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Solving the Fokker-Planck equation for high-dimensional complex dynamical systems remains a pivotal yet challenging task due to the intractability of analytical solutions and the limitations of traditional numerical methods. In this work,…
Surface partial differential equations arise in numerous scientific and engineering applications. Their numerical solution on static and evolving surfaces remains challenging due to geometric complexity and, for evolving geometries, the…
In this paper, we investigate neural networks applied to multiscale simulations and discuss a design of a novel deep neural network model reduction approach for multiscale problems. Due to the multiscale nature of the medium, the fine-grid…
Understanding the behavior of stochastic gradient methods is a central problem in modern machine learning. Recent work has highlighted diagonal linear networks as a simplified yet expressive setting for analyzing the optimization and…
We introduce Monte Carlo methods to compute the solution of elliptic equations with pure Neumann boundary conditions. We first prove that the solution obtained by the stochastic representation has a zero mean value with respect to the…
A tremendous range of design tasks in materials, physics, and biology can be formulated as finding the optimum of an objective function depending on many parameters without knowing its closed-form expression or the derivative. Traditional…
We introduce forward-backward stochastic differential equations, highlighting the connection between solutions of these and solutions of partial differential equations, related by the Feynman-Kac theorem. We review the technique of…
Simulating stochastic differential equations (SDEs) in bounded domains, presents significant computational challenges due to particle exit phenomena, which requires accurate modeling of interior stochastic dynamics and boundary…
We propose a predictor-corrector adaptive method for the simulation of hyperbolic partial differential equations (PDEs) on networks under general uncertainty in parameters, initial conditions, or boundary conditions. The approach is based…
A new penalty-free neural network method, PFNN-2, is presented for solving partial differential equations, which is a subsequent improvement of our previously proposed PFNN method [1]. PFNN-2 inherits all advantages of PFNN in handling the…
We consider the problem of solving partial differential equations (PDEs) in domains with complex microparticle geometry that is impractical, or intractable, to model explicitly. Drawing inspiration from volume rendering, we propose tackling…
Motivated by many applications in complex domains with boundaries exposed to large topological changes or deformations, fictitious domain methods regard the actual domain of interest as being embedded in a fixed Cartesian background. This…
We present a domain decomposition-based deep learning method for solving elliptic and parabolic interface problems with discontinuous coefficients in two to ten dimensions. Our Multi-Activation Function (MAF) approach employs two…
We propose and analyze a new discretization technique for a linear-quadratic optimal control problem involving the fractional powers of a symmetric and uniformly elliptic second oder operator; control constraints are considered. Since these…
Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high…
For the Poisson equation posed in a domain containing a large number of polygonal perforations, we propose a low-dimensional coarse approximation space based on a coarse polygonal partitioning of the domain. Similarly to other multiscale…
The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations…
Modern recording techniques enable neuroscientists to simultaneously study neural activity across large populations of neurons, with capturing predictor-dependent correlations being a fundamental challenge in neuroscience. Moreover, the…
We present and discuss a framework for computer-aided multiscale analysis, which enables models at a "fine" (microscopic/stochastic) level of description to perform modeling tasks at a "coarse" (macroscopic, systems) level. These…
In this paper, we propose a mesh-free method to solve full stokes equation which models the glacier movement with nonlinear rheology. Our approach is inspired by the Deep-Ritz method proposed in [12]. We first formulate the solution of…