# Physics-integrated hybrid framework for model form error identification in nonlinear dynamical systems

arXiv: https://arxiv.org/abs/2109.00538 · HTML: https://mlsanity.com/paper/2109.00538/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 55 references that are in the map.
- First posted on arXiv 1 Sep 2021.

## Builds on (most cited first, 13 of 13)

- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Transfer learning based multi-fidelity physics informed deep neural network](https://mlsanity.com/paper/2005.10614/) (2020)
- [Multi-fidelity Bayesian Neural Networks: Algorithms and Applications](https://mlsanity.com/paper/2012.13294/) (2020)
- [On spike-and-slab priors for Bayesian equation discovery of nonlinear dynamical systems via sparse linear regression](https://mlsanity.com/paper/2012.01937/) (2020)
- [The role of surrogate models in the development of digital twins of dynamic systems](https://mlsanity.com/paper/2001.09292/) (2020)
- [A Gaussian process latent force model for joint input-state estimation in linear structural systems](https://mlsanity.com/paper/1904.00093/) (2019)
- [Simulation free reliability analysis: A physics-informed deep learning based approach](https://mlsanity.com/paper/2005.01302/) (2020)
- [Machine learning based digital twin for stochastic nonlinear multi-degree of freedom dynamical system](https://mlsanity.com/paper/2103.15636/) (2021)

## Built on by (most cited first, 7 of 7)

- [Explainable, Interpretable & Trustworthy AI for Intelligent Digital Twin: Case Study on Remaining Useful Life](https://mlsanity.com/paper/2301.06676/) (2023)
- [Improved generalization with deep neural operators for engineering systems: Path towards digital twin](https://mlsanity.com/paper/2301.06701/) (2023)
- [Variational Bayes Deep Operator Network: A data-driven Bayesian solver for parametric differential equations](https://mlsanity.com/paper/2206.05655/) (2022)
- [Deep Physics Corrector: A physics enhanced deep learning architecture for solving stochastic differential equations](https://mlsanity.com/paper/2209.09750/) (2022)
- [Probabilistic machine learning based predictive and interpretable digital twin for dynamical systems](https://mlsanity.com/paper/2212.09240/) (2022)
- [DPA-WNO: A gray box model for a class of stochastic mechanics problem](https://mlsanity.com/paper/2309.15128/) (2023)
- [Machine learning assisted state prediction of misspecified linear dynamical system via modal reduction](https://mlsanity.com/paper/2601.05297/) (2026)

## Abstract (arXiv)

For real-life nonlinear systems, the exact form of nonlinearity is often not known and the known governing equations are often based on certain assumptions and approximations. Such representation introduced model-form error into the system. In this paper, we propose a novel gray-box modeling approach that not only identifies the model-form error but also utilizes it to improve the predictive capability of the known but approximate governing equation. The primary idea is to treat the unknown model-form error as a residual force and estimate it using duel Bayesian filter based joint input-state estimation algorithms. For improving the predictive capability of the underlying physics, we first use machine learning algorithm to learn a mapping between the estimated state and the input (model-form error) and then introduce it into the governing equation as an additional term. This helps in improving the predictive capability of the governing physics and allows the model to generalize to unseen environment. Although in theory, any machine learning algorithm can be used within the proposed framework, we use Gaussian process in this work. To test the performance of proposed framework, case studies discussing four different dynamical systems are discussed; results for which indicate that the framework is applicable to a wide variety of systems and can produce reliable estimates of original system's states.