# Combining physics-based and data-driven techniques for reliable hybrid analysis and modeling using the corrective source term approach

arXiv: https://arxiv.org/abs/2206.03451 · HTML: https://mlsanity.com/paper/2206.03451/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 28 references that are in the map.
- First posted on arXiv 7 Jun 2022.

## Builds on (most cited first, 10 of 10)

- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Digital Twin: Values, Challenges and Enablers](https://mlsanity.com/paper/1910.01719/) (2019)
- [Fusing Physics-based and Deep Learning Models for Prognostics](https://mlsanity.com/paper/2003.00732/) (2020)
- [Multifidelity Modeling for Physics-Informed Neural Networks (PINNs)](https://mlsanity.com/paper/2106.13361/) (2021)
- [Physics guided machine learning using simplified theories](https://mlsanity.com/paper/2012.13343/) (2020)
- [Physics guided neural networks for modelling of non-linear dynamics](https://mlsanity.com/paper/2205.06858/) (2022)
- [Deep neural network enabled corrective source term approach to hybrid analysis and modeling](https://mlsanity.com/paper/2105.11521/) (2021)
- [Data-Driven Discovery of Coarse-Grained Equations](https://mlsanity.com/paper/2002.00790/) (2020)
- [Deep-learning based discovery of partial differential equations in integral form from sparse and noisy data](https://mlsanity.com/paper/2011.11981/) (2020)
- [Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution](https://mlsanity.com/paper/2103.14629/) (2021)

## Built on by (most cited first, 6 of 6)

- [A novel corrective-source term approach to modeling unknown physics in aluminum extraction process](https://mlsanity.com/paper/2209.10861/) (2022)
- [Artificial intelligence-driven digital twin of a modern house demonstrated in virtual reality](https://mlsanity.com/paper/2212.07102/) (2022)
- [Sparse deep neural networks for modeling aluminum electrolysis dynamics](https://mlsanity.com/paper/2209.05832/) (2022)
- [Solving PDEs with Unmeasurable Source Terms Using Coupled Physics-Informed Neural Network with Recurrent Prediction for Soft Sensors](https://mlsanity.com/paper/2301.08618/) (2023)
- [Hybrid Modeling, Sim-to-Real Reinforcement Learning, and Large Language Model Driven Control for Digital Twins](https://mlsanity.com/paper/2510.23882/) (2025)
- [Physics-Informed Framework for Impact Identification in Aerospace Composites](https://mlsanity.com/paper/2603.28593/) (2026)

## Abstract (arXiv)

Upcoming technologies like digital twins, autonomous, and artificial intelligent systems involving safety-critical applications require models which are accurate, interpretable, computationally efficient, and generalizable. Unfortunately, the two most commonly used modeling approaches, physics-based modeling (PBM) and data-driven modeling (DDM) fail to satisfy all these requirements. In the current work, we demonstrate how a hybrid approach combining the best of PBM and DDM can result in models which can outperform them both. We do so by combining partial differential equations based on first principles describing partially known physics with a black box DDM, in this case, a deep neural network model compensating for the unknown physics. First, we present a mathematical argument for why this approach should work and then apply the hybrid approach to model two dimensional heat diffusion problem with an unknown source term. The result demonstrates the method's superior performance in terms of accuracy, and generalizability. Additionally, it is shown how the DDM part can be interpreted within the hybrid framework to make the overall approach reliable.