Inverse source problems are central to many applications in acoustics, geophysics, non-destructive testing, and more. Traditional imaging methods suffer from the resolution limit, preventing distinction of sources separated by less than the emitted wavelength. In this work we propose a method based on physically informed neural-networks for solving the source refocusing problem, constructing a novel loss term which promotes super-resolving capabilities of the network and is based on the physics of wave propagation. We demonstrate the approach in the setup of imaging an a priori unknown number of point sources in a two-dimensional rectangular waveguide from measurements of wavefield recordings along a vertical cross section. The results show the ability of the method to approximate the locations of sources with high accuracy, even when placed close to each other.

1.
U.
Albocher
,
A. A.
Oberai
,
P. E.
Barbone
, and
I.
Harari
, “
Adjoint-weighted equation for inverse problems of incompressible plane-stress elasticity
,”
Comput. Methods Appl. Mech. Eng.
198
(
30-32
),
2412
2420
(
2009
).
2.
P. E.
Barbone
,
A. A.
Oberai
, and
I.
Harari
, “
Adjoint-weighted variational formulation for a direct computational solution of an inverse heat conduction problem
,”
Inv. Problems
23
(
6
),
2325
2342
(
2007
).
3.
D.
Colton
and
R.
Kress
, “
Inverse acoustic and electromagnetic scattering theory
,” in
Applied Mathematical Sciences
(
Springer
,
New York
,
2013
), Vol. 93.
4.
V.
Isakov
, “
Inverse problems for partial differential equations
,” in
Applied Mathematical Sciences
(
Springer
,
New York
,
1998
), Vol. 127.
5.
A.
Tarantola
,
Inverse Problem Theory and Methods for Model Parameter Estimation
(
SIAM
,
Philadelphia
,
2005
).
6.
C. R.
Vogel
,
Computational Methods for Inverse Problems
(
SIAM
,
Philadelphia
,
2002
).
7.
R. V.
Allen
, “
Automatic earthquake recognition and timing from single traces
,”
Bull. Seismol. Soc. Am.
68
(
5
),
1521
1532
(
1978
).
8.
M.
Baer
and
U.
Kradolfer
, “
An automatic phase picker for local and teleseismic events
,”
Bull. Seismol. Soc. Am.
77
(
4
),
1437
1445
(
1987
).
9.
L.
Küperkoch
,
T.
Meier
,
J.
Lee
,
W.
Friederich
, and
E. W.
Group
, “
Automated determination of p-phase arrival times at regional and local distances using higher order statistics
,”
Geophys. J. Int.
181
(
2
),
1159
1170
(
2010
).
10.
R.
Sleeman
and
T.
Van Eck
, “
Robust automatic P-phase picking: An on-line implementation in the analysis of broadband seismogram recordings
,”
Phys. Earth Planet. Int.
113
(
1-4
),
265
275
(
1999
).
11.
H.
Niu
,
E.
Reeves
, and
P.
Gerstoft
, “
Source localization in an ocean waveguide using supervised machine learning
,”
J. Acoust. Soc. Am.
142
(
3
),
1176
1188
(
2017
).
12.
H.
Niu
,
Z.
Gong
,
E.
Ozanich
,
P.
Gerstoft
,
H.
Wang
, and
Z.
Li
, “
Deep-learning source localization using multi-frequency magnitude-only data
,”
J. Acoust. Soc. Am.
146
(
1
),
211
222
(
2019
).
13.
M.
Raissi
,
P.
Perdikaris
, and
G. E.
Karniadakis
, “
Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
,”
J. Comput. Phys.
378
,
686
707
(
2019
).
14.
W.
Zhu
and
G. C.
Beroza
, “
Phasenet: A deep-neural-network-based seismic arrival-time picking method
,”
Geophys. J. Int.
216
(
1
),
261
273
(
2018
).
15.
S.
Arridge
,
P.
Maass
,
O.
Öktem
, and
C.-B.
Schönlieb
, “
Solving inverse problems using data-driven models
,”
Acta Numerica
28
,
1
174
(
2019
).
16.
Y.
Wang
and
H.
