Related papers: The deep multi-FBSDE method: a robust deep learnin…
In the present work, a multi-scale framework for neural network enhanced methods is proposed for approximation of function and solution of partial differential equations (PDEs). By introducing the multi-scale concept, the total solution of…
Federated learning has attracted increasing attention with the emergence of distributed data. While extensive federated learning algorithms have been proposed for the non-convex distributed problem, federated learning in practice still…
Within recent years, considerable progress has been made regarding high-performance solvers for Partial Differential Equations (PDEs), yielding potential gains in efficiency compared to industry standard tools. However, the latter largely…
Stochastic differential equations (SDEs) offer powerful and accessible mathematical models for capturing both deterministic and probabilistic aspects of dynamic behavior across a wide range of physical, financial, and social systems.…
The Helmholtz equation is fundamental to wave modeling in acoustics, electromagnetics, and seismic imaging, yet high-frequency regimes remain challenging due to the ``pollution effect''. We propose FD-MGDL, an adaptive framework integrating…
In this paper, we consider the solvability problems for the fully coupled forward-backward stochastic difference equations (FBS{\Delta}Es) on spaces related to discrete time, finite state processes. On one hand, we provide the necessary and…
We propose a novel numerical approach for nonlocal diffusion equations [8] with integrable kernels, based on the relationship between the backward Kolmogorov equation and backward stochastic differential equations (BSDEs) driven by L\`{e}vy…
We analyze the DQN reinforcement learning algorithm as a stochastic approximation scheme using the o.d.e. (for 'ordinary differential equation') approach and point out certain theoretical issues. We then propose a modified scheme called…
A class of backward doubly stochastic differential equations (BDSDEs in short) with continuous coefficients is studied. We give the comparison theorems, the existence of the maximal solution and the structure of solutions for BDSDEs with…
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…
(Working Paper) Using a purely probabilistic argument, we prove the global well-posedness of multidimensional superquadratic backward stochastic differential equations (BSDEs) without Markovian assumption. The key technique is the interplay…
Machine learning for partial differential equations (PDEs) is a hot topic. In this paper we introduce and analyse a Deep BSDE scheme for nonlinear integro-PDEs with unbounded nonlocal operators -problems arising in e.g. stochastic control…
Recent work on Path-Dependent Partial Differential Equations (PPDEs) has shown that PPDE solutions can be approximated by a probabilistic representation, implemented in the literature by the estimation of conditional expectations using…
Deep unfolding neural networks derived from iterative optimization schemes and numerical ordinary/partial differential equations (ODEs/PDEs) have attracted much attention in data science over the last decade. Therein, numerous important…
The question of how to parallelize the stochastic gradient descent (SGD) method has received much attention in the literature. In this paper, we focus instead on batch methods that use a sizeable fraction of the training set at each…
Mathematical mean-field approaches have been used in many fields, not only in Physics and Chemistry, but also recently in Finance, Economics, and Game Theory. In this paper we will study a new special mean-field problem in a purely…
Very few methods for hybrid federated learning, where clients only hold subsets of both features and samples, exist. Yet, this scenario is extremely important in practical settings. We provide a fast, robust algorithm for hybrid federated…
Stochastic approximation with multiple coupled sequences (MSA) has found broad applications in machine learning as it encompasses a rich class of problems including bilevel optimization (BLO), multi-level compositional optimization (MCO),…
To solve optimization problems with parabolic PDE constraints, often methods working on the reduced objective functional are used. They are computationally expensive due to the necessity of solving both the state equation and a…
We demonstrate that backward stochastic differential equations (BSDE) may be reformulated as ordinary functional differential equations on certain path spaces. In this framework, neither It\^{o}'s integrals nor martingale representation…