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A main puzzle of deep neural networks (DNNs) revolves around the apparent absence of "overfitting", defined in this paper as follows: the expected error does not get worse when increasing the number of neurons or of iterations of gradient…
Unsupervised neural networks, such as restricted Boltzmann machines (RBMs) and deep belief networks (DBNs), are powerful tools for feature selection and pattern recognition tasks. We demonstrate that overfitting occurs in such models just…
Neural networks (NNs) have been widely used to solve partial differential equations (PDEs) in the applications of physics, biology, and engineering. One effective approach for solving PDEs with a fixed differential operator is learning…
Two aspects of neural networks that have been extensively studied in the recent literature are their function approximation properties and their training by gradient descent methods. The approximation problem seeks accurate approximations…
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…
In this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating PDEs with two characteristic scales. From a continuous perspective, our formulation…
Over the past few years, neural network methods have evolved in various directions for approximating partial differential equations (PDEs). A promising new development is the integration of neural networks with classical numerical…
The use of neural networks to solve differential equations, as an alternative to traditional numerical solvers, has increased recently. However, error bounds for the obtained solutions have only been developed for certain equations. In this…
A key task in actuarial modelling involves modelling the distributional properties of losses. Classic (distributional) regression approaches like Generalized Linear Models (GLMs; Nelder and Wedderburn, 1972) are commonly used, but…
We analyze neural network solutions to partial differential equations obtained with Physics Informed Neural Networks. In particular, we apply tools of classical finite element error analysis to obtain conclusions about the error of the Deep…
Over the last few years deep artificial neural networks (DNNs) have very successfully been used in numerical simulations for a wide variety of computational problems including computer vision, image classification, speech recognition,…
Solving partial differential equations (PDEs) is the canonical approach for understanding the behavior of physical systems. However, large scale solutions of PDEs using state of the art discretization techniques remains an expensive…
We introduce a probability distribution, combined with an efficient sampling algorithm, for weights and biases of fully-connected neural networks. In a supervised learning context, no iterative optimization or gradient computations of…
This paper proposes a rank inspired neural network (RINN) to tackle the initialization sensitivity issue of physics informed extreme learning machines (PIELM) when numerically solving partial differential equations (PDEs). Unlike PIELM…
Deep neural networks (DNNs) are powerful machine learning models and have succeeded in various artificial intelligence tasks. Although various architectures and modules for the DNNs have been proposed, selecting and designing the…
In recent studies, several asymptotic upper bounds on generalization errors on deep neural networks (DNNs) are theoretically derived. These bounds are functions of several norms of weights of the DNNs, such as the Frobenius and spectral…
We present a novel framework combining Deep Operator Networks (DeepONets) with Physics-Informed Neural Networks (PINNs) to solve partial differential equations (PDEs) and estimate their unknown parameters. By integrating data-driven…
This work studies approximation based on single-hidden-layer feedforward and recurrent neural networks with randomly generated internal weights. These methods, in which only the last layer of weights and a few hyperparameters are optimized,…
In the present work, a multi-scale framework for neural network enhanced methods is proposed for approximation of function and solution of partial differential equations (PDEs). By introducing the multi-scale concept, the total solution of…
Can neural networks learn to solve partial differential equations (PDEs)? We investigate this question for two (systems of) PDEs, namely, the Poisson equation and the steady Navier--Stokes equations. The contributions of this paper are…