This study presents a general framework, namely, Sparse Spatiotemporal System Discovery ( S 3 d), for discovering dynamical models given by Partial Differential Equations (PDEs) from spatiotemporal data. S 3 d is built on the recent development of sparse Bayesian learning, which enforces sparsity in the estimated PDEs. This approach enables a balance between model complexity and fitting error with theoretical guarantees. The proposed framework integrates Bayesian inference and a sparse priori distribution with the sparse regression method. It also introduces a principled iterative re-weighted algorithm to select dominant features in PDEs and solve for the sparse coefficients. We have demonstrated the discovery of the complex Ginzburg–Landau equation from a traveling-wave convection experiment, as well as several other PDEs, including the important cases of Navier–Stokes and sine-Gordon equations, from simulated data.

1
J. D.
Farmer
and
J. J.
Sidorowich
,
Phys. Rev. Lett.
59
,
845
(
1987
).
2
W.-X.
Wang
,
R.
Yang
,
Y.-C.
Lai
,
V.
Kovanis
, and
C.
Grebogi
,
Phys. Rev. Lett.
106
,
154101
(
2011
).
3
B.
Dorneanu
,
S.
Zhang
,
H.
Ruan
,
M.
Heshmat
,
R.
Chen
,
V. S.
Vassiliadis
, and
H.
Arellano-Garcia
,
Front. Eng. Manag.
9
,
623
(
2022
).
4
I. G.
Kevrekidis
,
C. W.
Gear
,
J. M.
Hyman
,
P. G.
Kevrekidid
,
O.
Runborg
, and
C.
Theodoropoulos
,
Commun. Math. Sci.
1
,
715
(
2003
).
5
J.
Bongard
and
H.
Lipson
,
Proc. Natl. Acad. Sci.
104
,
9943
(
2007
).
6
Z.
Shen
,
W.-X.
Wang
,
Y.
Fan
,
Z.
Di
, and
Y.-C.
Lai
,
Nat. Commun.
5
,
4323
(
2014
).
7
I.
Mezić
,
Annu. Rev. Fluid Mech.
45
,
357
(
2013
).
8
M. O.
Williams
,
I. G.
Kevrekidis
, and
C. W.
Rowley
,
J. Nonlinear Sci.
25
,
1307
(
2015
).
9
J. H.
Tu
,
C. W.
Rowley
,
D. M.
Luchtenburg
,
S. L.
Brunton
, and
J. N.
Kutz
,
J. Comput. Dyn.
1
,
391
(
2014
).
10
D.
Giannakis
,
Appl. Comput. Harmon.
(
2019
).
11
L.
Ljung
,
System Identification: Theory for the User
(
Prentice-Hall, Inc.
,
Upper Saddle River, NJ
,
1986
). ISBN 0-138-81640-9.
12
A.
Lindquist
and
G.
Picci
,
Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification
(
Springer
,
2015
), Vol. 1.
13
D.
Xu
and
O.
Khanmohamadi
,
Chaos
18
,
043122
(
2008
).
14
O.
Khanmohamadi
and
D.
Xu
,
Chaos
19
,
033117
(
2009
).
15
M.
Bär
,
R.
Hegger
, and
H.
Kantz
,
Phys. Rev. E
59
,
337
(
1999
).
16
L.
Guo
and
S. A.
Billings
,
IEEE Trans. Circuits Syst. II: Express Briefs
53
,
657
(
2006
).
17
T.
Müller
and
J.
Timmer
,
Int. J. Bifurcation Chaos
14
,
2053
(
2004
).
18
L.
Breiman
and
J. H.
Friedman
,
J. Am. Stat. Assoc.
80
,
580
(
1985
).
19
H. U.
Voss
,
P.
Kolodner
,
M.
Abel
, and
J.
Kurths
,
Phys. Rev. Lett.
83
,
3422
(
1999
).
20
H.
Voss
,
M.
Bünner
, and
M.
Abel
,
Phys. Rev. E
57
,
2820
(
1998
).
21
R.
Tibshirani
,
M.
Wainwright
, and
T.
Hastie
,
Statistical Learning with Sparsity: The Lasso and Generalizations
(
Chapman and Hall/CRC
,
2015
).
22
W.
Pan
,
Y.
Yuan
,
J.
Gonçalves
, and
G.-B.
Stan
, in 2012 IEEE 51st Annual Conference on Decision and Control (CDC) (IEEE, 2012), pp. 2334–2339.
23
S. L.
Brunton
,
J. L.
Proctor
, and
J. N.
Kutz
,
Proc. Natl. Acad. Sci.
113
,
312
(
2016
).
24
S. H.
Rudy
,
S. L.
Brunton
,
J. L.
Proctor
, and
J. N.
Kutz
,
Sci. Adv.
3
,
e1602614
(
2017
).
25
H.
Schaeffer
,
Proc. R. Soc. A
473
,
20160446
(
2017
).
