Using the SEKF to Transfer NN Models of Dynamical Systems with Limited Data

Joshua E. Hammond; Tyler A. Soderstrom; Brian A. Korgel; Michael Baldea

cs.LGMar 2, 2026

Using the SEKF to Transfer NN Models of Dynamical Systems with Limited Data

Authors: Joshua E. Hammond, Tyler A. Soderstrom, Brian A. Korgel, Michael Baldea

Organizations: McKetta Department of Chemical Engineering, The University of Texas at Austin, 200 East Dean Keeton St., Stop C0400, Austin, 78712, TX, United States · Energy Institute, The University of Texas at Austin, 2304 Whitis Ave., Stop C2400, Austin, 78712, TX, United States · Texas Materials Institute, The University of Texas at Austin, 204 E. Dean Keeton St., Stop C2201, Austin, 78712, TX, United States · ExxonMobil Technology and Engineering, Spring, TX, United States ·

Abstract

Data-driven models of dynamical systems require extensive amounts of training data. For many practical applications, gathering sufficient data is not feasible due to cost or safety concerns. This work uses the Subset Extended Kalman Filter (SEKF) to adapt pre-trained neural network models to new, similar systems with limited data available. Experimental validation across damped spring and continuous stirred-tank reactor systems demonstrates that small parameter perturbations to the initial model capture target system dynamics while requiring as little as 1% of original training data. In addition, finetuning requires less computational cost and reduces generalization error.

Figures & tables

	Source System	Target System
System Space	$\frac{d\bm{x}}{dt}=f^{\mathcal{S}}(\bm{x},\bm{u},p^{\mathcal{S}})$	$\frac{d\bm{x}}{dt}=f^{\mathcal{T}}(\bm{x},\bm{u},p^{\mathcal{T}})$
Data Space	$\mathcal{D}^{\mathcal{S}}$	$\mathcal{D}^{\mathcal{T}}$	$\dim\mathcal{D}^{\mathcal{S}}\gg\dim\mathcal{D}^{\mathcal{T}}$
Model Space	$\tilde{\bm{x}}_{k+1\ldots k+n_{out}}=\phi^{\mathcal{S}}(\bm{x}_{k},\bm{u}_{k\ldots k+n_{out}-1},\bm{\pi}^{\mathcal{S}})$	$\tilde{\bm{x}}_{k+1\ldots k+n_{out}}=\phi^{\mathcal{T}}(\bm{x}_{k},\bm{u}_{k\ldots k+n_{out}-1},\bm{\pi}^{\mathcal{T}})$	$\phi^{\mathcal{S}}$ and $\phi^{\mathcal{T}}$ have identical structures

Table 1: Summary of source and target dynamical systems used for transfer learning experiments.

Figure 1: Example trajectory of a damped spring-mass system (solid line) given initial position (blue triangle) and velocity (black arrow). A neural network trained on noisy measurements (blue

+

) makes predictions (orange

\times

) of future positions given the initial position and velocity. A target system with slightly different damping coefficient (dashed line) illustrates the transfer learning objective.

Metric	Purpose
Mean Squared Error (MSE)	Quantify model prediction error
Train-Test Gap	Measure overfitting and generalization
Normalized MSE	Compare target model accuracy to source model
Normalized Convergence Time	Compare training efficiency to source model
Wasserstein Distance	Quantify similarity of source and target data distributions
Cosine Similarity (CS)	Quantify similarity of source and target functional spaces and models

Table 2: Metrics to quantify similarity between source and target systems.

Manipulated
Parameter Initialization:	Finetune, Retrain
Optimization Methods:	SEKF, Adam, LBFGS
Dataset Sizes:	10, 50, 100, 500, 1,000 (Damped Spring)
	0.5 hr, 1 hr, 4 hr, 12 hr, 24 hr (10-second sampling frequency) (TCLab)
Transfer:	$m$ , $k$ , $c$ , $u$ $\pm 10\%$ (Damped Spring)
	Simulated $\rightarrow$ Real (TCLab)

Table 3: Summary of TL experiments

Figure 2: Normalized test loss versus target data size for finetuning and retraining initialization methods. (a) Damped spring system. (b) TCLab system. Error bars indicate standard error across replicates. The dashed line indicates source model performance. Finetuning achieves lower test loss, especially with limited data.

