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.
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.
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.
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.
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.
FNOs learn solution operators of dissipative equations efficiently via spectral methods.
problem Learning and approximation of solution operators for dissipative equations.
method Introducing spectral methods and deriving FNO approximation bounds and sample complexity guarantees.
result Polynomial sample complexity guarantees for FNOs learning solution operators of dissipative equations.
Many neural speech enhancement and source separation systems operate in the time-frequency domain. Such models often benefit from making their Short-Time Fourier Transform (STFT) front-ends trainable. In current literature, these are implemented as large Discrete Fourier Transform matrices; which are prohibitively inef…
Optimizes deep learning models for ocean dynamics using Fourier neural operators.
problem Efficiently training deep learning models for ocean dynamics with optimal hyperparameters.
method Multiobjective hyperparameter optimization with DeepHyper for Fourier neural operators.
result Optimal hyperparameters significantly improved model performance in ocean dynamics forecasting.
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.
New neural operators learn structured patterns efficiently.
problem Learning and representing complex, structured patterns in data.
method Sparse autoencoder neural operators (SAE-NOs) parameterize concepts as functions, enabling efficient and structured representation.
result SAE-FNOs learn localized patterns and generalize across different scales and discretizations.
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.
This note introduces a regression technique for finding a class of nonlinear integro-differential operators from data. The method parametrizes the spatial operator with neural networks and Fourier transforms such that it can fit a class of nonlinear operators without needing a library of a priori selected operators. We…
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.
G-FuNK learns solutions for nonlinear PDEs on multiple domains and parameters.
problem Predicting time-dependent dynamics of complex systems governed by nonlinear PDEs with varying parameters and domains.
method Graph Fourier Neural Kernels combining domain-adapted and transferable components for non-diffusive and diffusive terms.
result G-FuNK achieves low relative errors on unseen domains and fiber fields, significantly accelerating predictions.
Polynomial Chaos Expansion improves operator learning for PDEs.
problem Approximating mappings between infinite-dimensional functional spaces.
method Polynomial Chaos Expansion (PCE) for operator learning.
result PCE achieves strong performance in operator learning and uncertainty quantification.
Devoted to multi-task learning and structured output learning, operator-valued kernels provide a flexible tool to build vector-valued functions in the context of Reproducing Kernel Hilbert Spaces. To scale up these methods, we extend the celebrated Random Fourier Feature methodology to get an approximation of operator-…
GIT-Net uses neural networks to approximate PDE operators efficiently.
problem Approximating PDE operators for complex geometries.
method Parametrizes adaptive generalized integral transforms with deep neural networks.
result GIT-Net outperforms existing neural network operators in multiple areas.
Kernel methods outperform neural nets in operator learning tasks.
problem Learning operators between Banach spaces from partial observations.
method Kernel-based framework with a priori error analysis and numerical comparisons.
result Kernel methods are competitive with neural nets in cost-accuracy trade-off.
Study evolution equations on Lie groupoids using Fourier integral operators.
problem Solving evolution equations on Lie groupoids.
method Developed calculus of Fourier integral operators and used them to study the fundamental solution of the evolution equation.
result Developed a method to find the fundamental solution of the evolution equation on Lie groupoids.
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.
As announced in [12], we develop a calculus of Fourier integral G-operators on any Lie groupoid G. For that purpose, we study convolability and invertibility of Lagrangian conic submanifolds of the symplectic groupoid T * G. We also identify those Lagrangian which correspond to equivariant families parametrized by the …
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.
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.
This work develops a fast-running ROM for MOOSE-based AM model using OL.
problem Achieving desired material properties in real-time manufacturing processes.
method Operator learning (OL) and Fourier neural operator for ROM development.
result OL-based ROM outperforms conventional deep neural network-based ROM in benchmark tests.
New methods improve translation-equivariant neural processes for modeling unknown functions.
problem Modeling unknown latent functions from irregularly sampled measurements.
method Volterra series and set Fourier convolutions to address translation-equivariance and efficiency.
result Improved translation-equivariant neural processes with analytical transparency and linear scalability.
Study generalizes Picard iteration for nonlinear PDEs, deriving bounds on error.
problem Generalize Picard iteration for nonlinear parabolic PDEs.
method Formulate Picard iteration as abstract state-transition model, derive generalization error bounds.
result Picard depth reduction reduces Picard truncation error without increasing estimation error.
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.
Initialization of parameters in deep neural networks has been shown to have a big impact on the performance of the networks (Mishkin & Matas, 2015). The initialization scheme devised by He et al, allowed convolution activations to carry a constrained mean which allowed deep networks to be trained effectively (He et al.…
LUNO linearizes neural operators to quantify their predictive uncertainty.
problem Quantifying the predictive error of neural operators for high-stakes simulations.
method Model linearization to push weight-space uncertainty forward to predictions.
result LUNO provides a practical and theoretically sound way to apply Bayesian methods to neural operators.
