This paper studies neural network operators and their convergence properties.
problem Understanding the approximation and convergence of neural network operators.
method Proves density results, convergence estimates, and Voronovskaya-type theorems.
result Establishes quantitative convergence estimates and derives Voronovskaya-type theorems.
The paper explores neural scaling laws for deep operator networks, offering a theoretical foundation.
problem Understanding neural scaling laws in deep operator networks.
method Theoretical analysis of approximation and generalization errors.
result Established a theoretical framework to quantify neural scaling laws for deep operator networks.
Deep neural networks are proven universally powerful using Koopman operator.
problem Proving the universality of deep neural networks.
method Formal deep network as a dual voice transform with Koopman operator, using group actions and Schur's lemma.
result Simple proof of the universality of DNNs.
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.
The paper introduces Causal Neural Operators to approximate operators in stochastic analysis.
problem Leveraging temporal structure in non-linear operators for deep learning models.
method Designing a deep learning model framework for infinite-dimensional linear metric spaces.
result Causal Neural Operators can uniformly approximate Hölder or smooth trace class operators.
Novel neural operator predicts complex spatiotemporal dynamics from partial observations.
problem Capturing complex operator dynamics in infinite-dimensional function spaces.
method Integrates Koopman operator theory with deep neural networks to approximate nonlinear operators between Banach spaces.
result BNO achieves robust zero-shot super-resolution in unsteady flow prediction and outperforms conventional methods.
Develops a new deep learning formulation using Mori-Zwanzig formalism.
problem Improves deep learning by introducing a new concept of memory.
method Uses Mori-Zwanzig formalism to propagate quantities of interest through neural networks.
result Rigorously transforms deep networks into shallow ones using decay property of memory operator.
NKN deep neural network learns governing equations and classifies images.
problem Learning governing equations and classifying images with deep neural networks.
method Nonlocal kernel network (NKN) that is resolution independent, deep, and handles various tasks.
result NKN outperforms baseline methods in learning governing equations and image classification tasks.
GNPs learn operators on non-Euclidean geometries using neural networks.
problem Learning operators on complex geometries like manifolds.
method Geometric Neural Operators (GNPs) that incorporate geometric properties.
result GNPs can estimate metrics, solve PDEs, and learn LB operators on manifolds.
Deep neural operators learn complex probabilistic models efficiently.
problem Learning complex probabilistic models with global Lipschitz conditions.
method Deep neural-operator framework under global Lipschitz conditions.
result Explicit network-size bounds for universal approximation of probabilistic models.
Metrics assess uncertainty structure and distribution for regression models.
problem Quantifying uncertainty in high-dimensional and nonlinear regression tasks.
method Two bounded comparison metrics for uncertainty structure and distribution.
result DNNs and DNOs provide encouraging uncertainty metric values in high dimensions.
Deep neural networks solve noisy, complex problems accurately.
problem Reconstructing solutions from noisy, high-dimensional, non-linear inverse problems.
method Restricting infinite-dimensional forward operators to finite-dimensional spaces, training neural networks to approximate these operators robustly to noise.
result Deep neural networks can accurately solve high-dimensional, noisy, non-linear inverse problems.
Novel method for nowcasting implied volatility using neural operators.
problem Dynamic and spatially changing option prices in financial markets.
method Operator Deep Smoothing using graph neural operators.
result Highly accurate implied volatility smoothing on ten years of S&P 500 options data.
Adaptive weights improve physics-informed neural networks and deep operator networks.
problem Training physics-informed neural networks and deep operator networks can be challenging, leading to unsatisfactory accuracy and efficiency.
method Proposes a pointwise adaptive weighting method that balances the residual decay rate across different training points.
result Our proposed approach of balanced residual decay rates offers advantages including bounded weights, high prediction accuracy, fast convergence rate, low training uncertainty, low computational cost, and ease of hyperparameter tuning.
Deep neural nets estimate operators between infinite-dimensional spaces with fast rates.
problem Estimating operators between infinite-dimensional spaces.
method Deep neural networks for nonparametric estimation of Lipschitz operators.
result Error bounds decay with fast rates depending on intrinsic dimension.
Physics-informed WNO learns PDE solutions without labeled data.
problem Data-hungry nature of WNO framework.
method Physics-informed WNO for learning PDE solutions.
result Validated and illustrated with four nonlinear systems.
RaNNDy uses randomized neural networks to learn transfer operators efficiently.
problem Efficiently learning transfer operators from data.
method Randomized neural network approach with randomly initialized hidden layers and trained output layer.
result Significant reduction in training time and resources with improved stability.
New ADANNs improve PDE approximations.
problem Approximating operators for parametric PDEs.
method Custom ANN architectures and initialization schemes.
result ADANNs significantly outperform existing methods.
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.
New neural processes use stacked Markov operators to improve flexibility.
problem Improving flexibility in neural processes.
method Stacking neural parameterized Markov transition operators in function space.
result MNPs outperform baseline models on various tasks.
RP-WNO extends WNO with uncertainty quantification, useful for scientists and engineers.
problem Uncertainty in predictions of deep learning models.
method Randomized Prior Wavelet Neural Operator (RP-WNO) with uncertainty quantification module.
result RP-WNO effectively estimates uncertainty in predictions.
New model outperforms Neural ODEs while being more efficient.
problem Stable convergence and existence guarantees for implicit-depth models.
method Developed Monotone Operator Equilibrium Network (monDEQ) based on monotone operator theory.
result MonDEQ models outperform Neural ODEs and are more computationally efficient.
