2.3k citations · 2.7k across the 51 of their papers we have counts for
74 papers
Kernel Limit for a Class of Recurrent Neural Networks Trained on Ergodic Data Sequences
Samuel Chun-Hei Lam, Justin Sirignano, Konstantinos Spiliopoulos
Mathematical methods are developed to characterize the asymptotics of recurrent neural networks (RNN) as the number of hidden units, data samples in the sequence, hidden state upda…
Transport map unadjusted Langevin algorithms: learning and discretizing perturbed samplers
Benjamin J. Zhang, Youssef M. Marzouk, Konstantinos Spiliopoulos
Langevin dynamics are widely used in sampling high-dimensional, non-Gaussian distributions whose densities are known up to a normalizing constant. In particular, there is strong in…
Quantitative fluctuation analysis of multiscale diffusion systems via Malliavin calculus
Solesne Bourguin, Konstantinos Spiliopoulos
We study fluctuations of small noise multiscale diffusions around their homogenized deterministic limit. We derive quantitative rates of convergence of the fluctuation processes to…
Normalization effects on deep neural networks
Jiahui Yu, Konstantinos Spiliopoulos
We study the effect of normalization on the layers of deep neural networks of feed-forward type. A given layer with hidden units is allowed to be normalized by $1/N_{i}…
Fluctuation analysis for particle-based stochastic reaction-diffusion models
Max Heldman, Samuel Isaacson, Jingwei Ma +1
Recent works have derived and proven the large-population mean-field limit for several classes of particle-based stochastic reaction-diffusion (PBSRD) models. These limits correspo…
Moderate deviation principle for multiscale systems driven by fractional Brownian motion
Solesne Bourguin, Thanh Dang, Konstantinos Spiliopoulos
In this paper we study the moderate deviations principle (MDP) for slow-fast stochastic dynamical systems where the slow motion is governed by small fractional Brownian motion (fBm…
Importance sampling for stochastic reaction-diffusion equations in the moderate deviation regime
Ioannis Gasteratos, Michael Salins, Konstantinos Spiliopoulos
We develop a provably efficient importance sampling scheme that estimates exit probabilities of solutions to small-noise stochastic reaction-diffusion equations from scaled neighbo…
Rate of homogenization for fully-coupled McKean-Vlasov SDEs
Zachary Bezemek, Konstantinos Spiliopoulos
We consider a fully-coupled slow-fast system of McKean-Vlasov SDEs with full dependence on the slow and fast component and on the law of the slow component and derive convergence r…
Disentangling positive and negative partisanship in social media interactions using a coevolving latent space network with attractors model
Xiaojing Zhu, Cantay Caliskan, Dino P. Christenson +3
We develop a broadly applicable class of coevolving latent space network with attractors (CLSNA) models, where nodes represent individual social actors assumed to lie in an unknown…
Geometry-informed irreversible perturbations for accelerated convergence of Langevin dynamics
Benjamin J. Zhang, Youssef M. Marzouk, Konstantinos Spiliopoulos
We introduce a novel geometry-informed irreversible perturbation that accelerates convergence of the Langevin algorithm for Bayesian computation. It is well documented that there e…
PDE-constrained Models with Neural Network Terms: Optimization and Global Convergence
Justin Sirignano, Jonathan MacArt, Konstantinos Spiliopoulos
Recent research has used deep learning to develop partial differential equation (PDE) models in science and engineering. The functional form of the PDE is determined by a neural ne…
Online Adjoint Methods for Optimization of PDEs
Justin Sirignano, Konstantinos Spiliopoulos
We present and mathematically analyze an online adjoint algorithm for the optimization of partial differential equations (PDEs). Traditional adjoint algorithms would typically solv…
Moderate deviations for systems of slow-fast stochastic reaction-diffusion equations
Ioannis Gasteratos, Michael Salins, Konstantinos Spiliopoulos
The goal of this paper is to study the Moderate Deviation Principle (MDP) for a system of stochastic reaction-diffusion equations with a time-scale separation in slow and fast comp…
Normalization effects on shallow neural networks and related asymptotic expansions
Jiahui Yu, Konstantinos Spiliopoulos
We consider shallow (single hidden layer) neural networks and characterize their performance when trained with stochastic gradient descent as the number of hidden units and gra…
Large deviations for interacting multiscale particle systems
Zachary Bezemek, Konstantinos Spiliopoulos
We consider a collection of weakly interacting diffusion processes moving in a two-scale locally periodic environment. We study the large deviations principle of the empirical dist…
How reaction-diffusion PDEs approximate the large-population limit of stochastic particle models
Samuel A Isaacson, Jingwei Ma, Konstantinos Spiliopoulos
Reaction-diffusion PDEs and particle-based stochastic reaction-diffusion (PBSRD) models are commonly-used approaches for modeling the spatial dynamics of chemical and biological sy…
Discrete-time inference for slow-fast systems driven by fractional Brownian motion
Solesne Bourguin, Siragan Gailus, Konstantinos Spiliopoulos
We study statistical inference for small-noise-perturbed multiscale dynamical systems where the slow motion is driven by fractional Brownian motion. We develop statistical estimato…
Mean Field Limits of Particle-Based Stochastic Reaction-Diffusion Models
Samuel A. Isaacson, Jingwei Ma, Konstantinos Spiliopoulos
Particle-based stochastic reaction-diffusion (PBSRD) models are a popular approach for studying biological systems involving both noise in the reaction process and diffusive transp…
Asymptotics of Reinforcement Learning with Neural Networks
Justin Sirignano, Konstantinos Spiliopoulos
We prove that a single-layer neural network trained with the Q-learning algorithm converges in distribution to a random ordinary differential equation as the size of the model and…
Scaling Limit of Neural Networks with the Xavier Initialization and Convergence to a Global Minimum
Justin Sirignano, Konstantinos Spiliopoulos
We analyze single-layer neural networks with the Xavier initialization in the asymptotic regime of large numbers of hidden units and large numbers of stochastic gradient descent tr…
Typical dynamics and fluctuation analysis of slow-fast systems driven by fractional Brownian motion
Solesne Bourguin, Siragan Gailus, Konstantinos Spiliopoulos
This article studies typical dynamics and fluctuations for a slow-fast dynamical system perturbed by a small fractional Brownian noise. Based on an ergodic theorem with explicit ra…
Selection of quasi-stationary states in the stochastically forced Navier-Stokes equation on the torus
Margaret Beck, Eric Cooper, Gabriel Lord +1
The stochastically forced vorticity equation associated with the two dimensional incompressible Navier-Stokes equation on is considered for …
Metastability and exit problems for systems of stochastic reaction-diffusion equations
Michael Salins, Konstantinos Spiliopoulos
In this paper we develop a metastability theory for a class of stochastic reaction-diffusion equations exposed to small multiplicative noise. We consider the case where the unpertu…
Mean Field Analysis of Deep Neural Networks
Justin Sirignano, Konstantinos Spiliopoulos
We analyze multi-layer neural networks in the asymptotic regime of simultaneously (A) large network sizes and (B) large numbers of stochastic gradient descent training iterations.…