Related papers: Solving time dependent Fokker-Planck equations via…
A common way to numerically solve Fokker-Planck equations is the Chang-Cooper method in space combined with one of the Euler methods in time. However, the explicit Euler method is only conditionally positive, leading to severe restrictions…
In this paper, we propose a novel machine learning method based on an adaptive tensor neural network subspace for solving quasiperiodic elliptic problems. To this end, we first provide a theoretical analysis of the associated quasiperiodic…
In this note we analyze a model for a unidirectional unsteady flow of a viscous incompressible fluid with time dependent viscosity. A possible way to take into account such behaviour is to introduce a memory formalism, including thus the…
Time series prediction underpins a broad range of downstream tasks across many scientific domains. Recent advances and increasing adoption of black-box machine learning models for time series prediction highlight the critical need for…
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 establish a time-stepping learning algorithm and apply it to predict the solution of the partial differential equation of motion in micromagnetism as a dynamical system depending on the external field as parameter. The data-driven…
We introduce a generalization of temporal-difference (TD) learning to networks of interrelated predictions. Rather than relating a single prediction to itself at a later time, as in conventional TD methods, a TD network relates each…
The study of parametric differential equations plays a crucial role in weather forecasting and epidemiological modeling. These phenomena are better represented using fractional derivatives due to their inherent memory or hereditary effects.…
Designing predictive models for subjective problems in natural language processing (NLP) remains challenging. This is mainly due to its non-deterministic nature and different perceptions of the content by different humans. It may be solved…
We propose a high order adaptive-rank implicit integrators for stiff time-dependent PDEs, leveraging extended Krylov subspaces to efficiently and adaptively populate low-rank solution bases. This allows for the accurate representation of…
It is commonly known that the Fokker-Planck equation is exactly solvable only for some particular systems, usually with time-independent drift coefficients. To extend the class of solvable problems, we use the intertwining relations of SUSY…
We reconstruct the velocity field of incompressible flows given a finite set of measurements. For the spatial approximation, we introduce the Sparse Fourier divergence-free (SFdf) approximation based on a discrete $L^2$ projection. Within…
In this paper, we train turbulence models based on convolutional neural networks. These learned turbulence models improve under-resolved low resolution solutions to the incompressible Navier-Stokes equations at simulation time. Our study…
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 work is concerned with solving high-dimensional Fokker-Planck equations with the novel perspective that solving the PDE can be reduced to independent instances of density estimation tasks based on the trajectories sampled from its…
Machine teaching is an algorithmic framework for teaching a target hypothesis via a sequence of examples or demonstrations. We investigate machine teaching for temporal logic formulas -- a novel and expressive hypothesis class amenable to…
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…
Probabilistic Temporal Tensor Factorization (PTTF) is an effective algorithm to model the temporal tensor data. It leverages a time constraint to capture the evolving properties of tensor data. Nowadays the exploding dataset demands a large…
We address the numerical challenge of solving the Hookean-type time-fractional Navier--Stokes--Fokker--Planck equation, a history-dependent system of PDEs defined on the Cartesian product of two $d$-dimensional spaces in the turbulent…
We propose a novel approach for modeling chemical reactions within the particle-based Fokker-Planck framework for gas flow simulations which conserves mass, momentum, and energy while retaining the performance advantages of the…