Figure 3: Train-test gap (test loss minus training loss) versus target data size, grouped by initialization method. Smaller gaps indicate better generalization. Finetuning exhibits consistently smaller train-test gaps than retraining across all data sizes in both systems. (a) Damped spring system. (b) TCLab system.

Figure 4: Distribution of cosine similarity between adapted model parameters and source model parameters. Dashed vertical lines indicate mean cosine similarity for each initialization method, with horizontal error bars showing

\pm 1

standard deviation. (a) Damped spring system. (b) TCLab system. Finetuned models cluster tightly near 1.0, indicating that successful adaptation requires only small perturbations from the source model. Retrained models exhibit much wider spread and lower similarity, occupying different regions of parameter space.

System	Method	Optimizer	Cosine Similarity
Damped Spring	Finetune	Adam	$0.9959\pm 0.0113$
		L-BFGS	$0.9984\pm 0.0156$
		SEKF	$0.9977\pm 0.0238$
	Retrain	Adam	$0.6519\pm 0.1205$
		L-BFGS	$0.6819\pm 0.1210$
		SEKF	$0.6047\pm 0.2385$

Table 4: Average cosine similarity between source and target parameters, broken down by system, initialization method, and optimizer.

Figure 5: Changes in model weights during transfer learning for the damped spring system when $c^{\mathcal{T}}=0.9c^{\mathcal{S}}$ and 1,000 target examples. Each subplot shows the distribution of weight changes in a specific layer of the neural network. Contrary to common heuristics in image classification, weight changes are distributed across all layers rather than concentrated in later layers. This suggests that transfer learning for dynamical systems requires holistic parameter adjustments.

Figure 6: Normalized convergence time by optimizer and initialization method. Left: Damped spring system. Right: TCLab system. SEKF exhibits the longest convergence times, especially when retraining. Finetuning reduces convergence time across all optimizers.

		Damped Spring		TCLab
Outcome	Factor	$F$	$p$	$F$	$p$
Test Loss	Data Size	8.29	$2\times 10^{-4}$	24.88	$2\times 10^{-4}$
	Init Method	6.58	$2\times 10^{-4}$	0.75	0.389
	Optimizer	1.05	0.394	6.46	0.002
Train-Test Gap	Data Size	6.02	$2\times 10^{-4}$	67.43	$2\times 10^{-4}$
	Init Method	4.73	$2\times 10^{-4}$	7.87	0.007

Table 5: Summary of Permutation ANOVA results (

F

-statistics and

p

-values) for main effects across both benchmark systems. Significant effects (

p<0.05

) are shown in bold.

Appendix figures & tables19 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter	Value	Units
Mass, $m$	1.0	kg
Damping Coefficient, $c$	0.5	Ns/m
Spring Constant, $k$	1.0	N/m
Force, $u$	0	N
Initial Position $x_{0}$	Uniform (-5, 5)	m
Initial Velocity $\dot{x}$	Uniform (-5, 5)	m/s

Appendix

Table 6: Source Damped Spring-Mass System Parameters

Hyperparameter	Values
Learning Rate	$10^{-6}\textnormal{--}10^{-1}$
Batch Size	16,32,64,128
LR Patience	10, 20, 50, 100
LR Factor	0.1 – 0.9
Maximum Epochs	1000
Num. Samples	100

Appendix

Table 7: Source Damped Spring-Mass System NN Hyperparameters

2	Parameter Initialization Methods: Finetune, Retrain
3	Optimization Methods: SEKF, Adam, LBFGS
8	Target Systems: $\pm 10\%$ change in $m$ , $c$ , $k$ , $u$
5	Training Dataset Sizes: 10, 50, 100, 500, 1,000
10	Replicates per Target System and Dataset Size
2,400	Total Trials

