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Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

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Training stiff neural ordinary differential equations with explicit rational Taylor series methods.

Colby Fronk¹, Linda Petzold^2,3

¹Department of Chemical Engineering, University of California, Santa Barbara, Santa Barbara, California 93106, USA.

Chaos (Woodbury, N.Y.)

|July 18, 2025

Summary

We developed new explicit rational Taylor methods for training stiff neural ordinary differential equations. These methods offer high efficiency and stability, reducing computational costs for complex dynamics modeling.

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Area of Science:

Numerical Analysis
Machine Learning
Dynamical Systems

Background:

Stiff neural ordinary differential equations (NODEs) present significant computational challenges.
Traditional implicit methods for stiff NODEs are often computationally intensive.
Efficient and stable training of stiff dynamics is crucial for data-driven simulations.

Purpose of the Study:

To introduce novel explicit rational Taylor series methods for directly training stiff NODEs.
To enhance the efficiency and numerical stability of learning stiff dynamical systems.
To provide a computationally reduced alternative to traditional implicit methods.

Main Methods:

Development of second and third-order explicit rational Taylor series schemes.
Demonstration of A-stability for the proposed explicit schemes.
Application of these methods to train stiff systems, including the van der Pol oscillator.

Main Results:

The explicit schemes achieve high efficiency with a single linear solve per time step.
Proposed methods exhibit strong numerical stability, even at large step sizes.
Effective learning of stiff dynamics was demonstrated without stability issues common to implicit schemes.
Significantly reduced computational cost compared to traditional implicit methods.

Conclusions:

Explicit rational Taylor methods provide an efficient and stable approach for training stiff NODEs.
These methods expand the capabilities of data-driven simulation for complex dynamics.
The findings support applications in mesh-based simulation and physics-informed neural networks.