Related papers: Tensor approximation of functional differential eq…
In this article, we introduce and analyze a deep learning based approximation algorithm for SPDEs. Our approach employs neural networks to approximate the solutions of SPDEs along given realizations of the driving noise process. If applied…
We propose and study a scheme combining the finite element method and machine learning techniques for the numerical approximations of coupled nonlinear forward-backward stochastic partial differential equations (FBSPDEs) with homogeneous…
In complex physical systems, conventional differential equations often fall short in capturing non-local and memory effects, as they are limited to local dynamics and integer-order interactions. This study introduces a stepwise data-driven…
Fault-tolerant quantum computing is a promising technology to solve linear partial differential equations that are classically demanding to integrate. It is still challenging to solve non-linear equations in fluid dynamics, such as the…
By the approximation method introduced in \cite{FYW}, the existence and uniqueness are proved for a class of distribution-dependent stochastic functional differential equations (DDSFDEs). Moreover, combining the Harnack and shift-Harnack…
This paper derives a free analog of the Euler-Maruyama method (fEMM) to numerically approximate solutions of free stochastic differential equations (fSDEs). Simply speaking fSDEs are stochastic differential equations in the context of…
The accurate representation of numerous physical, chemical, and biological processes relies heavily on differential equations (DEs), particularly nonlinear differential equations (NDEs). While understanding these complex systems…
We study a general class of singular degenerate parabolic stochastic partial differential equations (SPDEs) which include, in particular, the stochastic porous medium equations and the stochastic fast diffusion equation. We propose a fully…
In this work, we concern with the high order numerical methods for coupled forward-backward stochastic differential equations (FBSDEs). Based on the FBSDEs theory, we derive two reference ordinary differential equations (ODEs) from the…
In this work, we study the numerical approximation of a class of singular fully coupled forward backward stochastic differential equations. These equations have a degenerate forward component and non-smooth terminal condition. They are…
Diffusion models have recently emerged as powerful stochastic frameworks for high-dimensional inference and generation. However, existing applications to partial differential equations (PDEs) predominantly rely on physics-informed training…
The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length and timescales. Often, it is computationally intractable to resolve the finest features…
Backward stochastic differential equation (BSDE) provides probabilistic solutions for a class of parabolic partial differential equations (PDEs). DeepBSDE and FBSNN are two deep learning approaches for solving high-dimensional PDEs through…
Fractional partial differential equations (FDEs) are used to describe phenomena that involve a "non-local" or "long-range" interaction of some kind. Accurate and practical numerical approximation of their solutions is challenging due to the…
Partial Differential Equations (PDEs) describe phenomena ranging from turbulence and epidemics to quantum mechanics and financial markets. Despite recent advances in computational science, solving such PDEs for real-world applications…
Quantum algorithms for partial differential equations (PDEs) face severe practical constraints on near-term hardware: limited qubit counts restrict spatial resolution to coarse grids, while circuit depth limitations prevent accurate…
Backward stochastic differential equations (BSDEs) appear in numeruous applications. Classical approximation methods suffer from the curse of dimensionality and deep learning-based approximation methods are not known to converge to the BSDE…
We present a stochastic method for efficiently computing the solution of time-fractional partial differential equations (fPDEs) that model anomalous diffusion problems of the subdiffusive type. After discretizing the fPDE in space, the…
We develop efficient and high-order accurate finite difference methods for elliptic partial differential equations in complex geometry in the Difference Potentials framework. The main novelty of the developed schemes is the use of local…
We introduce a novel numerical scheme for solving the Fokker-Planck equation of discretized Dean-Kawasaki models with a functional tensor network ansatz. The Dean-Kawasaki model describes density fluctuations of interacting particle…