TaylorNet: Learning PDEs from Non-Grid Data

Andrzej Dulny, Paul Heinisch, Andreas Hotho, Anna Krause
Proceedings of the 2nd ECAI Workshop on "Machine Learning Meets Differential Equations: From Theory to Applications", PMLR 277:26-46, 2025.

Abstract

Modeling data obtained from dynamical systems has gained attention in recent years as a challenging task for machine learning models. Previous approaches assume the measurements to be distributed on a grid. However, for real-world applications like weather prediction, the observations are taken from arbitrary locations within the spatial domain. In this paper, we propose TaylorNet - a novel machine learning method that is designed to overcome this challenge. Our algorithm uses the multidimensional Taylor expansion of a dynamical system at each observation point to estimate the spatial derivatives to perform predictions. TaylorNet is able to accomplish two objectives simultaneously: accurately forecast the evolution of a complex dynamical system and explicitly reconstruct the underlying differential equation describing the system. We evaluate our model on a variety of advection-diffusion equations with different parameters and show that it performs similarly to equivalent approaches on grid-structured data while being able to process unstructured data as well.

Cite this Paper

BibTeX
@InProceedings{pmlr-v277-dulny25a, title = {TaylorNet: Learning PDEs from Non-Grid Data}, author = {Dulny, Andrzej and Heinisch, Paul and Hotho, Andreas and Krause, Anna}, booktitle = {Proceedings of the 2nd ECAI Workshop on "Machine Learning Meets Differential Equations: From Theory to Applications"}, pages = {26--46}, year = {2025}, editor = {Coelho, Cecı́lia and Zimmering, Bernd and Costa, M. Fernanda P. and Ferrás, Luı́s L. and Niggemann, Oliver}, volume = {277}, series = {Proceedings of Machine Learning Research}, month = {26 Oct}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v277/main/assets/dulny25a/dulny25a.pdf}, url = {https://proceedings.mlr.press/v277/dulny25a.html}, abstract = {Modeling data obtained from dynamical systems has gained attention in recent years as a challenging task for machine learning models. Previous approaches assume the measurements to be distributed on a grid. However, for real-world applications like weather prediction, the observations are taken from arbitrary locations within the spatial domain. In this paper, we propose TaylorNet - a novel machine learning method that is designed to overcome this challenge. Our algorithm uses the multidimensional Taylor expansion of a dynamical system at each observation point to estimate the spatial derivatives to perform predictions. TaylorNet is able to accomplish two objectives simultaneously: accurately forecast the evolution of a complex dynamical system and explicitly reconstruct the underlying differential equation describing the system. We evaluate our model on a variety of advection-diffusion equations with different parameters and show that it performs similarly to equivalent approaches on grid-structured data while being able to process unstructured data as well.} }
Endnote
%0 Conference Paper %T TaylorNet: Learning PDEs from Non-Grid Data %A Andrzej Dulny %A Paul Heinisch %A Andreas Hotho %A Anna Krause %B Proceedings of the 2nd ECAI Workshop on "Machine Learning Meets Differential Equations: From Theory to Applications" %C Proceedings of Machine Learning Research %D 2025 %E Cecı́lia Coelho %E Bernd Zimmering %E M. Fernanda P. Costa %E Luı́s L. Ferrás %E Oliver Niggemann %F pmlr-v277-dulny25a %I PMLR %P 26--46 %U https://proceedings.mlr.press/v277/dulny25a.html %V 277 %X Modeling data obtained from dynamical systems has gained attention in recent years as a challenging task for machine learning models. Previous approaches assume the measurements to be distributed on a grid. However, for real-world applications like weather prediction, the observations are taken from arbitrary locations within the spatial domain. In this paper, we propose TaylorNet - a novel machine learning method that is designed to overcome this challenge. Our algorithm uses the multidimensional Taylor expansion of a dynamical system at each observation point to estimate the spatial derivatives to perform predictions. TaylorNet is able to accomplish two objectives simultaneously: accurately forecast the evolution of a complex dynamical system and explicitly reconstruct the underlying differential equation describing the system. We evaluate our model on a variety of advection-diffusion equations with different parameters and show that it performs similarly to equivalent approaches on grid-structured data while being able to process unstructured data as well.
APA
Dulny, A., Heinisch, P., Hotho, A. & Krause, A.. (2025). TaylorNet: Learning PDEs from Non-Grid Data. Proceedings of the 2nd ECAI Workshop on "Machine Learning Meets Differential Equations: From Theory to Applications", in Proceedings of Machine Learning Research 277:26-46 Available from https://proceedings.mlr.press/v277/dulny25a.html.

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