Appendix

Table 8: Summary of all Damped Spring-Mass Trials

All	Mini-batches per Epoch	50
	Max. Epochs	100
Adam	Learning Rate	$10^{-6}\textnormal{--}10^{-1}$
	Mini-batch Size	$1$ – $8,000$
	LR Patience	10
	LR Factor	0.1 – 0.9
LBFGS	Learning Rate	$10^{-6}\textnormal{--}1$

Appendix

Table 9: Transfer Learning Hyperparameters

Figure 7: The Temperature Control Lab (TCLab) hardware platform. Image courtesy of APMonitor.com.

Parameter	Value	Units
Mass, $m$	0.004	kg
Heat Capacity, $c_{p}$	500	J/(kg $\cdot$ K)
Heat Transfer Coefficient, $U$	10	W/(m ${}^{2}\cdot$ K)
Exposed Surface Area, $A$	$1.0\times 10^{-3}$	m 2
Inter-heater Surface Area, $A_{s}$	$2.0\times 10^{-4}$	m 2
Emissivity, $\epsilon$	0.9	–

Appendix

Table 10: TCLab Model Parameters

2	Parameter Initialization Methods: Finetune, Retrain
3	Optimization Methods: SEKF, Adam, LBFGS
5	Training Dataset Sizes: 0.5 hours, 1 hour, 4 hours, 12 hours, 24 hours
5	Replicates
150	Total Trials

Appendix

Table 11: Summary of all TCLab Trials

Figure 8: Example prediction trajectories for the damped spring system at the 90th, 50th, and 10th percentiles of test loss for the source model. Target system has

c^{\mathcal{T}}=0.45

(10% reduction from source). Finetuned models (middle row) maintain accuracy comparable to the source model, while retrained models (bottom row) exhibit larger prediction errors, especially in worst-case scenarios.

Figure 9: Example prediction trajectories for the TCLab system at the 90th, 50th, and 10th percentiles of test loss for the source model. Transfer learning adapts a simulation-trained model to physical hardware using 4 hours of experimental data. Finetuned models (middle row) accurately predict both heater temperatures, while retrained models (bottom row) show degraded performance, particularly for the second heater.

Figure 10: Complete scatter plots of normalized test loss for all individual trials across both experimental systems. Each point represents a single trial, colored by initialization method (blue: finetune, orange: retrain). Panels are organized by target data size.

Figure 11: Normalized test loss broken down by target system for the damped spring experiments. Each of the 8 target systems (varying

m

c

k

u

\pm 10\%

) is shown separately to illustrate consistency of transfer learning benefits across different parameter perturbations.

Figure 12: Complete scatter plots of train-test gap (difference between training and test loss) for all trials. Lower values indicate better generalization. Finetuning consistently shows smaller gaps, particularly at low data sizes.

Figure 13: Normalized convergence time broken down by target data size for each optimizer. This visualization reveals how optimizer efficiency scales with data availability and whether the relative performance of optimizers changes across data regimes.

Effect	Coef.	Std.Err.	$z$	$p$
Intercept	1.120	0.125	8.93	$4.11\times 10^{-19}$
L-BFGS	0.352	0.177	1.99	$4.70\times 10^{-2}$
Retrain	1.887	0.177	10.64	$1.88\times 10^{-26}$
50 Samples	$-$ 0.844	0.177	$-$ 4.76	$1.94\times 10^{-6}$
100 Samples	$-$ 1.005	0.177	$-$ 5.67	$1.47\times 10^{-8}$
500 Samples	$-$ 1.093	0.177	$-$ 6.17	$7.00\times 10^{-10}$

Appendix

Table 12: GLM (Gamma) significant coefficients for normalized test loss—Damped Spring. Reference levels: Adam optimizer, Finetune initialization, 10 training samples.

