Related papers: The Neural Network Approach to Inverse Problems in…

Inverse problems involving differential equations often require identifying unknown parameters or functions from data. Existing approaches, such as Physics-Informed Neural Networks (PINNs), Universal Differential Equations (UDEs) and…

We develop a randomized Newton's method for solving differential equations, based on a fully connected neural network discretization. In particular, the randomized Newton's method randomly chooses equations from the overdetermined nonlinear…

Solving inverse problems is a fundamental component of science, engineering and mathematics. With the advent of deep learning, deep neural networks have significant potential to outperform existing state-of-the-art, model-based methods for…

Deep neural networks are widely used for nonlinear function approximation with applications ranging from computer vision to control. Although these networks involve the composition of simple arithmetic operations, it can be very challenging…

In this paper, based on the combination of tensor neural network and a posteriori error estimator, a novel type of machine learning method is proposed to solve high-dimensional boundary value problems with homogeneous and non-homogeneous…

Neural ordinary differential equations (Neural ODEs) propose the idea that a sequence of layers in a neural network is just a discretisation of an ODE, and thus can instead be directly modelled by a parameterised ODE. This idea has had…

We explore unique considerations involved in fitting ML models to data with very high precision, as is often required for science applications. We empirically compare various function approximation methods and study how they scale with…

Deep learning, a branch of artificial intelligence, is a data-driven method that uses multiple layers of interconnected units or neurons to learn intricate patterns and representations directly from raw input data. Empowered by this…

Deep neural networks work well at approximating complicated functions when provided with data and trained by gradient descent methods. At the same time, there is a vast amount of existing functions that programmatically solve different…

Over the year, people have been using deep learning to tackle inversion problems, and we see the framework has been applied to build relationship between recording wavefield and velocity (Yang et al., 2016). Here we will extend the work…

Deep neural networks have significantly alleviated the burden of feature engineering, but comparable efforts are now required to determine effective architectures for these networks. Furthermore, as network sizes have become excessively…

Rectifying the orientation of images represents a daily task for every photographer. This task may be complicated even for the human eye, especially when the horizon or other horizontal and vertical lines in the image are missing. In this…

This work presents a new algorithm for training recurrent neural networks (although ideas are applicable to feedforward networks as well). The algorithm is derived from a theory in nonconvex optimization related to the diffusion equation.…

Despite the tremendous success of deep neural networks in various learning problems, it has been observed that adding an intentionally designed adversarial perturbation to inputs of these architectures leads to erroneous classification with…

We present an approach to adaptively utilize deep neural networks in order to reduce the evaluation time on new examples without loss of accuracy. Rather than attempting to redesign or approximate existing networks, we propose two schemes…

The numerical solution of differential equations using neural networks has become a central topic in scientific computing, with Physics-Informed Neural Networks (PINNs) emerging as a powerful paradigm for both forward and inverse problems.…

We develop a minimax rate analysis to describe the reason that deep neural networks (DNNs) perform better than other standard methods. For nonparametric regression problems, it is well known that many standard methods attain the minimax…

Feedforward neural networks offer a promising approach for solving differential equations. However, the reliability and accuracy of the approximation still represent delicate issues that are not fully resolved in the current literature.…

In this document, a neural network is employed in order to estimate the solution of the initial value problem in the context of non linear trajectories. Such trajectories can be subject to gravity, thrust, drag, centrifugal force,…

Machine Learning models incorporating multiple layered learning networks have been seen to provide effective models for various classification problems. The resulting optimization problem to solve for the optimal vector minimizing the…