Related papers: Meshless Shape Optimization using Neural Networks …
The loss surface of deep neural networks has recently attracted interest in the optimization and machine learning communities as a prime example of high-dimensional non-convex problem. Some insights were recently gained using spin glass…
In this paper, we propose a multilevel stochastic framework for the solution of nonconvex unconstrained optimization problems. The proposed approach uses random regularized first-order models that exploit an available hierarchical…
In this paper, we propose a level set-based shape optimization method for acoustic wave propagation problems with a deformable structure. First, we propose a mathematical model for acoustic wave propagation with a deformed structure based…
This paper proposes a mesh-free computational framework and machine learning theory for solving elliptic PDEs on unknown manifolds, identified with point clouds, based on diffusion maps (DM) and deep learning. The PDE solver is formulated…
3D shape analysis has been largely focused on traditional 3D representations of point clouds and meshes, but the discrete nature of these data makes the analysis susceptible to variations in input resolutions. Recent development of neural…
Coordinate-based neural networks parameterizing implicit surfaces have emerged as efficient representations of geometry. They effectively act as parametric level sets with the zero-level set defining the surface of interest. We present a…
We propose a method that morphs high-orger meshes such that their boundaries and interfaces coincide/align with implicitly defined geometries. Our focus is particularly on the case when the target surface is prescribed as the zero…
Optimization problems with norm-bounding constraints arise in a variety of applications, including portfolio optimization, machine learning, and feature selection. A common approach to these problems involves relaxing the norm constraint…
Solving inverse and optimization problems over solutions of nonlinear partial differential equations (PDEs) on complex spatial domains is a long-standing challenge. Here we introduce a method that parameterizes the solution using spectral…
Models incorporating uncertain inputs, such as random forces or material parameters, have been of increasing interest in PDE-constrained optimization. In this paper, we focus on the efficient numerical minimization of a convex and smooth…
This work introduces a new approach for accelerating the numerical analysis of time-domain partial differential equations (PDEs) governing complex physical systems. The methodology is based on a combination of a classical reduced-order…
Neural networks are powerful tools for approximating high dimensional data that have been used in many contexts, including solution of partial differential equations (PDEs). We describe a solver for multiscale fully nonlinear elliptic…
This article revolves around shape and topology optimization, in the applicative context where the objective and constraint functionals depend on the solution to a physical boundary value problem posed on the optimized domain. We introduce…
Polygonal meshes have become the standard for discretely approximating 3D shapes, thanks to their efficiency and high flexibility in capturing non-uniform shapes. This non-uniformity, however, leads to irregularity in the mesh structure,…
This article investigates the application of computer vision and graph-based models in solving mesh-based partial differential equations within high-performance computing environments. Focusing on structured, graded structured, and…
We derive several numerical methods for designing optimized first-order algorithms in unconstrained convex optimization settings. Our methods are based on the Performance Estimation Problem (PEP) framework, which casts the worst-case…
A novel multi-level method for partial differential equations with uncertain parameters is proposed. The principle behind the method is that the error between grid levels in multi-level methods has a spatial structure that is by good…
We propose a neural network-based algorithm for solving forward and inverse problems for partial differential equations in unsupervised fashion. The solution is approximated by a deep neural network which is the minimizer of a cost…
In this paper, we propose a mesh-free numerical method for solving elliptic PDEs on unknown manifolds, identified with randomly sampled point cloud data. The PDE solver is formulated as a spectral method where the test function space is the…
We present a new approach to discretizing shape optimization problems that generalizes standard moving mesh methods to higher-order mesh deformations and that is naturally compatible with higher-order finite element discretizations of…