Study examines the geometry of a group of Fourier-integral operators related to Diff(S^1).
problem Investigating the geometry of a specific group of Fourier-integral operators.
method Defined a right-invariant pseudo-Riemannian metric on the group using renormalized traces of pseudo-differential operators.
result Extended the Hilbert-Schmidt Riemannian metric to the group.
A hybrid framework uses machine learning to price options faster and more accurately.
problem Rapid recalibration of option pricing models in dynamic markets.
method Integrates smooth offset algorithm with supervised machine learning models.
result Surrogate pricing operators achieve up to 1000x speedup over direct SOA evaluation.
Novel neural network layer improves long-range interactions in point clouds.
problem Efficiently treating long-range interactions in point clouds.
method Long-range convolutional (LRC)-layer that incorporates Fourier transforms.
result Global all-to-all convolution operation performed in nearly-linear time.
We endow the group of invertible Fourier integral operators on an open}manifold with the structure of an ILH Lie group. This is done by establishing such structures for the groups of invertible pseudodifferential operators and contact transformations on an open manifold of bounded geometry, and gluing those together vi…
Structured CNN designed using the prior information of problems potentially improves efficiency over conventional CNNs in various tasks in solving PDEs and inverse problems in signal processing. This paper introduces BNet2, a simplified Butterfly-Net and inline with the conventional CNN. Moreover, a Fourier transform i…
HS-FNO models non-Markovian PDEs by learning history and future states.
problem Non-Markovian dynamics where future states depend on past history.
method History-Space Fourier Neural Operator (HS-FNO) for delay and memory-driven PDEs.
result HS-FNO achieves lowest aggregate errors across various PDE families.
We show that the index of an elliptic Fourier integral operator associated to a contact diffeomorphism φ of cosphere bundles of two Riemannian manifolds X and Y is given by ∫B∗XA^(T∗X)expθ−∫B∗YA^(T∗Y)expθ. Here B∗ stands for the unit coball bundle and θ is a certain characteristic…
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.
The covariance of a stationary process X is diagonalized by a Fourier transform. It does not take into account the complex Fourier phase and defines Gaussian maximum entropy models. We introduce a general family of phase harmonic covariance moments, which rely on complex phases to capture non-Gaussian properties. The…
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.
Study explores how neural networks and Transformers learn modular arithmetic with multiple inputs.
problem Understanding how neural networks and Transformers learn modular arithmetic with multiple inputs.
method Analytical characterization of features learned by neural networks and Transformers, focusing on margin maximization and Fourier spectra.
result Neural networks and Transformers require a minimum neuron count of \( m \geq 2^{2k-2} \cdot (p-1) \) to solve modular addition problems with \( k \) inputs and modulus \( p \).
Physics-informed neural networks and neural operators speed up solving parametric PDEs by orders of magnitude.
problem Solving PDEs for varying parameters is computationally expensive.
method Physics-informed neural networks and neural operators learn solution mappings across parameter spaces.
result Neural operators achieve computational speedups of 10^3 to 10^5 times faster than traditional methods.
ARBITER learns SPX-VIX term structures without arbitrage constraints.
problem Arbitrage-free modeling of SPX-VIX term structures.
method Risk-neutral neural operator mapping market states to operator outputs enforcing static arbitrage constraints.
result ARBITER outperforms other models in derivatives term structure evaluation metrics.
New proof of Zelditch's generalization using Riemannian geometry.
problem Asymptotic formula for eigenfunction sums on compact manifolds.
method Riemannian geometry methods.
result Proof of Zelditch's generalization without Fourier integral operators.
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.
This work develops fast and accurate ROMs for AM models using OL methods.
problem Achieving specific material properties in AM by manipulating process parameters increases computational load.
method Operator learning (OL) approach with Fourier neural operator (FNO) and DeepONet.
result OL methods offer comparable performance and outperform DNN in accuracy and generalizability.
FNO model predicts GCS pressure fields with 81% less data, even with limited high-fidelity data.
problem Accurate prediction of complex physical behaviors in large-scale 3D geological carbon storage problems with limited data.
method Multi-fidelity Fourier Neural Operator (FNO) for efficient training with multi-fidelity datasets.
result Multi-fidelity FNO model predicts pressure fields with reasonable accuracy even with limited high-fidelity data.
Quantum computers can enhance spectral methods in machine learning.
problem Spectral methods are fundamental but challenging for classical models.
method Utilizing quantum Fourier Transform for spectral manipulations.
result Quantum computing can offer more efficient spectral design.
Infrastructure monitoring is critical for safe operations and sustainability. Water distribution networks (WDNs) are large-scale networked critical systems with complex cascade dynamics which are difficult to predict. Ubiquitous monitoring is expensive and a key challenge is to infer the contaminant dynamics from parti…