Related papers: Solving Fokker-Planck equation using deep learning
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 this work, we investigated the feasibility of applying deep learning techniques to solve Poisson's equation. A deep convolutional neural network is set up to predict the distribution of electric potential in 2D or 3D cases. With proper…
We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of multi-dimensional non-linear stochastic differential equations, which relies upon discrete-time observations of the state. The…
In this paper, we present a deep learning-based numerical method for approximating high dimensional stochastic partial differential equations (SPDEs). At each time step, our method relies on a predictor-corrector procedure. More precisely,…
We propose the Factorized Fourier Neural Operator (F-FNO), a learning-based approach for simulating partial differential equations (PDEs). Starting from a recently proposed Fourier representation of flow fields, the F-FNO bridges the…
In this article, we study the nonlinear Fokker-Planck (FP) equation that arises as a mean-field (macroscopic) approximation of bounded confidence opinion dynamics, where opinions are influenced by environmental noises and opinions of…
Recent efforts to develop trustworthy AI systems have increased interest in learning problems with explicit requirements, or constraints. In deep learning, however, such problems are often handled through fixed weighted-sum penalization:…
Differential equations are used to model and predict the behaviour of complex systems in a wide range of fields, and the ability to solve them is an important asset for understanding and predicting the behaviour of these systems.…
Deep Neural Networks (DNNs) have performed admirably in classification tasks. However, the characterization of their classification uncertainties, required for certain applications, has been lacking. In this work, we investigate the issue…
Solving the stationary nonlinear Fokker-Planck equations is important in applications and examples include the Poisson-Boltzmann equation and the two layer neural networks. Making use of the connection between the interacting particle…
We build upon Estrin et al. (2019) to develop a general constrained nonlinear optimization algorithm based on a smooth penalty function proposed by Fletcher (1970, 1973b). Although Fletcher's approach has historically been considered…
In this paper, we investigate the unsupervised deep representation learning issue and technically propose a novel framework called Deep Self-representative Concept Factorization Network (DSCF-Net), for clustering deep features. To improve…
Deep neural network approaches show promise in solving partial differential equations. However, unlike traditional numerical methods, they face challenges in enforcing essential boundary conditions. The widely adopted penalty-type methods,…
We consider three classes of linear non-symmetric Fokker-Planck equations having a unique steady state and establish exponential convergence of solutions towards the steady state with explicit (estimates of) decay rates. First,…
Federated Learning (FL) enables edge devices to collaboratively learn a global model, but it may not perform well when clients have high data heterogeneity. In this paper, we propose a dynamic clustering algorithm for personalized federated…
In this paper we suggest a consistent approach to derivation of generalized Fokker-Planck equation (GFPE) for Gaussian non-Markovian processes with stationary increments. This approach allows us to construct the probability density function…
We study optimization algorithms for the finite sum problems frequently arising in machine learning applications. First, we propose novel variants of stochastic gradient descent with a variance reduction property that enables linear…
Particle-based stochastic approximations of the Boltzmann equation are popular tools for simulations of non-equilibrium gas flows, for which the Navier-Stokes-Fourier equations fail to provide accurate description. However, these numerical…
This letter aims at justifying the stochastic equations in terms of the number density variable, which are still controversial, via complementing Dean's approach [Dean D S 1996 {\itshape J. Phys. A} {\bf 29} L613]. Our course is twofold:…
Calculating perturbation response properties of materials from first principles provides a vital link between theory and experiment, but is bottlenecked by the high computational cost. Here a general framework is proposed to perform density…