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High-frequency issues have been remarkably challenges in numerical methods for partial differential equations. In this paper, a learning based numerical method (LbNM) is proposed for Helmholtz equation with high frequency. The main novelty…
Recurrent neural networks (RNNs), especially long short-term memory (LSTM) RNNs, are effective network for sequential task like speech recognition. Deeper LSTM models perform well on large vocabulary continuous speech recognition, because…
In this paper, we investigate the logarithmic nonlinear Schr\"odinger (LNLS) equation with the parity-time (PT)-symmetric harmonic potential, which is an important physical model in many fields such as nuclear physics, quantum optics, magma…
A supervised learning approach is proposed for regularization of large inverse problems where the main operator is built from noisy data. This is germane to superresolution imaging via the sampling indicators of the inverse scattering…
Inverse scattering problems have many important applications. In this paper, given limited aperture data, we propose a Bayesian method for the inverse acoustic scattering to reconstruct the shape of an obstacle. The inverse problem is…
Microwave imaging is commonly based on the solution of linearized inverse scattering problems by matched filtering algorithms, i.e., by applying the adjoint of the forward scattering operator to the observation data. A more rigorous…
This paper is dedicated to addressing the simultaneous inversion problem involving the initial value and space-dependent source term in a time-fractional diffusion-wave equation. Firstly, we establish the uniqueness of the inverse problem…
We present a hybrid approach combining isogeometric analysis with deep operator networks to solve electromagnetic scattering problems. The neural network takes a computer-aided design representation as input and predicts the electromagnetic…
This paper re-visits the spectral method for learning latent variable models defined in terms of observable operators. We give a new perspective on the method, showing that operators can be recovered by minimizing a loss defined on a finite…
We present a proof of concept machine learning model resting on a convolutional neural network capable to yield accurate scattering s-wave phase shifts caused by different three-dimensional spherically symmetric potentials at fixed…
In this chapter we provide a theoretically founded investigation of state-of-the-art learning approaches for inverse problems from the point of view of spectral reconstruction operators. We give an extended definition of regularization…
This letter adopts long short-term memory(LSTM) to predict sea surface temperature(SST), which is the first attempt, to our knowledge, to use recurrent neural network to solve the problem of SST prediction, and to make one week and one…
This paper investigates the inverse scattering problem of time-harmonic plane waves incident on a perfectly reflecting random periodic structure. To simulate random perturbations arising from manufacturing defects and surface wear in…
We present a continuous formulation of machine learning, as a problem in the calculus of variations and differential-integral equations, in the spirit of classical numerical analysis. We demonstrate that conventional machine learning models…
Integer quantization of neural networks can be defined as the approximation of the high precision computation of the canonical neural network formulation, using reduced integer precision. It plays a significant role in the efficient…
Nonlinearity arising from mutual interactions is one of the two main difficulties to be addressed in inverse scattering. In this paper, we review and describe under a common rationale some approaches which have been introduced in literature…
This paper is concerned with the time-domain stochastic acoustic scattering problem driven by a spatially white additive Gaussian noise. The main contributions of the work are twofold. First, we prove the existence and uniqueness of the…
Training RNNs to learn long-term dependencies is difficult due to vanishing gradients. We explore an alternative solution based on explicit memorization using linear autoencoders for sequences, which allows to maximize the short-term memory…
Regularization techniques are widely employed in optimization-based approaches for solving ill-posed inverse problems in data analysis and scientific computing. These methods are based on augmenting the objective with a penalty function,…
We consider the inverse scattering problem to reconstruct a local perturbation of a given inhomogeneous periodic layer in $\mathbb{R}^d$, $d=2,3$, using near field measurements of the scattered wave on an open set of the boundary above the…