Related papers: A Deep Learning Approach to Nonconvex Energy Minim…

This paper presents a novel approach that combines the Deep Ritz Method (DRM) with Fourier feature mapping to solve minimization problems comprised of multi-well, non-convex energy potentials. These problems present computational challenges…

In the setting of continuum elasticity, phase transformations involving martensitic variants are modeled by a free energy density function that is non-convex in strain space. Here, we adopt an existing mathematical model in which we…

In materials that undergo martensitic phase transformation, macroscopic loading often leads to the creation and/or rearrangement of elastic domains. This paper considers an example {involving} a single-crystal slab made from two martensite…

We present a practical algorithm for partially relaxing multiwell energy densities such as pertain to materials undergoing martensitic phase transitions. The algorithm is based on sequential lamination, but the evolution of the…

In this paper, we propose a novel mesh-free numerical method for solving the elliptic interface problems based on deep learning. We approximate the solution by the neural networks and, since the solution may change dramatically across the…

We introduce two new particle-based algorithms for learning latent variable models via marginal maximum likelihood estimation, including one which is entirely tuning-free. Our methods are based on the perspective of marginal maximum…

We introduce a novel variant of the recursive multilevel trust-region (RMTR) method, called MASTR. The method is designed for solving non-convex bound-constrained minimization problems, which arise from the finite element discretization of…

In this paper, a shallow Ritz-type neural network for solving elliptic equations with delta function singular sources on an interface is developed. There are three novel features in the present work; namely, (i) the delta function…

We present an approach to numerical homogenization of the elastic response of microstructures. Our work uses deep neural network representations trained on data obtained from direct numerical simulation (DNS) of martensitic phase…

Recent progress in robust statistical learning has mainly tackled convex problems, like mean estimation or linear regression, with non-convex challenges receiving less attention. Phase retrieval exemplifies such a non-convex problem,…

In this work, we present a framework for the matrix-free solution to a monolithic quasi-static phase-field fracture model with geometric multigrid methods. Using a standard matrix based approach within the Finite Element Method requires…

The multi-scale and non-linear nature of phase-field models of solidification requires fine spatial and temporal discretization, leading to long computation times. This could be overcome with artificial-intelligence approaches. Surrogate…

Residual minimization is a widely used technique for solving Partial Differential Equations in variational form. It minimizes the dual norm of the residual, which naturally yields a saddle-point (min-max) problem over the so-called trial…

In this paper, we propose a mesh-free method to solve interface problems using the deep learning approach. Two interface problems are considered. The first one is an elliptic PDE with a discontinuous and high-contrast coefficient. While the…

We explore the potential of the deep Ritz method to learn complex fracture processes such as quasistatic crack nucleation, propagation, kinking, branching, and coalescence within the unified variational framework of phase-field modeling of…

In this paper, we study the statistical limits of deep learning techniques for solving elliptic partial differential equations (PDEs) from random samples using the Deep Ritz Method (DRM) and Physics-Informed Neural Networks (PINNs). To…

Although deep learning has shown its powerful performance in many applications, the mathematical principles behind neural networks are still mysterious. In this paper, we consider the problem of learning a one-hidden-layer neural network…

We consider a model of energy minimization arising in the study of the mechanical behavior caused by cell contraction within a fibrous biological medium. The macroscopic model is based on the theory of non rank-one convex nonlinear…

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…

Multi-user multiple-input multiple-output (MU-MIMO) beamforming design is typically formulated as a non-convex weighted sum rate (WSR) maximization problem that is known to be NP-hard. This problem is solved either by iterative algorithms,…