On Data-Driven Discovery Of Symbolic Differential Equations From Unsuitable Coordinates Using SINDy-Autoencoders

(from 'Data-driven discovery of coordinates and governing equations' by K. Champion et al., edited)

Abstract

"Machine Learning Methods have evolved to a powerful tool in many fields of science, including physics. While methods have become fast and sophisticated enough to learn a wide variety of tasks, they often come at the cost of interpretability. This problem can be partially overcome by making use of symbolic regression as shown in [1] and [2], but it requires data in suitable coordinates. In my work, I assess and improve the SINDY-Autoencoder proposed by K. Champion et al. [3], which is designed to learn not only symbolic equations of motion from high-dimensional data, but also the coordinates in which the equations would be most conveniently formulated. I conduct a verification and replication of the proposed method before creating and evaluating variants which eventually lead to better and more reliable results."

Results

Non-Linear Pendulum

SR-Method	Variant	Source	FVU_x [10^-4] ↓	FVU_ddx [10^-4] ↓	FVU_ddz [10^-2] ↓
SINDy	N/A	K. Champ. et al. [3]	(8)	(3)	(2)
SINDy	O1	Output [13]	(3)	(4)	(2.1)
SINDy	O1	Verified	1.1	2.0	0.4
SINDy	O2	Output [13]	(9)	(10)	(21)
SINDy	O2	Verified	3	8	13
SINDy	V {1-10}	Verified	7±6	27±22	30±40
SINDy	R {1-10}	Replicated	7±6	26±21	50±40
SINDy	PTAT {1-10}	Modified	9±13	8±6	2.1±0.9
pySR	pAE {1-10}	Modified	500±260	16k±14k	40±80

Chaotic Lorenz System

In-Distribution

SR-Method	Variant	Source	FVU_x [10^-5] ↓	FVU_dx [10^-4] ↓	FVU_dz [10^-4] ↓
SINDy	O1	K. Champ. et al. [3]	(3)	(2)	(7)
SINDy	O1	Output [13]	(2.7)	(5)	(7)
SINDy	O1	Verified	2.7	1.8	10
SINDy	O2	K. Champ. et al. [3]	(0.2)	(0.6)	(3)
SINDy	O2	Output [13]	(0.2)	(0.9)	(5)
SINDy	O2	Verified	0.2	1.1	6
SINDy	V {1-10}	Verified	5±4	11±8	45±20
SINDy	R {1-10}	Replicated	5±4	10±6	39±15
SINDy	PTAT {1-10}	Modified	2.7±1.5	7.2±2.0	36±9
pySR	pAE {1-10}	Modified	3±3	5.0k±1.9k	4.0k±1.4k

Out-Of-Distribution

SR-Method	Variant	Source	FVU_x [10^-2] ↓	FVU_dx [10^-2] ↓	FVU_dz [10^-2] ↓
SINDy	O1	K. Champ. et al. [3]	-	-	-
SINDy	O1	Output [13]	(1.3)	(11)	(8)
SINDy	O1	Verified	1	9	8
SINDy	O2	K. Champ. et al. [3]	-	-	-
SINDy	O2	Output [13]	(1.5)	(10)	(6)
SINDy	O2	Verified	2.1	14	8
SINDy	V {1-10}	Verified	1.6±0.4	16±3	24±5
SINDy	R {1-10}	Replicated	1.6±0.5	13.1±2.5	18±5
SINDy	PTAT {1-10}	Modified	1.9±0.7	15±5	22±6
pySR	pAE {1-10}	Modified	510±120	6.7k±2.1k	9k±3k

Reaction-Diffusion System

SR-Method	Variant	Source	FVU_x [10^-2] ↓	FVU_dx [10^-2] ↓	FVU_dz [10^-2] ↓
SINDy	O1	K. Champ. et al. [3]	(1.6)	(1.6)	(0.2)
SINDy	O1	Output [13]	(1.6)	(1.6)	(0.2)
SINDy	O1	Verified	1.6	1.6	0.21
SINDy	O2	K. Champ. et al. [3]	-	-	-
SINDy	O2	Output [13]	(1.6)	(1.6)	(0.8)
SINDy	O2	Verified	1.6	1.6	0.8
SINDy	V {1-10}	Verified	1.69±0.24	2.1±1.2	14±29
SINDy	R {1-10}	Replicated	1.64±0.1	1.9±0.6	14±22
SINDy	PTAT {1-10}	Modified	0.16±0.05	0.18±0.04	0.103±0.023
pySR	pAE {1-10}	Modified	0.067±0.010	0.28±0.12	0.26±0.12

Code

https://gitlab.com/psaegert/sindy-autoencoders-improvements

Citation

@misc{sindy-ae-improvements-saegert-22,
    author = {Paul Saegert},
    title = {On Data-Driven Discovery Of Symbolic Differential Equations From Unsuitable Coordinates Using SINDy-Autoencoders},
    month = may,
    year = 2022,
    publisher = {GitHub},
    school = "Heidelberg University",
    url = {https://github.com/psaegert/sindy-autoencoders-thesis}
}