Taylor-Model Physics-Informed Neural Networks (PINNs) for Ordinary Differential Equations

Chandra Kanth Nagesh, Sriram Sankaranarayanan, Ramneet Kaur, Tuhin Sahai, Susmit Jha
Proceedings of the International Conference on Neuro-symbolic Systems, PMLR 288:621-642, 2025.

Abstract

We study the problem of learning neural network models for Ordinary Differential Equations (ODEs) with parametric uncertainties. Such neural network models capture the solution to the ODE over a given set of parameters, initial conditions, and range of times. Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for learning such models that combine data-driven deep learning with symbolic physics models in a principled manner. However, the accuracy of PINNs degrade when they are used to solve an entire family of initial value problems characterized by varying parameters and initial conditions. In this paper, we combine symbolic differentiation and Taylor series methods to propose a class of higher-order models for capturing the solutions to ODEs. These models combine neural networks and symbolic terms: they use higher order Lie derivatives and a Taylor series expansion obtained symbolically, with the remainder term modeled as a neural network. The key insight is that the remainder term can itself be modeled as a solution to a first-order ODE. We show how the use of these higher order PINNs can improve accuracy using interesting, but challenging ODE benchmarks. We also show that the resulting model can be quite useful for situations such as controlling uncertain physical systems modeled as ODEs.

Cite this Paper

BibTeX
@InProceedings{pmlr-v288-nagesh25a, title = {Taylor-Model Physics-Informed Neural Networks (PINNs) for Ordinary Differential Equations}, author = {Nagesh, Chandra Kanth and Sankaranarayanan, Sriram and Kaur, Ramneet and Sahai, Tuhin and Jha, Susmit}, booktitle = {Proceedings of the International Conference on Neuro-symbolic Systems}, pages = {621--642}, year = {2025}, editor = {Pappas, George and Ravikumar, Pradeep and Seshia, Sanjit A.}, volume = {288}, series = {Proceedings of Machine Learning Research}, month = {28--30 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v288/main/assets/nagesh25a/nagesh25a.pdf}, url = {https://proceedings.mlr.press/v288/nagesh25a.html}, abstract = {We study the problem of learning neural network models for Ordinary Differential Equations (ODEs) with parametric uncertainties. Such neural network models capture the solution to the ODE over a given set of parameters, initial conditions, and range of times. Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for learning such models that combine data-driven deep learning with symbolic physics models in a principled manner. However, the accuracy of PINNs degrade when they are used to solve an entire family of initial value problems characterized by varying parameters and initial conditions. In this paper, we combine symbolic differentiation and Taylor series methods to propose a class of higher-order models for capturing the solutions to ODEs. These models combine neural networks and symbolic terms: they use higher order Lie derivatives and a Taylor series expansion obtained symbolically, with the remainder term modeled as a neural network. The key insight is that the remainder term can itself be modeled as a solution to a first-order ODE. We show how the use of these higher order PINNs can improve accuracy using interesting, but challenging ODE benchmarks. We also show that the resulting model can be quite useful for situations such as controlling uncertain physical systems modeled as ODEs.} }
Endnote
%0 Conference Paper %T Taylor-Model Physics-Informed Neural Networks (PINNs) for Ordinary Differential Equations %A Chandra Kanth Nagesh %A Sriram Sankaranarayanan %A Ramneet Kaur %A Tuhin Sahai %A Susmit Jha %B Proceedings of the International Conference on Neuro-symbolic Systems %C Proceedings of Machine Learning Research %D 2025 %E George Pappas %E Pradeep Ravikumar %E Sanjit A. Seshia %F pmlr-v288-nagesh25a %I PMLR %P 621--642 %U https://proceedings.mlr.press/v288/nagesh25a.html %V 288 %X We study the problem of learning neural network models for Ordinary Differential Equations (ODEs) with parametric uncertainties. Such neural network models capture the solution to the ODE over a given set of parameters, initial conditions, and range of times. Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for learning such models that combine data-driven deep learning with symbolic physics models in a principled manner. However, the accuracy of PINNs degrade when they are used to solve an entire family of initial value problems characterized by varying parameters and initial conditions. In this paper, we combine symbolic differentiation and Taylor series methods to propose a class of higher-order models for capturing the solutions to ODEs. These models combine neural networks and symbolic terms: they use higher order Lie derivatives and a Taylor series expansion obtained symbolically, with the remainder term modeled as a neural network. The key insight is that the remainder term can itself be modeled as a solution to a first-order ODE. We show how the use of these higher order PINNs can improve accuracy using interesting, but challenging ODE benchmarks. We also show that the resulting model can be quite useful for situations such as controlling uncertain physical systems modeled as ODEs.
APA
Nagesh, C.K., Sankaranarayanan, S., Kaur, R., Sahai, T. & Jha, S.. (2025). Taylor-Model Physics-Informed Neural Networks (PINNs) for Ordinary Differential Equations. Proceedings of the International Conference on Neuro-symbolic Systems, in Proceedings of Machine Learning Research 288:621-642 Available from https://proceedings.mlr.press/v288/nagesh25a.html.

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