相关论文: A deep learning method for solving Fokker-Planck e…

Nonlinear Fokker-Planck equations play a major role in modeling large systems of interacting particles with a proved effectiveness in describing real world phenomena ranging from classical fields such as fluids and plasma to social and…

This work is concerned with the existence of mild solutions and the uniqueness of distributional solutions to nonlinear Fokker-Planck equations with nonlocal operators $\Psi(-\Delta)$, where $\Psi$ is a Bernstein function. As applications,…

In this paper, we propose a general framework for solving high-dimensional partial differential equations with tensor networks. Our approach uses Monte-Carlo simulations to update the solution and re-estimates the new solution from samples…

We show how the Equation-Free approach for multi-scale computations can be exploited to systematically study the dynamics of neural interactions on a random regular connected graph under a pairwise representation perspective. Using an…

We study a discrete-time portfolio selection problem with partial information and maxi\-mum drawdown constraint. Drift uncertainty in the multidimensional framework is modeled by a prior probability distribution. In this Bayesian framework,…

This paper explores the use of a discrete singular convolution algorithm as a unified approach for numerical integration of the Fokker-Planck equation. The unified features of the discrete singular convolution algorithm are discussed. It is…

We present a neural network-based method for solving linear and nonlinear partial differential equations, by combining the ideas of extreme learning machines (ELM), domain decomposition and local neural networks. The field solution on each…

We utilize the Feature Decoupling Distributed (FDD) method to enhance the capability of deep learning to fit the Nonlinear Schrodinger Equation (NLSE), significantly reducing the NLSE loss compared to non decoupling model.

Nonlinear differential equations are encountered as models of fluid flow, spiking neurons, and many other systems of interest in the real world. Common features of these systems are that their behaviors are difficult to describe exactly and…

We study the asymptotic behavior of Fokker-Planck equations with spatially inhomogeneous nonlinear diffusion, based on the energy dissipation law. First, we consider the Fokker-Planck equation with porous-medium-type nonlinear diffusion…

We propose a neural network-based algorithm for solving forward and inverse problems for partial differential equations in unsupervised fashion. The solution is approximated by a deep neural network which is the minimizer of a cost…

A machine learning technique is proposed for quantifying uncertainty in power system dynamics with spatiotemporally correlated stochastic forcing. We learn one-dimensional linear partial differential equations for the probability density…

We introduce a self-consistent deep-learning framework which, for a noisy deterministic time series, provides unsupervised filtering, state-space reconstruction, identification of the underlying differential equations and forecasting.…

Increasingly larger data sets of processes in space and time ask for statistical models and methods that can cope with such data. We show that the solution of a stochastic advection-diffusion partial differential equation provides a…

While there is currently a lot of enthusiasm about "big data", useful data is usually "small" and expensive to acquire. In this paper, we present a new paradigm of learning partial differential equations from {\em small} data. In…

We obtain equilibration rates for a one-dimensional nonlocal Fokker-Planck equation with time-dependent diffusion coefficient and drift, modeling the relaxation of a large swarm of robots, feeling each other in terms of their distance,…

Despite its empirical success, deep learning still lacks a comprehensive theoretical understanding of model fitting and generalization. This paper proposes the probability distribution (PD) learning framework to analyze the optimization and…

In insurance mathematics optimal control problems over an infinite time horizon arise when computing risk measures. Their solutions correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In…

We introduce a deep neural network-based numerical method for solving kinetic Fokker Planck equations, including both linear and nonlinear cases. Building upon the conservative dissipative structure of Vlasov-type equations, we formulate a…

In this paper, based on the combination of finite element mesh and neural network, a novel type of neural network element space and corresponding machine learning method are designed for solving partial differential equations. The…