Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysis

Seyyed Shaho Alaviani; Yongzhi Qu; Gregory W. Vogl

cs.LGSep 3, 2026

Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysis

Authors: Seyyed Shaho Alaviani, Yongzhi Qu, Gregory W. Vogl

Organizations: Department of Mechanical Engineering University of Minnesota-Twin Cities Minneapolis, MN 55455, USA · Department of Mechanical Engineering University of Utah Salt Lake City, UT 84112, USA · National Institute of Standards and Technology Gaithersburg, MD 20899, USA

Abstract

In this paper, the problem of data-driven discovery of nonlinear ordinary differential equations (ODEs) is recast, and a new interpretable machine learning (ML) method is proposed. The proposed method aims to learn the unknown vector field of nonlinear dynamics without prior knowledge of the system's physics from only one single state trajectory's data. The proposed method has two fundamental differences with existing methods: 1) the formulation presented in this method is derived based on Functional Analysis and Operator Theory, and 2) the cost function is constructed in the function space as a distance between two functions as an integral, instead of the discrete-sum of errors used in existing ML approaches. An incremental learning algorithm is proposed to learn the unknown vector field to handle new training samples in an online manner. The proposed method can discover the unknown vector field from both forced and unforced autonomous and non-autonomous (or time-varying) dynamical systems. The proposed method is able to simultaneously discover unknown external forces as a function of time and unknown underlying dynamics. Finally, numerical examples are given to demonstrate the advantages of the proposed method.

Figures & tables

Figure 1: Difference between FINS and existing methods for data-driven dynamics learning, with NODEs as an example. (Top: Cost evaluation and potential learned function of NODEs; Bottom: Cost evaluation and potential learned function of FINS).

Dynamical systems	Equations	Parameters
Duffing equation (Unforced)	$\begin{pmatrix}\dot{x}\\ \dot{y}\end{pmatrix}=\begin{pmatrix}y\\ -\delta y-\alpha x-\beta x^{3}\end{pmatrix}$	$\delta=\alpha=\beta=1$
Van der Pol	$\begin{pmatrix}\dot{x}\\ \dot{y}\end{pmatrix}=\begin{pmatrix}y\\ -\alpha x+\varepsilon_{1}y-\varepsilon_{2}x^{2}y\end{pmatrix}$	$\alpha=1,\varepsilon_{1}=\varepsilon_{2}=0.05$
Duffing equation with known force	$\begin{pmatrix}\dot{x}\\ \dot{y}\end{pmatrix}=\begin{pmatrix}y\\ -\delta y-\alpha x-\beta x^{3}\end{pmatrix}+\begin{pmatrix}0\\ 1000\sin(2t)\end{pmatrix}$	$\delta=\alpha=\beta=1$
Van der Pol with unknown force	$\begin{pmatrix}\dot{x}\\ \dot{y}\end{pmatrix}=\begin{pmatrix}y\\ -\alpha x+\varepsilon_{1}y-\varepsilon_{2}x^{2}y+\delta t^{2}\end{pmatrix}$	$\alpha=1,\varepsilon_{1}=\varepsilon_{2}=0.02,\delta=0.01$
Lorenz chaotic system	$\begin{pmatrix}\dot{x}\\ \dot{y}\\ \dot{z}\end{pmatrix}=\begin{pmatrix}10(y-x)\\ x(28-z)-y\\ xy-\frac{8}{3}z\end{pmatrix}$	$\alpha=10,\rho=28,\beta=8/3$
Mathieu-Hill’s equation	$\begin{pmatrix}\dot{x}\\ \dot{y}\end{pmatrix}=\begin{pmatrix}y\\ -(\delta+\epsilon\cos(t))x\end{pmatrix}$	$\delta=1,\epsilon=0.2$

Table 1: Dynamical systems considered in Examples 1-7

Duffing equation	EDMD	SINDy	FINS
no noise	13.72 $\pm$ 11.30	0.48 $\pm$ 0.68 (np) 0.09 $\pm$ 0.03 (sp)	0.03 $\pm$ 0.06 (np) 0.05 $\pm$ 0.09 (sp)
noisy	8.87 $\pm$ 5.34	4.88 $\pm$ 3.84 (np) 6.36 $\pm$ 5.02 (sp)	2.91 $\pm$ 2.11 (np) 5.20 $\pm$ 3.97 (sp)

Table 2: Comparison of errors of prediction time (i.e.,

[3,25]

and

[6,10]

seconds for no noise and noisy cases, respectively) for learned models of different methods for Duffing system in Example 1 under non-uniform sampling time