Bayesian Neural Networks help quantify uncertainty in deep learning predictions.
problem Uncertainty quantification in deep learning predictions.
method Bayesian statistics applied to neural networks.
result Design, implementation, training, and evaluation of Bayesian Neural Networks.
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.
User response prediction makes a crucial contribution to the rapid development of online advertising system and recommendation system. The importance of learning feature interactions has been emphasized by many works. Many deep models are proposed to automatically learn high-order feature interactions. Since most featu…
Explaining neural network computation in terms of probabilistic/fuzzy logical operations has attracted much attention due to its simplicity and high interpretability. Different choices of logical operators such as AND, OR and XOR give rise to another dimension for network optimization, and in this paper, we study the o…
This paper aims to propose a novel deep learning-integrated framework for deriving reliable simulation input models through incorporating multi-source information. The framework sources and extracts multisource data generated from construction operations, which provides rich information for input modeling. The framewor…
New theory for local parameterization of deep ReLU networks.
problem Determining local parameters of deep ReLU neural networks.
method Introducing local lifting operators and charts of a manifold, deriving necessary and sufficient conditions for local identifiability.
result Sharp and testable conditions for local identifiability of deep ReLU networks.
Business analytics refers to methods and practices that create value through data for individuals, firms, and organizations. This field is currently experiencing a radical shift due to the advent of deep learning: deep neural networks promise improvements in prediction performance as compared to models from traditional…
This article introduces machine learning methods for solving PDEs.
problem Approximating solutions of partial differential equations.
method Machine learning methods, including physics-informed neural networks and deep operator learning.
result Recent advances in machine learning have made PDE solutions more accessible.
KNF uses Koopman theory to forecast time series with changing dynamics.
problem Temporal distributional shifts in time series data.
method KNF combines DNNs with Koopman theory to learn dynamic operators.
result KNF outperforms alternatives on time series datasets with distributional shifts.
Deep learning framework for kernel methods using RKHM and Perron-Frobenius operators.
problem Kernel methods in deep learning with potential overfitting issues.
method Combining RKHM and Perron-Frobenius operator to derive a new Rademacher bound and analyze deep kernel methods.
result Theoretical interpretation of benign overfitting and milder dependency on output dimension.
We propose a new method to solve eigenvalue problems for linear and semilinear second order differential operators in high dimensions based on deep neural networks. The eigenvalue problem is reformulated as a fixed point problem of the semigroup flow induced by the operator, whose solution can be represented by Feynman…
Paper introduces CRP-O framework for uncertainty quantification in deep operators.
problem Uncertainty quantification in energy-efficient deep learning algorithms, especially in SNNs.
method CRP-O framework using RP networks and SCP, with Gaussian Process Regression for super-resolution.
result Enhanced uncertainty bounds improve UQ estimates compared to existing methods.
Random feature method approximates operators with theoretical guarantees and reduced computation.
problem Approximating operators between infinite dimensional Banach spaces using machine learning.
method Random feature operator learning method with theoretical guarantees and error bounds.
result The random feature method can achieve similar or better test errors than kernel-based methods and neural networks with significantly reduced training times.
CMCO provides robust uncertainty estimates for neural operators without retraining.
problem Uncertainty quantification in deep learning for real-time virtual sensing.
method Unified Monte Carlo dropout and split conformal prediction in DeepONet.
result Near-nominal empirical coverage in diverse applications.
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.
NON model improves tabular data classification accuracy.
problem Tabular data classification in real-world applications.
method Field-wise network, across field network, operation fusion network.
result NON significantly outperforms state-of-the-art models.
Novel framework explains generalization in deep neural networks.
problem Understanding and improving generalization in deep neural networks.
method Topological Quantum Neural Networks as the semi-classical limit of Deep Neural Networks.
result Demonstrates that the perceptron, viewed as the semi-classical limit, achieves similar results to standard neural networks without training.
Generative operators solve many convex problems with minimal parameters.
problem Worst-case parameter bounds limit the practical use of neural operators.
method Developed generative equilibrium operators (GEOs) using realizable finite-dimensional layers.
result GEOs can uniformly approximate solutions to convex optimization problems with logarithmic growth in parameters.
We consider the use of look-up tables (LUT) to simplify the hardware implementation of a deep learning network for inferencing after weights have been successfully trained. The use of LUT replaces the matrix multiply and add operations with a small number of LUTs and addition operations resulting in a completely multip…
Deep neural networks reduce portfolio tail-risk by 99% in crisis-era simulations.
problem Managing tail risk in financial portfolios.
method Parameterizing convex-risk minimization with deep neural networks.
result Significant reduction in one-day 99% CVaR.
Random sampling improves DeepONet training efficiency without sacrificing accuracy.
problem Training DeepONet models with high computational and memory costs.
method Random sampling of inputs in the trunk network of DeepONet.
result Significant reduction in training time with comparable accuracy.
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.
Deep neural networks are powerful learning models that achieve state-of-the-art performance on many computer vision, speech, and language processing tasks. In this paper, we study a fundamental question that arises when designing deep network architectures: Given a target network architecture can we design a smaller ne…
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.
DQA efficiently quantizes deep neural network activations for resource-constrained devices.
problem Efficiently quantizing deep neural network activations for resource-constrained devices.
method DQA uses simple shifting-based operations and Huffman coding for sub-6-bit quantization.
result DQA achieves significantly better accuracy than direct quantization and state-of-the-art methods.
DeepONets improve surrogate modeling for engineering systems.
problem Accurately modeling complex PDEs for engineering systems.
method DeepONets specialize in approximating mathematical operators for PDEs.
result DeepONets achieve high prediction accuracy and zero-shot capability.