operator behavior for PDE simulations.
method Approximation of operator symbols in Fourier domain using semi-norms.
result Identifies conditions for achieving predefined approximation error.
Improved electrical load forecasting model using Fourier-enhanced RNN.
problem Electrical load time series downscaling with high accuracy and low error.
method Combines recurrent neural network with Fourier seasonal embeddings and self-attention.
result Significantly reduces RMSE across different time horizons compared to existing methods.
NFM models time-series data directly in the Fourier domain, achieving state-of-the-art performance.
problem Traditional time-series analysis focuses on the time domain, limiting flexibility.
method NFM models time-series data in the Fourier domain, using frequency extrapolation and interpolation.
result NFM achieves state-of-the-art performance on various time-series tasks.
In this paper, we present a reverberation removal approach for speaker verification, utilizing dual-label deep neural networks (DNNs). The networks perform feature mapping between the spectral features of reverberant and clean speech. Long short term memory recurrent neural networks (LSTMs) are trained to map corrupted…
Efficient trainable front-end for neural speech enhancement.
problem Inefficient STFT front-ends in neural speech enhancement models.
method Butterfly mechanism for Fast Fourier Transform, trainable STFT window.
result Accuracy and efficiency improvements for low-compute systems.
New proof of timelike minimal surfaces using split-harmonic maps.
problem Interpolating a split-Fourier curve to a timelike minimal surface.
method Using split-harmonic maps to solve the singular Björling problem.
result Solved the interpolation problem for timelike minimal surfaces.
DeepPhaseCut uses neural networks to improve Fourier phase retrieval.
problem Fourier phase retrieval from magnitude data.
method Unsupervised feed-forward neural network with cycleGAN training.
result Outperforms existing methods in Fourier phase retrieval.
Study identifies and analyzes three types of errors in learning Fourier operators.
problem Statistical, discretization, and truncation errors in learning Fourier operators.
method Analysis of a Discrete Fourier Transform (DFT) based least squares estimator.
result Established upper and lower bounds on statistical, discretization, and truncation errors.
New random feature maps for Laplacian and related kernels.
problem Challenges in approximating the Laplacian kernel and its generalizations.
method Developed random feature maps for Laplacian and related kernels, providing efficient sampling schemes.
result Demonstrated the efficacy of these random feature maps on real datasets.
UNOT solves optimal transport problems efficiently using neural networks.
problem Computational expense in solving optimal transport problems.
method UNOT (Universal Neural Optimal Transport) uses Fourier Neural Operators to predict OT distances and plans accurately and efficiently.
result UNOT achieves up to 7.4x speedup over the Sinkhorn algorithm while maintaining accuracy.
RNNs solve modular addition tasks using low rank and sparse Fourier structures.
problem Solving modular addition tasks with recurrent neural networks.
method Identified low rank structures and sparse Fourier representations in RNN weights.
result RNNs robust to removing individual frequencies but degrade with more ablation.
FNOs improve spatio-temporal forecasting without needing PDE details.
problem Complex spatio-temporal dynamics in physical and biological phenomena.
method Fourier Neural Operators (FNOs) for dynamic spatio-temporal modeling.
result FNO forecasts are accurate and capture complex real-world dependencies.
Given two compact hyperkähler surfaces X and Y and a holomorphic vector bundle Q on X×Y, which is a generalized instanton, one can define a Fourier-Mukai transform, which, under suitable assumptions, maps vector bundles on X to vector bundles on Y. If X and Y are dual complex tori, this transform …
Researchers use operator learning to predict cardiac activation and repolarization times.
problem Computational demands and need for clear, interpretable information in cardiac electrophysiology.
method Exploiting Fourier Neural Operators (FNO) and Kernel Operator Learning (KOL) to learn operator mappings.
result Both FNO and KOL approaches are computationally efficient and robust to hyperparameters.
FourNet approximates financial transition densities using Fourier transforms.
problem Approximating transition densities in finance with high accuracy.
method FourNet is a novel FFNN with Gaussian activation, learning from characteristic functions.
result FourNet can approximate transition densities arbitrarily well with finite neurons.
A fast calibration method for rough volatility models with jumps.
problem Calibrating stochastic volatility models to market data efficiently.
method Structure-preserving approach: split pricing formula, precompute data-independent integrals, and approximate market-dependent remainder with neural networks.
result Calibration achieves high accuracy and speed, and a pure-jump rough volatility model adequately captures VIX dynamics.
FNSDA adapts to new dynamics via Fourier space adaptation.
problem Generalizing to unseen dynamical systems with limited data.
method Automatic partitioning of known environments in Fourier modes and adaptation of specific modes for new environments.
result FNSDA achieves superior or competitive generalization performance with reduced parameter cost.
Fourier Neural Operators accurately predict dynamics of high-dimensional ionic models.
problem Approximating stiff, multiscale ionic models using neural networks.
method Fourier Neural Operators for learning dynamics of high-dimensional ionic models.
result Fourier Neural Operators can accurately predict dynamics of high-dimensional ionic models.
Improved sampling strategy reduces Fourier measurements for neural network signals.
problem Efficiently sampling signals from neural networks with random Fourier matrices.
method Model-adapted sampling strategy with improved sample complexity.
result Reduced sample complexity from O(kdnα∞²) to O(kdα²₂) measurements.