Parameters	$\delta$	$\alpha$	$\beta$	$\gamma$	$\eta$	constant1	constant2
Duffing equation	1	1	1	1	0	0	0
SINDy	1.0051	0.7393	1.0007	0.9995	0	-0.1005	-1.2393
KLT	0.9999	1.0002	0.9999	1.0000	0	0	0.0005
FINS	1.0004	0.9709	1.0003	1.0000	-0.0006	-0.0002	-0.0077
FINS_s	1.0001	0.9991	1.0000	1.0000	0	0	0

Table 3: Learned models by several methods in Example 1 without noise under uniform sampling time

Duffing equation	EDMD	KLT	SINDy	FINS
no noise	1.5141 $\times 10^{99}\pm$ 1.7720 $\times 10^{100}$	0.0006 $\pm$ 0.0020	0.8988 $\pm$ 0.1949	0.0113 $\pm$ 0.0326 (np) 0.0003 $\pm$ 0.0040 (sp)

Table 4: Comparison of errors of learned models of different methods for Duffing system in Example 1 under uniform sampling time

Figure 7: 3D plot of trajectories of learned models by several methods for Van der Pol equation in Example 2

Parameters	$\beta$	$\alpha$	$\varepsilon_{1}$	$\varepsilon_{2}$	const1	const2
Van der Pol	1	1	0.05	0.05	0	0
SINDy	0.9999	0.9991	0.0500	0.0499	0	0
SINDy_np	0.9995	0.9983	0.0517	0.0501	-0.0015	0.0068
FINS	1.0001	1	0.0500	0.0500	0.0001	0
FINS_s	0.9999	1	0.0500	0.0500	0	0

Table 5: Learned models by SINDy and FINS in Example 2 under non-uniform sampling time

	EDMD	SINDy	FINS
Van Der Pol	4.0798 $\pm$ 1.5080	0.0829 $\pm$ 0.0224 (np) 0.0345 $\pm$ 0.0110 (sp)	0.0044 $\pm$ 0.0010 (np) 0.0046 $\pm$ 0.0010 (sp)

Table 6: Comparison of errors of learned models of different methods in Example 2 under non-uniform sampling time

Parameters	$\beta$	$\alpha$	$\varepsilon_{1}$	$\varepsilon_{2}$
Van der Pol	1	1	0.05	0.05
SINDy	0.9994	0.9994	0.0494	0.0499
KLT	1.0000	0.9999	0.0499	0.0499
FINS	1.0000	1.0001	0.0500	0.0500
FINS_s	0.9999	1.0000	0.0499	0.0499

Table 7: Learned models by several methods in Example 2 under uniform sampling time

	EDMD	KLT	SINDy	FINS
Van der Pol	0.7817 $\pm$ 0.4096	0.0068 $\pm$ 0.0017	0.0337 $\pm$ 0.0105	0.0038 $\pm$ 0.0009 (np) 0.0067 $\pm$ 0.0016 (sp)

Table 8: Comparison of errors of learned models of different methods for Van der Pol system in Example 2 under uniform sampling time

Figure 10: Prediction error v.s. sampling time interval of learned models by SINDY_c, FINS, and FINS_s in Example 3

	FINS, N=2	FINS_s, N=2	FINS, N=3	FINS_s, N=3
Training error	0.0055 $\pm$ 0.0033	0 $\pm$ 0	5.4768 $\pm$ 4.6148	7.2563 $\pm$ 4.4988
Prediction error	0.0156 $\pm$ 0.0102	0 $\pm$ 0	8.9985 $\pm$ 4.6787	10.3255 $\pm$ 4.5012

Table 9: Effect of the selection of polynomial order on training and prediction error for Example 4

Monomials	Lorenz system	SINDy	FINS	FINS_s
1	0	0	-0.0005	0
x	-10	-9.9986	-9.9999	-9.9999
y	10	9.9991	10	9.9999
z	0	0	0	0
xy	0	0	0	0
xz	0	0	0	0

Table 10: Coefficients of the monomials of

\dot{x}(t)

of learned models by SINDy, FINS, and FINS_s in Example 5 under non-uniform sampling time

Monomials	Lorenz system	SINDy	FINS	FINS_s
1	0	0	-0.0005	0
x	28	27.9913	28.0002	28.0013
y	-1	-0.9993	-0.9999	-1.0003
z	0	0	0	0
xy	0	0	0	0
xz	-1	-0.9996	-1	-1

Table 11: Coefficients of monomials of

\dot{y}(t)

of learned models by SINDy, FINS, and FINS_s in Example 5 under non-uniform sampling time

Monomials	Lorenz system	SINDy	FINS	FINS_s
1	0	0	-0.0010	0
x	0	0	0.0002	0
y	0	0	-0.0001	0
z	$-\frac{8}{3}$	-2.6667	-2.6666	-2.6666
xy	1	0.9999	1	1
xz	0	0	0	0

Table 12: Coefficients of monomials of

\dot{z}(t)

of learned models by SINDy, FINS, and FINS_s in Example 5 under non-unifrom sampling time

