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We develop a framework for estimating unknown partial differential equations from noisy data, using a deep learning approach. Given noisy samples of a solution to an unknown PDE, our method interpolates the samples using a neural network,…
Sparse deep learning has become a popular technique for improving the performance of deep neural networks in areas such as uncertainty quantification, variable selection, and large-scale network compression. However, most existing research…
Neural networks (NNs) can achieved high performance in various fields such as computer vision, and natural language processing. However, deploying NNs in resource-constrained safety-critical systems has challenges due to uncertainty in the…
We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate…
Motivated by recent research on Physics-Informed Neural Networks (PINNs), we make the first attempt to introduce the PINNs for numerical simulation of the elliptic Partial Differential Equations (PDEs) on 3D manifolds. PINNs are one of the…
The problem of automatic and accurate forecasting of time-series data has always been an interesting challenge for the machine learning and forecasting community. A majority of the real-world time-series problems have non-stationary…
Designing an optimal deep neural network for a given task is important and challenging in many machine learning applications. To address this issue, we introduce a self-adaptive algorithm: the adaptive network enhancement (ANE) method,…
The use of deep learning methods for solving PDEs is a field in full expansion. In particular, Physical Informed Neural Networks, that implement a sampling of the physical domain and use a loss function that penalizes the violation of the…
Classical methods of solving spatiotemporal dynamical systems include statistical approaches such as autoregressive integrated moving average, which assume linear and stationary relationships between systems' previous outputs. Development…
This paper establishes a comprehensive approximation result for deep fully-connected neural networks with commonly-used and general activation functions in Sobolev spaces $W^{n,\infty}$, with errors measured in the $W^{m,p}$-norm for $m <…
We present a subspace method based on neural networks (SNN) for solving the partial differential equation with high accuracy. The basic idea of our method is to use some functions based on neural networks as base functions to span a…
Deep neural networks (DNNs) are increasingly being used in a variety of traditional radiofrequency (RF) problems. Previous work has shown that while DNN classifiers are typically more accurate than traditional signal processing algorithms,…
Recent theoretical studies proved that deep neural network (DNN) estimators obtained by minimizing empirical risk with a certain sparsity constraint can attain optimal convergence rates for regression and classification problems. However,…
Bayesian neural networks (BNN) and deep ensembles are principled approaches to estimate the predictive uncertainty of a deep learning model. However their practicality in real-time, industrial-scale applications are limited due to their…
With the development of high computational devices, deep neural networks (DNNs), in recent years, have gained significant popularity in many Artificial Intelligence (AI) applications. However, previous efforts have shown that DNNs were…
In the past decade, deep neural networks (DNNs) came to the fore as the leading machine learning algorithms for a variety of tasks. Their raise was founded on market needs and engineering craftsmanship, the latter based more on trial and…
Physical systems whose dynamics are governed by partial differential equations (PDEs) find applications in numerous fields, from engineering design to weather forecasting. The process of obtaining the solution from such PDEs may be…
Deep learning takes advantage of large datasets and computationally efficient training algorithms to outperform other approaches at various machine learning tasks. However, imperfections in the training phase of deep neural networks make…
We derive quantitative error bounds for deep neural networks (DNNs) approximating option prices on a $d$-dimensional risky asset as functions of the underlying model parameters, payoff parameters and initial conditions. We cover a general…
The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length and timescales. Often, it is computationally intractable to resolve the finest features…