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A new and efficient neural-network and finite-difference hybrid method is developed for solving Poisson equation in a regular domain with jump discontinuities on embedded irregular interfaces. Since the solution has low regularity across…
In this work, we develop an efficient solver based on neural networks for second-order elliptic equations with variable coefficients and singular sources. This class of problems covers general point sources, line sources and the combination…
We present PFNN, a penalty-free neural network method, to efficiently solve a class of second-order boundary-value problems on complex geometries. To reduce the smoothness requirement, the original problem is reformulated to a weak form so…
This study presents a two-level Deep Domain Decomposition Method (Deep-DDM) augmented with a coarse-level network for solving boundary value problems using physics-informed neural networks (PINNs). The addition of the coarse level network…
In this paper, we present a parallel higher-order boundary integral method to solve the linear Poisson-Boltzmann (PB) equation. In our method, a well-posed boundary integral formulation is used to ensure the fast convergence of Krylov…
In this paper, we introduce a type of tensor neural network based machine learning method to solve elliptic multiscale problems. Based on the special structure, we can do the direct and highly accurate high dimensional integrations for the…
In this paper, we present a new iterative approximate method of solving boundary value problems. The idea is to compute approximate polynomial solutions in the Bernstein form using least squares approximation combined with some properties…
Elliptic partial differential equations are important both from application and analysis points of views. In this paper we apply the Closest Point Method to solving elliptic equations on general curved surfaces. Based on the closest point…
In this paper, we present BERN-NN as an efficient tool to perform bound propagation of Neural Networks (NNs). Bound propagation is a critical step in wide range of NN model checkers and reachability analysis tools. Given a bounded input…
A novel and scalable geometric multi-level algorithm is presented for the numerical solution of elliptic partial differential equations, specially designed to run with high occupancy of streaming processors inside Graphics Processing…
Algorithms based on deep neural networks (DNNs) have attracted increasing attention from the scientific computing community. DNN based algorithms are easy to implement, natural for nonlinear problems, and have shown great potential to…
The Maximum Balanced Biclique Problem (MBBP) is a prominent model with numerous applications. Yet, the problem is NP-hard and thus computationally challenging. We propose novel ideas for designing effective exact algorithms for MBBP.…
In this paper, a two-grid method is proposed to linearize and symmetrize the steady-state Poisson-Nernst-Planck equations. The computational system is decoupled to linearize and symmetrize equations by using this method, which can improve…
In non-linear estimations, it is common to assess sampling uncertainty by bootstrap inference. For complex models, this can be computationally intensive. This paper combines optimization with resampling: turning stochastic optimization into…
We provide an a priori analysis of collocation methods for solving elliptic boundary value problems. They begin with information in the form of point values of the data and utilize only this information to numerically approximate the…
We develop a randomized Newton's method for solving differential equations, based on a fully connected neural network discretization. In particular, the randomized Newton's method randomly chooses equations from the overdetermined nonlinear…
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…
This paper proposes a meshless deep learning algorithm, enriched physics-informed neural networks (EPINNs), to solve dynamic Poisson-Nernst-Planck (PNP) equations with strong coupling and nonlinear characteristics. The EPINNs takes the…
Bayesian Physics Informed Neural Networks (BPINN) have attracted considerable attention for inferring the system states and physical parameters of differential equations according to noisy observations. However, in practice, Hamiltonian…
This paper presents the benefits of using randomized neural networks instead of standard basis functions or deep neural networks to approximate the solutions of optimal stopping problems. The key idea is to use neural networks, where the…