Lorenz System	SINDy	FINS	FINS_s
Interpolation	0.0101 $\pm$ 0.0054	0.0016 $\pm$ 0.0010	0.0010 $\pm$ 0.0006
Extrapolation1	0.038 $\pm$ 0.016	0.008 $\pm$ 0.004	0.003 $\pm$ 0.002
Extrapolation2	0.13 $\pm$ 0.08	0.04 $\pm$ 0.03	0.02 $\pm$ 0.02

Table 13: Comparison of errors of learned models by SINDy, FINS, and FINS_s in Example 5 under non-uniform sampling time

Lorenz System	SINDy	FINS	FINS_s
k = 26k, N = 1	N/A	0.0375 $\pm$ 0.0063	0.0403 $\pm$ 0.0103
k = 260k, N = 1	N/A	0.3527 $\pm$ 0.0117	0.3500 $\pm$ 0.0093
k = 2.6M, N = 1	N/A	3.5580 $\pm$ 0.0240	3.5850 $\pm$ 0.0157
k = 26k, N = 2	0.0139 $\pm$ 0.0083	0.0787 $\pm$ 0.0130	0.0805 $\pm$ 0.0113
k = 260k, N = 2	0.1771 $\pm$ 0.0115	0.7625 $\pm$ 0.0108	0.7585 $\pm$ 0.0099
k = 2.6M, N = 2	1.9897 $\pm$ 0.0221	7.6285 $\pm$ 0.0414	7.6977 $\pm$ 0.0492

Table 14: Comparison of computational time (seconds) (for 25 independent runs) for discovering Lorenz system by SINDy, FINS, and FINS_s ( if the cardinality of

\mathcal{D}_{t}

is fixed for the training time ) where N is highest degree of monomials in SINDy and of Hermite polynomials here

Lorenz System	FINS	FINS_s
k = 26k, N = 1	9.1228 $\times 10^{-6}$ $\pm$ 3.2566 $\times 10^{-5}$	5.7904 $\times 10^{-5}$ $\pm$ 2.0895 $\times 10^{-4}$
k = 26k, N = 2	6.8872 $\times 10^{-4}$ $\pm$ 9.7060 $\times 10^{-4}$	0.0015 $\pm$ 0.0021
k = 26k, N = 3	7.7463 $\times 10^{-4}$ $\pm$ 8.1156 $\times 10^{-4}$	0.0024 $\pm$ 0.0028
k = 26k, N = 4	9.8189 $\times 10^{-4}$ $\pm$ 8.3949 $\times 10^{-4}$	0.0028 $\pm$ 0.0025

Table 15: Comparison of computational time (seconds) for discovering Lorenz system for incremental learning (i.e., update time per new datum) of FINS and FINS_s where N is highest degree of Hermite polynomials

	Duffing n=2	Lorenz n=3	Lorenz 96 n=4	Method
k = 26k, N = 2	0.0114 $\pm$ 0.0036	0.0139 $\pm$ 0.0083	0.0209 $\pm$ 0.0048	SINDy
k = 260k, N = 2	0.0896 $\pm$ 0.0042	0.1771 $\pm$ 0.0115	0.2457 $\pm$ 0.0111	SINDy
k = 2.6M, N = 2	0.8026 $\pm$ 0.0058	1.9897 $\pm$ 0.0221	3.6259 $\pm$ 0.0314	SINDy
k = 26k, N = 2	0.0489 $\pm$ 0.0058	0.0787 $\pm$ 0.0130	0.3802 $\pm$ 0.0394	FINS
k = 260k, N = 2	0.4639 $\pm$ 0.0108	0.7625 $\pm$ 0.0108	3.8330 $\pm$ 0.0402	FINS
k = 2.6M, N = 2	4.7110 $\pm$ 0.0459	7.6285 $\pm$ 0.0414	37.8838 $\pm$ 0.2629	FINS

Table 16: Evaluation of computational time (seconds) ( if the cardinality of

\mathcal{D}_{t}

is fixed for the training time ) w.r.t. the dimension of the states, i.e., n , (for 25 independent runs) where N is highest degrees of monomials in SINDy and of Hermite polynomials

	Duffing n=2	Lorenz n=3	Lorenz 96 n=4
FINS	5.8375 $\times 10^{-4}$ $\pm$ 7.8622 $\times 10^{-4}$	6.8872 $\times 10^{-4}$ $\pm$ 9.7060 $\times 10^{-4}$	5.1372 $\times 10^{-4}$ $\pm$ 4.4352 $\times 10^{-4}$
FINS_s	0.0014 $\pm$ 0.0020	0.0015 $\pm$ 0.0021	0.0081 $\pm$ 0.0053

Table 17: Evaluation of computational time (seconds) for incremental learning (i.e., update time per new datum) w.r.t. the dimension of the states, i.e., n , where

k

= 26k and N=2

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