13 research outputs found
Solving the Wide-band Inverse Scattering Problem via Equivariant Neural Networks
This paper introduces a novel deep neural network architecture for solving
the inverse scattering problem in frequency domain with wide-band data, by
directly approximating the inverse map, thus avoiding the expensive
optimization loop of classical methods. The architecture is motivated by the
filtered back-projection formula in the full aperture regime and with
homogeneous background, and it leverages the underlying equivariance of the
problem and compressibility of the integral operator. This drastically reduces
the number of training parameters, and therefore the computational and sample
complexity of the method. In particular, we obtain an architecture whose number
of parameters scale sub-linearly with respect to the dimension of the inputs,
while its inference complexity scales super-linearly but with very small
constants. We provide several numerical tests that show that the current
approach results in better reconstruction than optimization-based techniques
such as full-waveform inversion, but at a fraction of the cost while being
competitive with state-of-the-art machine learning methods.Comment: 21 pages, 9 figures, and 4 table
Solving Inverse Obstacle Scattering Problem with Latent Surface Representations
We propose a novel iterative numerical method to solve the three-dimensional
inverse obstacle scattering problem of recovering the shape of the obstacle
from far-field measurements. To address the inherent ill-posed nature of the
inverse problem, we advocate the use of a trained latent representation of
surfaces as the generative prior. This prior enjoys excellent expressivity
within the given class of shapes, and meanwhile, the latent dimensionality is
low, which greatly facilitates the computation. Thus, the admissible manifold
of surfaces is realistic and the resulting optimization problem is less
ill-posed. We employ the shape derivative to evolve the latent surface
representation, by minimizing the loss, and we provide a local convergence
analysis of a gradient descent type algorithm to a stationary point of the
loss. We present several numerical examples, including also backscattered and
phaseless data, to showcase the effectiveness of the proposed algorithm
A Neural Network Warm-Start Approach for the Inverse Acoustic Obstacle Scattering Problem
We consider the inverse acoustic obstacle problem for sound-soft star-shaped
obstacles in two dimensions wherein the boundary of the obstacle is determined
from measurements of the scattered field at a collection of receivers outside
the object. One of the standard approaches for solving this problem is to
reformulate it as an optimization problem: finding the boundary of the domain
that minimizes the distance between computed values of the scattered
field and the given measurement data. The optimization problem is
computationally challenging since the local set of convexity shrinks with
increasing frequency and results in an increasing number of local minima in the
vicinity of the true solution. In many practical experimental settings, low
frequency measurements are unavailable due to limitations of the experimental
setup or the sensors used for measurement. Thus, obtaining a good initial guess
for the optimization problem plays a vital role in this environment.
We present a neural network warm-start approach for solving the inverse
scattering problem, where an initial guess for the optimization problem is
obtained using a trained neural network. We demonstrate the effectiveness of
our method with several numerical examples. For high frequency problems, this
approach outperforms traditional iterative methods such as Gauss-Newton
initialized without any prior (i.e., initialized using a unit circle), or
initialized using the solution of a direct method such as the linear sampling
method. The algorithm remains robust to noise in the scattered field
measurements and also converges to the true solution for limited aperture data.
However, the number of training samples required to train the neural network
scales exponentially in frequency and the complexity of the obstacles
considered. We conclude with a discussion of this phenomenon and potential
directions for future research
Reinforced Inverse Scattering
Inverse wave scattering aims at determining the properties of an object using
data on how the object scatters incoming waves. In order to collect
information, sensors are put in different locations to send and receive waves
from each other. The choice of sensor positions and incident wave frequencies
determines the reconstruction quality of scatterer properties. This paper
introduces reinforcement learning to develop precision imaging that decides
sensor positions and wave frequencies adaptive to different scatterers in an
intelligent way, thus obtaining a significant improvement in reconstruction
quality with limited imaging resources. Extensive numerical results will be
provided to demonstrate the superiority of the proposed method over existing
methods
Conditional Injective Flows for Bayesian Imaging
Most deep learning models for computational imaging regress a single
reconstructed image. In practice, however, ill-posedness, nonlinearity, model
mismatch, and noise often conspire to make such point estimates misleading or
insufficient. The Bayesian approach models images and (noisy) measurements as
jointly distributed random vectors and aims to approximate the posterior
distribution of unknowns. Recent variational inference methods based on
conditional normalizing flows are a promising alternative to traditional MCMC
methods, but they come with drawbacks: excessive memory and compute demands for
moderate to high resolution images and underwhelming performance on hard
nonlinear problems. In this work, we propose C-Trumpets -- conditional
injective flows specifically designed for imaging problems, which greatly
diminish these challenges. Injectivity reduces memory footprint and training
time while low-dimensional latent space together with architectural innovations
like fixed-volume-change layers and skip-connection revnet layers, C-Trumpets
outperform regular conditional flow models on a variety of imaging and image
restoration tasks, including limited-view CT and nonlinear inverse scattering,
with a lower compute and memory budget. C-Trumpets enable fast approximation of
point estimates like MMSE or MAP as well as physically-meaningful uncertainty
quantification.Comment: 23 pages, 23 figure
An overview on deep learning-based approximation methods for partial differential equations
It is one of the most challenging problems in applied mathematics to
approximatively solve high-dimensional partial differential equations (PDEs).
Recently, several deep learning-based approximation algorithms for attacking
this problem have been proposed and tested numerically on a number of examples
of high-dimensional PDEs. This has given rise to a lively field of research in
which deep learning-based methods and related Monte Carlo methods are applied
to the approximation of high-dimensional PDEs. In this article we offer an
introduction to this field of research, we review some of the main ideas of
deep learning-based approximation methods for PDEs, we revisit one of the
central mathematical results for deep neural network approximations for PDEs,
and we provide an overview of the recent literature in this area of research.Comment: 23 page
IAE-Net: Integral Autoencoders for Discretization-Invariant Learning
Discretization invariant learning aims at learning in the
infinite-dimensional function spaces with the capacity to process heterogeneous
discrete representations of functions as inputs and/or outputs of a learning
model. This paper proposes a novel deep learning framework based on integral
autoencoders (IAE-Net) for discretization invariant learning. The basic
building block of IAE-Net consists of an encoder and a decoder as integral
transforms with data-driven kernels, and a fully connected neural network
between the encoder and decoder. This basic building block is applied in
parallel in a wide multi-channel structure, which are repeatedly composed to
form a deep and densely connected neural network with skip connections as
IAE-Net. IAE-Net is trained with randomized data augmentation that generates
training data with heterogeneous structures to facilitate the performance of
discretization invariant learning. The proposed IAE-Net is tested with various
applications in predictive data science, solving forward and inverse problems
in scientific computing, and signal/image processing. Compared with
alternatives in the literature, IAE-Net achieves state-of-the-art performance
in existing applications and creates a wide range of new applications