Effect	Coef.	Std.Err.	$z$	$p$
Intercept	$-$ 2.929	0.122	$-$ 23.99	$3.85\times 10^{-127}$
Retrain	1.077	0.173	6.24	$4.50\times 10^{-10}$
50 Samples	$-$ 0.366	0.173	$-$ 2.12	$3.40\times 10^{-2}$
100 Samples	$-$ 0.431	0.173	$-$ 2.49	$1.27\times 10^{-2}$
500 Samples	$-$ 0.475	0.173	$-$ 2.75	$6.00\times 10^{-3}$
1000 Samples	$-$ 0.481	0.173	$-$ 2.78	$5.40\times 10^{-3}$

Appendix

Table 13: GLM (Gamma) significant coefficients for train-test gap—Damped Spring. Reference levels: Adam optimizer, Finetune initialization, 10 training samples.

Effect	Coef.	Std.Err.	$z$	$p$
Intercept	$-$ 5.161	0.209	$-$ 24.67	$2.33\times 10^{-134}$
SEKF	2.138	0.296	7.23	$5.01\times 10^{-13}$
Retrain	$-$ 1.036	0.296	$-$ 3.50	$4.65\times 10^{-4}$
SEKF $\times$ Retrain	3.418	0.442	7.74	$9.99\times 10^{-15}$
L-BFGS $\times$ 50 Samples	2.036	0.418	4.87	$1.14\times 10^{-6}$
SEKF $\times$ 50 Samples	1.543	0.418	3.69	$2.26\times 10^{-4}$

Appendix

Table 14: GLM (Gamma) significant coefficients for normalized convergence time—Damped Spring. Reference levels: Adam optimizer, Finetune initialization, 10 training samples.

Effect	Coef.	Std.Err.	$z$	$p$
Intercept	1.554	0.131	11.89	$1.32\times 10^{-32}$
Retrain	0.384	0.185	2.08	$3.79\times 10^{-2}$
12 hr	$-$ 1.076	0.185	$-$ 5.82	$5.82\times 10^{-9}$
24 hr	$-$ 1.205	0.185	$-$ 6.52	$7.00\times 10^{-11}$
4 hr	$-$ 0.792	0.185	$-$ 4.28	$1.84\times 10^{-5}$
SEKF $\times$ 12 hr	1.081	0.261	4.13	$3.57\times 10^{-5}$

Appendix

Table 15: GLM (Gamma) significant coefficients for normalized test loss—TCLab. Reference levels: Adam optimizer, Finetune initialization, 0.5 hr training data.

Effect	Coef.	Std.Err.	$z$	$p$
Intercept	1.210	0.091	13.35	$1.19\times 10^{-40}$
Retrain	0.288	0.128	2.25	$2.45\times 10^{-2}$
12 hr	$-$ 0.657	0.128	$-$ 5.13	$2.95\times 10^{-7}$
24 hr	$-$ 0.738	0.128	$-$ 5.76	$8.62\times 10^{-9}$
4 hr	$-$ 0.541	0.128	$-$ 4.22	$2.42\times 10^{-5}$
SEKF $\times$ 12 hr	$-$ 0.388	0.181	$-$ 2.14	$3.24\times 10^{-2}$

Appendix

Table 16: GLM (Gamma) significant coefficients for train-test gap—TCLab. Reference levels: Adam optimizer, Finetune initialization, 0.5 hr training data.

Effect	Coef.	Std.Err.	$z$	$p$
Intercept	$-$ 4.168	0.449	$-$ 9.28	$1.65\times 10^{-20}$
SEKF	$-$ 1.800	0.635	$-$ 2.83	$4.59\times 10^{-3}$
SEKF $\times$ Retrain	2.025	0.898	2.25	$2.42\times 10^{-2}$
SEKF $\times$ 12 hr	$-$ 2.197	0.898	$-$ 2.45	$1.44\times 10^{-2}$
SEKF $\times$ Retrain $\times$ 12 hr	3.106	1.270	2.45	$1.45\times 10^{-2}$

Appendix

Table 17: GLM (Gamma) significant coefficients for normalized convergence time—TCLab. Reference levels: Adam optimizer, Finetune initialization, 0.5 hr training data.

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