26
M. E.
Tipping
,
J. Mach. Learn. Res.
1
,
211
(
2001
).
27
A. C.
Faul
and
M. E.
Tipping
, in Advances in Neural Information Processing Systems (The MIT Press, 2002), pp. 383–389.
28
B. D.
Rao
,
K.
Engan
,
S. F.
Cotter
,
J.
Palmer
, and
K.
Kreutz-Delgado
,
IEEE Trans. Signal Process.
51
,
760
(
2003
).
29
J.
Palmer
,
K.
Kreutz-Delgado
,
B. D.
Rao
, and
D. P.
Wipf
, in Advances in Neural Information Processing Systems (The MIT Press, 2006), pp. 1059–1066.
30
M.
Figueiredo
, in Advances in Neural Information Processing Systems (The MIT Press, 2002), pp. 697–704.
31
D. P.
Wipf
and
B. D.
Rao
,
IEEE Trans. Signal Process.
52
,
2153
(
2004
).
32
D. P.
Wipf
,
B. D.
Rao
, and
S.
Nagarajan
,
IEEE Trans. Inf. Theory
57
,
6236
(
2011
).
33
M.
Raissi
,
P.
Perdikaris
, and
G. E.
Karniadakis
,
J. Comput. Phys.
378
,
686
(
2019
).
34
P.
Kolodner
,
Phys. Rev. Lett.
69
,
2519
(
1992
).
35
P.
Kolodner
,
S.
Slimani
,
N.
Aubry
, and
R.
Lima
,
Physica D
85
,
165
(
1995
).
36
C.
Shu
and
B. E.
Richards
,
Int. J. Numer. Methods Fluids
15
,
791
(
1992
).
38
E.
Schrödinger
,
Phys. Rev.
28
,
1049
(
1926
).
39
R. J.
LeVeque
,
Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems
(
SIAM
,
2007
), Vol. 98.
40
J.
Li
and
Y.-T.
Chen
,
Computational Partial Differential Equations Using MATLAB
(
Chapman and Hall/CRC
,
2008
).
41
L. N.
Trefethen
,
Spectral Methods in MATLAB
(
SIAM
,
2000
), Vol. 10.
42
A. G.
Baydin
,
B. A.
Pearlmutter
,
A. A.
Radul
, and
J. M.
Siskind
,
J. Mach. Learn. Res.
18
,
5595
(
2017
).
43
D. P.
Wipf
and
S. S.
Nagarajan
, in Advances in Neural Information Processing Systems (The MIT Press, 2008), pp. 1625–1632.
44
M. C.
Cross
and
P. C.
Hohenberg
,
Rev. Mod. Phys.
65
,
851
(
1993
).
45
M. C.
Cross
,
Phys. Rev. A
38
,
3593
(
1988
).
46
A. C.
Newell
, in Lectures in Applied Mathematics (American Mathematical Society, Providence, RI, 1974), Vol. 15, p. 237.
47
P.
Kolodner
,
J. A.
Glazier
, and
H.
Williams
,
Phys. Rev. Lett.
65
,
1579
(
1990
).
48
P.
Kolodner
,
Phys. Rev. A
44
,
6466
(
1991
).
49
P.
Kolodner
and
H.
Williams
, in Proceedings of the NATO Advanced Research Workshop on Nonlinear Evolution of Spatio-temporal Structures in Dissipative Continuous Systems (
Springer Science & Business Media
,
1990
).
50
G.
Strang
,
Computational Science and Engineering
(
Wellesley-Cambridge Press, Wellesley
,
2007
), Vol. 791.
51
S.
Linz
and
M.
Lücke
,
Phys. Rev. A
35
,
3997
(
1987
).
52
W.
Schöpf
and
W.
Zimmermann
,
EPL (Europhys. Lett.)
8
,
41
(
1989
).
53
W.
Schöpf
and
L.
Kramer
,
Phys. Rev. Lett.
66
,
2316
(
1991
).
54
G. E.
Uhlenbeck
and
L. S.
Ornstein
,
Phys. Rev.
36
,
823
(
1930
).
55
D. J.
Higham
,
SIAM Rev.
43
,
525
(
2001
).
56
G.
Strang
,
Computational Science and Engineering
(
Wellesley-Cambridge Press, Wellesley
,
2007
), Vol. 791.
57
G.
Berkooz
,
P.
Holmes
, and
J. L.
Lumley
,
Annu. Rev. Fluid Mech.
25
,
539
(
1993
).
58
L.
Sirovich
,
Q. Appl. Math.
45
,
561
(
1987
).
59
M.
Schmidt
and
H.
Lipson
,
Science
324
,
81
(
2009
).
60
Y.
Yuan
, et al. (
2018
)
Matlab code for 'Machine discovery of partial differential equations from spatiotemporal data
Github.
https://github.com/HAIRLAB/S3d
You do not currently have access to this content.