Peng
, “
Underwater acoustic source localization using generalized regression neural network
,”
J. Acoust. Soc. Am.
143
(
4
),
2321
2331
(
2018
).
17.
P.-A.
Grumiaux
,
S.
Kitić
,
L.
Girin
, and
A.
Guérin
, “
A survey of sound source localization with deep learning methods
,”
J. Acoust. Soc. Am.
152
(
1
),
107
151
(
2022
).
18.
Y.
Huang
,
X.
Wu
, and
T.
Qu
, “
Dnn-based sound source localization method with microphone array
,” in
International Conference on Information, Electronic and Communication Engineering
(
2018
), pp.
191
197
.
19.
D.
Suvorov
,
G.
Dong
, and
R.
Zhukov
, “
Deep residual network for sound source localization in the time domain
,”
J. Eng. Appl. Sci.
13
(
13
),
5096
5104
(
2018
), available at https://scholar.google.com/scholar?cluster=16621617040146518240&hl=en&as_sdt=0,40&as_vis=1.
20.
G.
Bologni
,
R.
Heusdens
, and
J.
Martinez
, “
Acoustic reflectors localization from stereo recordings using neural networks
,” in
ICASSP 2021–2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
(
IEEE
,
Piscataway, NJ
,
2021
), pp.
1
5
.
21.
M.
Li
,
L.
Demanet
, and
L.
Zepeda-Núñez
, “
Accurate and robust deep learning framework for solving wave-based inverse problems in the super-resolution regime
,” arXiv:2106.01143 (
2021
).
22.
A. G.
Baydin
,
B. A.
Pearlmutter
,
A. A.
Radul
, and
J. M.
Siskind
, “
Automatic differentiation in machine learning: A survey
,”
J. Machine Learn. Res.
18
,
1
43
(
2018
).
23.
G. E.
Karniadakis
,
I. G.
Kevrekidis
,
L.
Lu
,
P.
Perdikaris
,
S.
Wang
, and
L.
Yang
, “
Physics-informed machine learning
,”
Nat. Rev. Phys.
3
(
6
),
422
440
(
2021
).
24.
H.
Sundar
,
W.
Wang
,
M.
Sun
, and
C.
Wang
, “
Raw waveform based end-to-end deep convolutional network for spatial localization of multiple acoustic sources
,” in
ICASSP 2020–2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
(
2020
).
25.
N.
Bleistein
,
Mathematical Methods for Wave Phenomena
(
Academic Press
,
New York
,
2012
).
26.
D.
Batenkov
,
L.
Demanet
, and
H. N.
Mhaskar
, “
Stable soft extrapolation of entire functions
,”
Inv. Probl.
35
(
1
),
015011
(
2019
).
27.
M.
Bertero
and
C.
de Mol
, “
III Super-resolution by data inversion
,”
Prog. Opt.
36
,
129
178
(
1996
).
28.
G.
De Villiers
and
E. R.
Pike
,
The Limits of Resolution
(
CRC Press
,
Boca Raton, FL
,
2016
).
29.
J.
Lindberg
, “
Mathematical concepts of optical superresolution
,”
J. Opt.
14
(
8
),
083001
(
2012
).
30.
I.
Daubechies
,
M.
Defrise
, and
C. D.
Mol
, “
Sparsity-enforcing regularisation and ISTA revisited
,”
Inv. Probl.
32
(
10
),
104001
(
2016
).
31.
E. J.
Candès
and
C.
Fernandez-Granda
, “
Towards a mathematical theory of super-resolution
,”
Commun. Pure Appl. Math.
67
(
6
),
906
956
(
2014
).
32.
D.
Batenkov
,
G.
Goldman
, and
Y.
Yomdin
, “
Super-resolution of near-colliding point sources
,”
Inf. Inference: J. IMA
10
(
2
),
515
572
(
2021
).
33.
D. P.
Kingma
and
J.
Ba
, “
Adam: A method for stochastic optimization
,” arXiv:1412.6980 (
2014
).
34.
L.
Borcea
,
G.
Papanicolaou
, and
F. G.
Vasquez
, “
Edge illumination and imaging of extended reflectors
,”
SIAM J. Imaging Sci.
1
(
1
),
75
114
(
2008
).
You do not currently have access to this content.