Related papers: Adaptive importance sampling for Deep Ritz

Physics-informed neural networks (PINNs) have shown to be an effective tool for solving forward and inverse problems of partial differential equations (PDEs). PINNs embed the PDEs into the loss of the neural network, and this PDE loss is…

Modeling dynamics in the form of partial differential equations (PDEs) is an effectual way to understand real-world physics processes. For complex physics systems, analytical solutions are not available and numerical solutions are…

Sampling is an important tool for estimating large, complex sums and integrals over high dimensional spaces. For instance, important sampling has been used as an alternative to exact methods for inference in belief networks. Ideally, we…

Importance sampling has been successfully used to accelerate stochastic optimization in many convex problems. However, the lack of an efficient way to calculate the importance still hinders its application to Deep Learning. In this paper,…

Physics-informed neural networks (PINNs) have emerged as powerful tools for solving a wide range of partial differential equations (PDEs). However, despite their user-friendly interface and broad applicability, PINNs encounter challenges in…

In this paper, we propose a new approach to model reduction of parameterized partial differential equations (PDEs) based on the concept of adaptive reduced bases. The presented approach is particularly suited for large-scale nonlinear…

In this paper we propose a new model-based unsupervised learning method, called VarNet, for the solution of partial differential equations (PDEs) using deep neural networks (NNs). Particularly, we propose a novel loss function that relies…

Variational inequalities are widely applied in mechanical engineering, fluid penetration, transportation, and other fields. In this paper, a Deep Ritz method based on Physics-Informed Neural Networks (PINNs) is proposed to enhance the…

Supervised deep learning requires a large amount of training samples with annotations (e.g. label class for classification task, pixel- or voxel-wised label map for segmentation tasks), which are expensive and time-consuming to obtain.…

Stochastic gradient algorithms are the main focus of large-scale optimization problems and led to important successes in the recent advancement of the deep learning algorithms. The convergence of SGD depends on the careful choice of…

An important step in the design of autonomous systems is to evaluate the probability that a failure will occur. In safety-critical domains, the failure probability is extremely small so that the evaluation of a policy through Monte Carlo…

Adaptive sampling algorithms are modern and efficient methods that dynamically adjust the sample size throughout the optimization process. However, they may encounter difficulties in risk-averse settings, particularly due to the challenge…

In this paper, inspired by the multigrid method, we propose a multi-level deep framework for deep solvers. Overall, it divides the entire training process into different levels of training. At each level of training, an adaptive sampling…

In solving simulation-based stochastic root-finding or optimization problems that involve rare events, such as in extreme quantile estimation, running crude Monte Carlo can be prohibitively inefficient. To address this issue, importance…

Machine learning is a rapidly advancing field with diverse applications across various domains. One prominent area of research is the utilization of deep learning techniques for solving partial differential equations(PDEs). In this work, we…

Randomized neural network (RaNN) methods have been proposed for solving various partial differential equations (PDEs), demonstrating high accuracy and efficiency. However, initializing the fixed parameters remains challenging. Additionally,…

The deep-learning-based least squares method has shown successful results in solving high-dimensional non-linear partial differential equations (PDEs). However, this method usually converges slowly. To speed up the convergence of this…

A classic approach for solving differential equations with neural networks builds upon neural forms, which employ the differential equation with a discretisation of the solution domain. Making use of neural forms for time-dependent…

Randomized neural networks (RaNNs) are attractive for partial differential equations (PDEs) because they replace expensive end-to-end training with a linear least-squares solve over randomized hidden features. Their practical performance,…

Adaptive importance sampling is a class of techniques for finding good proposal distributions for importance sampling. Often the proposal distributions are standard probability distributions whose parameters are adapted based on the…