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Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find…
Many nonlinear differential equations arising from practical problems may permit nontrivial multiple solutions relevant to applications, and these multiple solutions are helpful to deeply understand these practical problems and to improve…
Multiple equilibrium states arise in many physical systems, including various types of liquid crystal structures. Having the ability to reliably compute such states enables more accurate physical analysis and understanding of experimental…
Nonlinear Partial Differential Equations (PDEs) are ubiquitous in mathematical physics and engineering. Although Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for solving PDE problems, they typically struggle to…
In this paper we generalize the technique of deflation to define two new methods to systematically find many local minima of a nonlinear least squares problem. The methods are based on the Gauss-Newton algorithm, and as such do not require…
Deep learning is a powerful tool for solving nonlinear differential equations, but usually, only the solution corresponding to the flattest local minimizer can be found due to the implicit regularization of stochastic gradient descent. This…
We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for the numerical solution of partial differential equations. We start with a detailed explanation of the method for the…
We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three…
In this paper, we propose a novel variable-separation (NVS) method for generic multivariate functions. The idea of NVS is extended to to obtain the solution in tensor product structure for stochastic partial differential equations (SPDEs).…
Variational inequalities can in general support distinct solutions. In this paper we study an algorithm for computing distinct solutions of a variational inequality, without varying the initial guess supplied to the solver. The central idea…
Deflation is an efficient numerical technique for identifying new branches of steady state solutions to nonlinear partial differential equations. Here, we demonstrate how to extend deflation to discover new periodic orbits in nonlinear…
The numerical solution methods for partial differential equation (PDE) solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods…
Stellarator optimization is a multi-objective, non-convex problem characterized by a complex objective landscape containing many local minima. The solution resulting from a single optimization is highly sensitive to factors such as the…
This paper presents a numerical method for variable coefficient elliptic PDEs with mostly smooth solutions on two dimensional domains. The PDE is discretized via a multi-domain spectral collocation method of high local order (order 30 and…
We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…
In this article we investigate the numerical solution of a scalar semilinear stochastic delay differential equation (SDDE) where the linear instantaneous feedback and nonlinear delayed feedback terms are perturbed by a pair of standard…
Spectral and spectral element methods using Galerkin type formulations are efficient for solving linear fractional PDEs (FPDEs) of constant order but are not efficient in solving nonlinear FPDEs and cannot handle FPDEs with variable-order.…
A multiscale numerical method is proposed for the solution of semi-linear elliptic stochastic partial differential equations with localized uncertainties and non-linearities, the uncertainties being modeled by a set of random parameters. It…
This paper proposes a domain decomposition subspace neural network method for efficiently solving linear and nonlinear partial differential equations. By combining the principles of domain decomposition and subspace neural networks, the…
Multiple solutions are common in various non-convex problems arising from industrial and scientific computing. Nonetheless, understanding the nontrivial solutions' qualitative properties seems limited, partially due to the lack of efficient…