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Kernel density estimation (KDE) is integral to a range of generative and discriminative tasks in machine learning. Drawing upon tools from the multidimensional calculus of variations, we derive an optimal weight function that reduces bias…
Machine learning models are increasingly used to predict material properties and accelerate atomistic simulations, but the reliability of their predictions depends on the representativeness of the training data. We present a scalable,…
The problem of system identification for the Kalman filter, relying on the expectation-maximization (EM) procedure to learn the underlying parameters of a dynamical system, has largely been studied assuming that observations are sampled at…
We propose a new algorithm to approach weakly the solution of a McKean-Vlasov SDE. Based on the cubature method of Lyons and Victoir 2004, the algorithm is deterministic differing from the the usual methods based on interacting particles.…
The performance of multivariate kernel density estimation (KDE) depends strongly on the choice of bandwidth matrix. The high computational cost required for its estimation provides a big motivation to develop fast and accurate methods. One…
The Fokker-Planck (FP) equation governing the evolution of the probability density function (PDF) is applicable to many disciplines but it requires specification of the coefficients for each case, which can be functions of space-time and…
This article presents a three-step framework for learning and solving partial differential equations (PDEs) using kernel methods. Given a training set consisting of pairs of noisy PDE solutions and source/boundary terms on a mesh, kernel…
In recent years we have witnessed a growth in mathematics for deep learning, which has been used to solve inverse problems of partial differential equations (PDEs). However, most deep learning-based inversion methods either require paired…
The reconstruction of smooth density fields from scattered data points is a procedure that has multiple applications in a variety of disciplines, including Lagrangian (particle-based) models of solute transport in fluids. In random walk…
In this work, we propose adaptive deep learning approaches based on normalizing flows for solving fractional Fokker-Planck equations (FPEs). The solution of a FPE is a probability density function (PDF). Traditional mesh-based methods are…
The probability density function of stochastic differential equations is governed by the Fokker-Planck (FP) equation. A novel machine learning method is developed to solve the general FP equations based on deep neural networks. The proposed…
Learning can be seen as approximating an unknown function by interpolating the training data. Kriging offers a solution to this problem based on the prior specification of a kernel. We explore a numerical approximation approach to kernel…
In this paper we propose a new methodology for decision-making under uncertainty using recent advancements in the areas of nonlinear stochastic optimal control theory, applied mathematics, and machine learning. Grounded on the fundamental…
Comparing differently sized data sets is one main task in model assessment and calibration. This is due to field data being generally sparse compared to simulated model results. We tackled this task by the application of a new…
Federated Learning (FL) is an emerging learning scheme that allows different distributed clients to train deep neural networks together without data sharing. Neural networks have become popular due to their unprecedented success. To the…
A kernel-based approach for the learning of the solution operator of general nonhomogeneous partial differential equations (PDEs) is proposed. The method incorporates physical priors, typically encoded through the PDE operator, into a…
Face Recognition (FR) has been the interest to several researchers over the past few decades due to its passive nature of biometric authentication. Despite high accuracy achieved by face recognition algorithms under controlled conditions,…
We propose a nonparametric method to learn the L\'evy density from probability density data governed by a nonlocal Fokker-Planck equation. We recast the problem as identifying the kernel in a nonlocal integral operator from discrete data,…
We obtain upper and lower Gaussian density estimates for the law of each component of the solution to a one-dimensional fully coupled forward-backward SDE (FBSDE). Our approach relies on the link between FBSDEs and quasilinear parabolic…
Physics-informed machine learning typically integrates physical priors into the learning process by minimizing a loss function that includes both a data-driven term and a partial differential equation (PDE) regularization. Building on the…