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Found 24 papers, 5 papers with code
Operator Learning: Algorithms and Analysis
This review article summarizes recent progress and the current state of our theoretical understanding of neural operators, focusing on an approximation theoretic point of view.
Modeling groundwater levels in California's Central Valley by hierarchical Gaussian process and neural network regression
Modeling groundwater levels continuously across California's Central Valley (CV) hydrological system is challenging due to low-quality well data which is sparsely and noisily sampled across time and space.
The Parametric Complexity of Operator Learning
The first contribution of this paper is to prove that for general classes of operators which are characterized only by their $C^r$- or Lipschitz-regularity, operator learning suffers from a ``curse of parametric complexity'', which is an infinite-dimensional analogue of the well-known curse of dimensionality encountered in high-dimensional approximation problems.
Learning Homogenization for Elliptic Operators
However, a major challenge in data-driven learning approaches for this problem has remained unexplored: the impact of discontinuities and corner interfaces in the underlying material.
The Nonlocal Neural Operator: Universal Approximation
A popular variant of neural operators is the Fourier neural operator (FNO).
Gradient Flows for Sampling: Mean-Field Models, Gaussian Approximations and Affine Invariance
The flow in the Gaussian space may be understood as a Gaussian approximation of the flow.
Second Order Ensemble Langevin Method for Sampling and Inverse Problems
We propose a sampling method based on an ensemble approximation of second order Langevin dynamics.
Convergence Rates for Learning Linear Operators from Noisy Data
This paper studies the learning of linear operators between infinite-dimensional Hilbert spaces.
A Framework for Machine Learning of Model Error in Dynamical Systems
For ergodic continuous-time systems, we prove that both excess risk and generalization error are bounded above by terms that diminish with the square-root of T, the time-interval over which training data is specified.
Ensemble Inference Methods for Models With Noisy and Expensive Likelihoods
The ensemble Kalman methods are shown to behave favourably in the presence of noise in the parameter-to-data map, whereas Langevin methods are adversely affected.
Iterated Kalman Methodology For Inverse Problems
In this paper, we work with the ExKI, EKI, and a variant on EKI which we term unscented Kalman inversion (UKI).
Numerical Analysis Numerical Analysis Dynamical Systems
Calibration and Uncertainty Quantification of Convective Parameters in an Idealized GCM
Here we demonstrate an approach to model calibration and uncertainty quantification that requires only $O(10^2)$ model runs and can accommodate internal climate variability.
Gaussian Processes
Statistics Theory
Statistics Theory
Posterior Consistency of Semi-Supervised Regression on Graphs
Graph-based semi-supervised regression (SSR) is the problem of estimating the value of a function on a weighted graph from its values (labels) on a small subset of the vertices.
Consistency of Empirical Bayes And Kernel Flow For Hierarchical Parameter Estimation
The purpose of this paper is to study two paradigms of learning hierarchical parameters: one is from the probabilistic Bayesian perspective, in particular, the empirical Bayes approach that has been largely used in Bayesian statistics; the other is from the deterministic and approximation theoretic view, and in particular the kernel flow algorithm that was proposed recently in the machine learning literature.
The Random Feature Model for Input-Output Maps between Banach Spaces
Well known to the machine learning community, the random feature model is a parametric approximation to kernel interpolation or regression methods.
Model Reduction and Neural Networks for Parametric PDEs
We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces.
Spectral Analysis Of Weighted Laplacians Arising In Data Clustering
Graph Laplacians computed from weighted adjacency matrices are widely used to identify geometric structure in data, and clusters in particular; their spectral properties play a central role in a number of unsupervised and semi-supervised learning algorithms.
Consistency of semi-supervised learning algorithms on graphs: Probit and one-hot methods
Graph-based semi-supervised learning is the problem of propagating labels from a small number of labelled data points to a larger set of unlabelled data.
Continuous Time Analysis of Momentum Methods
Firstly we show that standard implementations of fixed momentum methods approximate a time-rescaled gradient descent flow, asymptotically as the learning rate shrinks to zero; this result does not distinguish momentum methods from pure gradient descent, in the limit of vanishing learning rate.
Ensemble Kalman Inversion: A Derivative-Free Technique For Machine Learning Tasks
The standard probabilistic perspective on machine learning gives rise to empirical risk-minimization tasks that are frequently solved by stochastic gradient descent (SGD) and variants thereof.
Large Data and Zero Noise Limits of Graph-Based Semi-Supervised Learning Algorithms
Scalings in which the graph Laplacian approaches a differential operator in the large graph limit are used to develop understanding of a number of algorithms for semi-supervised learning; in particular the extension, to this graph setting, of the probit algorithm, level set and kriging methods, are studied.
Dimension-Robust MCMC in Bayesian Inverse Problems
One popular formulation of such problems is as Bayesian inverse problems, where a prior distribution is used to regularize inference on a high-dimensional latent state, typically a function or a field.
Uncertainty quantification in graph-based classification of high dimensional data
In this paper we introduce, develop algorithms for, and investigate the properties of, a variety of Bayesian models for the task of binary classification; via the posterior distribution on the classification labels, these methods automatically give measures of uncertainty.
Posterior Consistency for Gaussian Process Approximations of Bayesian Posterior Distributions
We prove error bounds on the Hellinger distance between the true posterior distribution and various approximations based on the Gaussian process emulator.
Numerical Analysis
