Interpretable polynomial neural ordinary differential equations
Open accessAbstract
Neural networks have the ability to serve as universal function approximators, but they are not interpretable and do not generalize well outside of their training region. Both of these issues are problematic when trying to apply standard neural ordinary differential equations (ODEs) to dynamical systems. We introduce the polynomial neural ODE, which is a deep polynomial neural network inside of the neural ODE framework. We demonstrate the capability of polynomial neural ODEs to predict outside of the training region, as well as to perform direct symbolic regression without using additional tools such as SINDy.
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Data-driven models can efficiently learn complex patterns from operational data but frequently lack physical consistency, interpretability, and generalizability beyond the training domain.
“In contrast, data-driven models can efficiently learn complex patterns from operational data but frequently lack physical consistency, interpretability, and generalizability beyond the training domain.”
From State of mission: Battery management with neural networks and electrochemical AI (2025) · cites this paper as [26]
The Introduction (pp. 1–2) says data-driven models process large amounts of data quickly and that conventional neural ODEs lack interpretability and reliable predictions outside their training region. The Damped Oscillatory System results (p. 9) show a conventional neural ODE failing to preserve the solution’s shape, but the paper does not establish that data-driven models generally lack physical consistency when learning patterns from operational data. (LLM, conf=0.84)
Experimental — accuracy not yet measuredData-driven models' effectiveness heavily depends on large, diverse, and high-quality datasets, and they may yield physically inconsistent results under unseen or noisy conditions.
“Their effectiveness heavily depends on large, diverse, and high-quality datasets, and they may yield physically inconsistent results under unseen or noisy conditions.26,27”
From State of mission: Battery management with neural networks and electrochemical AI (2025) · cites this paper as [26]
The Introduction (p. 2) says conventional neural ODEs do not make reliable predictions outside their training region, and the Results (p. 9) show distortion of a model’s vector field there. But the paper does not establish that effectiveness heavily depends on large, diverse, high-quality datasets; the Discussion (p. 16) explicitly defers analysis of noise and training-data requirements to future work. (LLM, conf=0.87)
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- Authors
- Fronk, C., & Petzold, L.
- Affiliations
- University of California, Santa Barbara (US)
- Venue
- Chaos An Interdisciplinary Journal of Nonlinear Science
- Publisher
- AIP Publishing
- Published
- 2023-04-01
- Volume
- 33
- Issue
- 4
- ISSN
- 1054-1500 (ISSN-L)
- 1089-7682
- PMC
- PMC10076068
- OpenAlex
- W4362579742
- Cited by
- 33
- References
- 34
- Citations by year
- 2026: 8 · 2025: 14 · 2024: 9 · 2023: 2
- FWCI
- 3.29
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Chaos An Interdisciplinary Journal of Nonlinear Science (publishedVersion)
- PubMed Central (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- eScholarship (California Digital Library) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Institute of Biomedical Imaging and Bioengineering (2-R01-EB014877-04A1)
- National Science Foundation (CNS-1725797, CNS-1730158, ACI-1540112, ACI-1541349, OAC-1826967, OAC-2112167, CNS-2120019)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Model Reduction and Neural Networks
- Physical Sciences › Computer Science › Artificial Intelligence › Neural Networks and Applications
- Physical Sciences › Mathematics › Numerical Analysis › Numerical methods for differential equations
- Keywords
- neural ordinary differential equations, polynomial neural network, interpretability, dynamical systems, symbolic regression
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