# Analysis of Numerical Integration in RNN-Based Residuals for Fault Diagnosis of Dynamic Systems

arXiv: https://arxiv.org/abs/2305.04670 · HTML: https://mlsanity.com/paper/2305.04670/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 15 references that are in the map.
- First posted on arXiv 8 May 2023.

## Builds on (most cited first, 7 of 7)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations](https://mlsanity.com/paper/1710.10121/) (2017)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [On Numerical Integration in Neural Ordinary Differential Equations](https://mlsanity.com/paper/2206.07335/) (2022)
- [Isolation and Localization of Unknown Faults Using Neural Network-Based Residuals](https://mlsanity.com/paper/1910.05626/) (2019)

## Built on by (most cited first, 2 of 2)

- [Stability-Informed Initialization of Neural Ordinary Differential Equations](https://mlsanity.com/paper/2311.15890/) (2023)
- [Probabilistic Machine Learning for Uncertainty-Aware Diagnosis of Industrial Systems](https://mlsanity.com/paper/2509.18810/) (2025)

## Abstract (arXiv)

Data-driven modeling and machine learning are widely used to model the behavior of dynamic systems. One application is the residual evaluation of technical systems where model predictions are compared with measurement data to create residuals for fault diagnosis applications. While recurrent neural network models have been shown capable of modeling complex non-linear dynamic systems, they are limited to fixed steps discrete-time simulation. Modeling using neural ordinary differential equations, however, make it possible to evaluate the state variables at specific times, compute gradients when training the model and use standard numerical solvers to explicitly model the underlying dynamic of the time-series data. Here, the effect of solver selection on the performance of neural ordinary differential equation residuals during training and evaluation is investigated. The paper includes a case study of a heavy-duty truck's after-treatment system to highlight the potential of these techniques for improving fault diagnosis performance.