# Neural modal ordinary differential equations: Integrating physics-based modeling with neural ordinary differential equations for modeling high-dimensional monitored structures

arXiv: https://arxiv.org/abs/2207.07883 · HTML: https://mlsanity.com/paper/2207.07883/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 57 references that are in the map.
- First posted on arXiv 16 Jul 2022.

## Builds on (most cited first, 19 of 19)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Structured Inference Networks for Nonlinear State Space Models](https://mlsanity.com/paper/1609.09869/) (2016)
- [Deep Kalman Filters](https://mlsanity.com/paper/1511.05121/) (2015)
- [Physics-guided Neural Networks (PGNN): An Application in Lake Temperature Modeling](https://mlsanity.com/paper/1710.11431/) (2017)
- [Dynamical Variational Autoencoders: A Comprehensive Review](https://mlsanity.com/paper/2008.12595/) (2020)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [Multiscale Simulations of Complex Systems by Learning their Effective Dynamics](https://mlsanity.com/paper/2006.13431/) (2020)
- [Scientific multi-agent reinforcement learning for wall-models of turbulent flows](https://mlsanity.com/paper/2106.11144/) (2021)
- [Physics-guided Convolutional Neural Network (PhyCNN) for Data-driven Seismic Response Modeling](https://mlsanity.com/paper/1909.08118/) (2019)
- [Modeling System Dynamics with Physics-Informed Neural Networks Based on Lagrangian Mechanics](https://mlsanity.com/paper/2005.14617/) (2020)
- [Learning physics-based reduced-order models for a single-injector combustion process](https://mlsanity.com/paper/1908.03620/) (2019)
- [Machine Learning Approach to Model Order Reduction of Nonlinear Systems via Autoencoder and LSTM Networks](https://mlsanity.com/paper/2109.11213/) (2021)
- [Physics-guided Deep Markov Models for Learning Nonlinear Dynamical Systems with Uncertainty](https://mlsanity.com/paper/2110.08607/) (2021)
- [Value of information from vibration-based structural health monitoring extracted via Bayesian model updating](https://mlsanity.com/paper/2103.07382/) (2021)
- [A local basis approximation approach for nonlinear parametric model order reduction](https://mlsanity.com/paper/2003.07716/) (2020)
- [An enhanced parametric nonlinear reduced order model for imperfect structures using Neumann expansion](https://mlsanity.com/paper/2102.01739/) (2021)

## Built on by (most cited first, 5 of 5)

- [Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applications](https://mlsanity.com/paper/2310.20425/) (2023)
- [Discover physical concepts and equations with machine learning](https://mlsanity.com/paper/2412.12161/) (2024)
- [Deep recurrent-convolutional neural network learning and physics Kalman filtering comparison in dynamic load identification](https://mlsanity.com/paper/2511.00100/) (2025)
- [Capturing reduced-order quantum many-body dynamics out of equilibrium via neural ordinary differential equations](https://mlsanity.com/paper/2512.13913/) (2025)
- [Stable Differentiable Modal Synthesis for Learning Nonlinear Dynamics](https://mlsanity.com/paper/2601.10453/) (2026)

## Abstract (arXiv)

The order/dimension of models derived on the basis of data is commonly restricted by the number of observations, or in the context of monitored systems, sensing nodes. This is particularly true for structural systems (e.g., civil or mechanical structures), which are typically high-dimensional in nature. In the scope of physics-informed machine learning, this paper proposes a framework -- termed Neural Modal ODEs -- to integrate physics-based modeling with deep learning for modeling the dynamics of monitored and high-dimensional engineered systems. Neural Ordinary Differential Equations -- Neural ODEs are exploited as the deep learning operator. In this initiating exploration, we restrict ourselves to linear or mildly nonlinear systems. We propose an architecture that couples a dynamic version of variational autoencoders with physics-informed Neural ODEs (Pi-Neural ODEs). An encoder, as a part of the autoencoder, learns the abstract mappings from the first few items of observational data to the initial values of the latent variables, which drive the learning of embedded dynamics via physics-informed Neural ODEs, imposing a modal model structure on that latent space. The decoder of the proposed model adopts the eigenmodes derived from an eigen-analysis applied to the linearized portion of a physics-based model: a process implicitly carrying the spatial relationship between degrees-of-freedom (DOFs). The framework is validated on a numerical example, and an experimental dataset of a scaled cable-stayed bridge, where the learned hybrid model is shown to outperform a purely physics-based approach to modeling. We further show the functionality of the proposed scheme within the context of virtual sensing, i.e., the recovery of generalized response quantities in unmeasured DOFs from spatially sparse data.