**4. Summary and Conclusions**

In this article we have presented the development and application of a GB modelling framework for lithium-ion batteries based on a coupling of NODEs and physics-based ODEs. The model was trained and tested using experimental data of an LFP battery cell used in home-storage applications. The main findings can be summarised as follows.

We showed how to derive a GB model from a physics-based ECM with appropriate choice of learnable functions and parameters. We emphasised the importance of normalisation and initialisation of the parametric parts of the model. The training was split into two training steps: first, a simplified static model was trained where the capacitance of the RC element was neglected. In the second step, the pre-trained parameters were used to train the short-term battery dynamics. When choosing the hyperparameters, especially the number of hidden neurons in *f* ∗ and *g*∗ and the number of training epochs, care had to be taken to avoid long training times and overfitting.

The model trained this way was able to reproduce the complete set of training data (CCCV charge and discharge curves as well as pulse tests) with good accuracy (typically < 1% deviation between predicted and measured voltage). In contrast to the GB model proposed in our previous work [17], the present model can approximate the fast (1 s to 30 s) dynamics of the battery. The model was tested against two data sets, half cycles and a synthetic load profile. The simulations showed good agreement with the experimental data. The highest but still acceptable errors occur in the area of low and high SOC values where the OCV curve is very steep. It is worth mentioning that the training database was rather small: only eight time series covering charging and discharging processes were available for training; and the test data sets spanned a much longer time duration than the training data sets.

As an outlook it would be interesting to use more training data, especially from pulse tests with different current steps. Additional data would also improve model validation. For example, a k-fold cross validation could deliver insights into the robustness of the model against the chosen training data. Moreover, the comparison of a WB model and a GB model using NODEs would be of interest.

In conclusion, we have shown that the use of NODEs can be a powerful methodology for modelling lithium-ion batteries.

**Author Contributions:** Conceptualisation, J.B., W.G.B. and R.G.; methodology, J.B., W.G.B. and R.G.; software, J.B.; validation, J.B., W.G.B. and R.G.; formal analysis, J.B.; investigation, J.B. and R.B.; resources, J.B.; data curation, J.B. and R.B.; writing—original draft preparation, J.B.; writing—review and editing, J.B., R.B., W.G.B. and R.G.; visualisation, J.B.; supervision, W.G.B. and R.G.; project administration, R.G.; funding acquisition, R.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** J.B. and R.G. acknowledge funding from the Carl Zeiss Foundation and R.B. acknowledges funding from the State of Baden-Württemberg in the framework of the Mittelbauprogramm 2019. Experiments were carried out in the Enerlab 4.0 laboratory which was funded by the Federal Ministry of Education and Research (BMBF) under grant no. 13FH091IN6. The article processing charge was funded by the Baden-Württemberg Ministry of Science, Research and Culture and the Offenburg University of Applied Sciences in the funding programme Open Access Publishing.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The code and measurement data presented in this study are openly available in Zenodo at https://doi.org/10.5281/zenodo.6138075.

**Acknowledgments:** J.B. is an associated member of the Research Training Group GRK 2218 SiMET— Simulation of mechano-electro-thermal processes in lithium-ion batteries, project number: 281041241. She thanks the German Research Foundation (DFG) for the cooperative support. The authors thank Simone Schede (Offenburg University of Applied Sciences) for proofreading.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
