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Accelerating the learning of Partial Differential Equations (PDEs) from experimental data will speed up the pace of scientific discovery. Previous randomized algorithms exploit sparsity in PDE updates for acceleration. However such methods…
Monte Carlo methods represent the "de facto" standard for approximating complicated integrals involving multidimensional target distributions. In order to generate random realizations from the target distribution, Monte Carlo techniques use…
Generative AI (GenAI) has revolutionized data-driven modeling by enabling the synthesis of high-dimensional data across various applications, including image generation, language modeling, biomedical signal processing, and anomaly…
This paper concerns the analysis of random second order linear differential equations. Usually, solving these equations consists of computing the first statistics of the response process, and that task has been an essential goal in the…
Diffusion-based generative models learn to iteratively transfer unstructured noise to a complex target distribution as opposed to Generative Adversarial Networks (GANs) or the decoder of Variational Autoencoders (VAEs) which produce samples…
In applied sciences, we often deal with deterministic simulation models that are too slow for simulation-intensive tasks such as calibration or real-time control. In this paper, an emulator for a generic dynamic model, given by a system of…
In this paper, we consider a class of finite-sum convex optimization problems defined over a distributed multiagent network with $m$ agents connected to a central server. In particular, the objective function consists of the average of $m$…
With the traditional equilibrium molecular simulations, it is usually difficult to efficiently visit the whole conformational space in complex systems, which are separated into some metastable conformational regions by high free energy…
Many problems arising in applications result in the need to probe a probability distribution for functions. Examples include Bayesian nonparametric statistics and conditioned diffusion processes. Standard MCMC algorithms typically become…
Resampling techniques are widely used in statistical inference and ensemble learning, in which estimators' statistical properties are essential. However, existing methods are computationally demanding, because repetitions of…
This paper investigates the problems large-scale distributed composite convex optimization, with motivations from a broad range of applications, including multi-agent systems, federated learning, smart grids, wireless sensor networks,…
Efficient and stable solution of partial differential equations (PDEs) is central to scientific and engineering applications, yet existing numerical solvers rely heavily on matrix based discretizations, while learning based methods require…
Stochastic Differential Equations (SDEs) are used as statistical models in many disciplines. However, intractable likelihood functions for SDEs make inference challenging, and we need to resort to simulation-based techniques to estimate and…
The accurate numerical solution of partial differential equations is a central task in numerical analysis allowing to model a wide range of natural phenomena by employing specialized solvers depending on the scenario of application. Here,…
Gradient flow in the 2-Wasserstein space is widely used to optimize functionals over probability distributions and is typically implemented using an interacting particle system with $n$ particles. Analyzing these algorithms requires showing…
In this study, we introduce a novel method for generating new synthetic samples that are independent and identically distributed (i.i.d.) from high-dimensional real-valued probability distributions, as defined implicitly by a set of Ground…
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…
A probability density function (pdf) encodes the entire stochastic knowledge about data distribution, where data may represent stochastic observations in robotics, transition state pairs in reinforcement learning or any other empirically…
The general pressure equation (GPE) is a new method proposed recently by Toutant (J. Comput. Phys., 374:822-842 (2018)) for incompressible flow simulation. It circumvents the Poisson equation for the pressure and performs better than the…
We extend the Deep Galerkin Method (DGM) introduced in Sirignano and Spiliopoulos (2018)} to solve a number of partial differential equations (PDEs) that arise in the context of optimal stochastic control and mean field games. First, we…