Study extends neural network approximation to time-varying PDEs using Fourier-Lebesgue spaces.
problem Limitation to static PDEs and different time-domain regularity.
method Extend spectral Barron spaces to anisotropic weighted Fourier-Lebesgue spaces, measure approximation error in Bochner-Sobolev norm.
result Established bound on approximation rate for functions in anisotropic weighted Fourier-Lebesgue spaces.
The paper derives and proves the Helgason Fourier transform for vector bundle-valued differential forms on homogeneous spaces.
problem Deriving the Helgason Fourier transform for vector bundle-valued differential forms on homogeneous spaces.
method Employing the perspective of the functional equation satisfied by the classical Fourier transform, the paper derives the Helgason Fourier transform map and proves its properties.
result The Fourier transform is explicitly given and proven to be a map from vector bundle-valued differential forms to another vector bundle-valued differential form on the product space.
Study bounds Rademacher complexity of Fourier neural operators.
problem Bounding Rademacher complexity for Fourier neural operators.
method Investigated using specific group norms and capacity.
result Inferred that group norms determine model information.
New method uses random features and Tikhonov regularization for operator learning from noisy data.
problem Accurate approximation of mappings between infinite-dimensional function spaces with reduced training time.
method Regularized random Fourier features (RRFF) coupled with finite element reconstruction (RRFF-FEM).
result The method achieves improved performance with reduced training time and noise robustness.
We construct a certain cross product of two copies of the braided dual H~ of a quasitriangular Hopf algebra H, which we call the elliptic double EH, and which we use to construct representations of the punctured elliptic braid group extending the well-known representations of the planar braid group attache…
A new CNN-based algorithm improves Fourier ptychography for faster, more robust image reconstruction.
problem Slow and inefficient Fourier ptychography reconstruction under system aberrations.
method A CNN-based iterative phase retrieval algorithm trained on GPUs.
result Significantly faster and more robust image reconstruction under system aberrations.
A neural network method estimates densities from characteristic functions.
problem Estimating fixed-horizon probability densities from empirical characteristic functions.
method Data-driven Fourier-mixture neural-network method trained in Fourier space.
result Competitive performance and clear gains on heavy-tailed targets.
Random SNNs are stable and simple, with low-frequency Fourier spectra.
problem Stability and robustness of spiking neural networks.
method Boolean function analysis and Fourier spectrum concentration.
result Random LIF-SNNs are stable and biased towards simple functions.
Improved Gaussian Process regression using TQFF over RFF and Gaussian QFF.
problem Limited performance of Quadrature Fourier Features (QFF) in approximating highly oscillatory functions.
method Developed Trigonometric Quadrature Fourier Features (TQFF) using a novel non-Gaussian quadrature rule.
result TQFF provides better approximation accuracy and fewer features compared to RFF and Gaussian QFF.
New PINN architectures learn high-frequency features using Fourier features.
problem PINNs struggle with high-frequency or multi-scale features.
method Employ spatio-temporal and multi-scale random Fourier features.
result Effective PINN models for multi-scale PDEs.
We establish a link between Fourier optics and a recent construction from the machine learning community termed the kernel mean map. Using the Fraunhofer approximation, it identifies the kernel with the squared Fourier transform of the aperture. This allows us to use results about the invertibility of the kernel mean m…
New bounds on ReLU networks for low-regular functions.
problem Bounding approximation error for ReLU networks on low-regular functions.
method Complexity analysis of Fourier features residual networks to ReLU networks.
result Approximation error bound proportional to target function norm and inversely proportional to network width and depth.
This paper establishes rates of universal approximation for the shallow neural tangent kernel (NTK): network weights are only allowed microscopic changes from random initialization, which entails that activations are mostly unchanged, and the network is nearly equivalent to its linearization. Concretely, the paper has …
This work extends Gaussian process priors to neural operators for function space mappings.
problem Improving uncertainty quantification in deep neural networks.
method Extending Gaussian process priors to neural operators with conditions for convergence and computation of covariance functions.
result Arbitrary-depth neural operators with Gaussian kernels converge to function-valued GPs, enabling posterior computation in regression scenarios.
Quantum method improves neural density estimation in high dimensions.
problem High-dimensional density estimation with poor performance and high computational complexity.
method Adaptive Fourier features based on quantum density matrices, integrated with neural networks.
result Competitive performance compared to state-of-the-art methods in various datasets.
Deep neural network approximates flow averages for rough walls in multiscale simulations.
problem Approximating flow averages in rough-wall Stokes flow simulations.
method Fourier neural operator for local averages, parameterized by local wall geometry.
result Stable and accurate HMM solution with reduced micro problem solving cost.
Fourier methods have a long and proven track record as an excellent tool in data processing. As memory and computational constraints gain importance in embedded and mobile applications, we propose to combine Fourier methods and recurrent neural network architectures. The short-time Fourier transform allows us to effici…
Deep neural-kernel models combine neural networks and kernel machines for scalable large datasets.
problem Combining neural networks and kernel machines for efficient large-scale learning.
method Hybrid neural-kernel architecture using explicit feature mapping and pooling layers.
result The deep neural-kernel models are effective and scalable on benchmark datasets.