Related papers: Machine Learning for semi linear PDEs
Partial differential equations (PDEs) play a crucial role in studying a vast number of problems in science and engineering. Numerically solving nonlinear and/or high-dimensional PDEs is often a challenging task. Inspired by the traditional…
Learning the solution of partial differential equations (PDEs) with a neural network is an attractive alternative to traditional solvers due to its elegance, greater flexibility and the ease of incorporating observed data. However, training…
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…
Recently, neural networks have been widely applied for solving partial differential equations (PDEs). Although such methods have been proven remarkably successful on practical engineering problems, they have not been shown, theoretically or…
This paper surveys the machine learning literature and presents in an optimization framework several commonly used machine learning approaches. Particularly, mathematical optimization models are presented for regression, classification,…
In this paper, we present a novel nonlinear programming-based approach to fine-tune pre-trained neural networks to improve robustness against adversarial attacks while maintaining high accuracy on clean data. Our method introduces…
Deep neural networks has been increasingly applied in fault diagnostics, where it uses historical data to capture systems behavior, bypassing the need for high-fidelity physical models. However, despite their competence in prediction tasks,…
Recent progress in building large-scale quantum devices for exploring quantum computing and simulation paradigms has relied upon effective tools for achieving and maintaining good experimental parameters, i.e. tuning up devices. In many…
In recent years, deep learning techniques have outperformed traditional models in many machine learning tasks. Deep neural networks have successfully been applied to address time series forecasting problems, which is a very important topic…
Following the development of digitization, a growing number of large Original Equipment Manufacturers (OEMs) are adapting computer vision or natural language processing in a wide range of applications such as anomaly detection and quality…
In various engineering and applied science applications, repetitive numerical simulations of partial differential equations (PDEs) for varying input parameters are often required (e.g., aircraft shape optimization over many design…
Numerical approximations of partial differential equations (PDEs) are routinely employed to formulate the solution of physics, engineering, and mathematical problems involving functions of several variables, such as the propagation of heat…
Physics-informed neural networks (PINNs) are extensively employed to solve partial differential equations (PDEs) by ensuring that the outputs and gradients of deep learning models adhere to the governing equations. However, constrained by…
Deep neural networks can achieve great successes when presented with large data sets and sufficient computational resources. However, their ability to learn new concepts quickly is limited. Meta-learning is one approach to address this…
Predicting outcomes and planning interactions with the physical world are long-standing goals for machine learning. A variety of such tasks involves continuous physical systems, which can be described by partial differential equations…
Deep neural networks (DNN) have achieved remarkable success in various fields, including computer vision and natural language processing. However, training an effective DNN model still poses challenges. This paper aims to propose a method…
A new method to solve computationally challenging (random) parametric obstacle problems is developed and analyzed, where the parameters can influence the related partial differential equation (PDE) and determine the position and surface…
Continual shrinking of pattern dimensions in the semiconductor domain is making it increasingly difficult to inspect defects due to factors such as the presence of stochastic noise and the dynamic behavior of defect patterns and types.…
We construct least squares formulations of PDEs with inhomogeneous essential boundary conditions, where boundary residuals are not measured in unpractical fractional Sobolev norms, but which formulations nevertheless are shown to yield a…
Thanks to their universal approximation properties and new efficient training strategies, Deep Neural Networks are becoming a valuable tool for the approximation of mathematical operators. In the present work, we introduce Mesh-Informed…