# Deep Grey-Box Modeling With Adaptive Data-Driven Models Toward Trustworthy Estimation of Theory-Driven Models

arXiv: https://arxiv.org/abs/2210.13103 · HTML: https://mlsanity.com/paper/2210.13103/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 30 references that are in the map.
- First posted on arXiv 24 Oct 2022.

## Builds on (most cited first, 16 of 16)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems](https://mlsanity.com/paper/1903.12394/) (2019)
- [Learning by Transduction](https://mlsanity.com/paper/1301.7375/) (2013)
- [Augmenting Physical Models with Deep Networks for Complex Dynamics Forecasting](https://mlsanity.com/paper/2010.04456/) (2020)
- [Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modeling](https://mlsanity.com/paper/2102.13156/) (2021)
- [Integrating Expert ODEs into Neural ODEs: Pharmacology and Disease Progression](https://mlsanity.com/paper/2106.02875/) (2021)
- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [Interpretable Sequence Learning for COVID-19 Forecasting](https://mlsanity.com/paper/2008.00646/) (2020)
- [Neural Dynamical Systems: Balancing Structure and Flexibility in Physical Prediction](https://mlsanity.com/paper/2006.12682/) (2020)
- [Robust Hybrid Learning With Expert Augmentation](https://mlsanity.com/paper/2202.03881/) (2022)
- [Controlling Neural Networks with Rule Representations](https://mlsanity.com/paper/2106.07804/) (2021)
- [Tailoring: encoding inductive biases by optimizing unsupervised objectives at prediction time](https://mlsanity.com/paper/2009.10623/) (2020)
- [Integrating Machine Learning with Physics-Based Modeling](https://mlsanity.com/paper/2006.02619/) (2020)
- [Incorporating Physical Knowledge into Machine Learning for Planetary Space Physics](https://mlsanity.com/paper/2006.01927/) (2020)
- [Half-Inverse Gradients for Physical Deep Learning](https://mlsanity.com/paper/2203.10131/) (2022)
- [Unifying Model-Based and Neural Network Feedforward: Physics-Guided Neural Networks with Linear Autoregressive Dynamics](https://mlsanity.com/paper/2209.12489/) (2022)

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

- [Variational Grey-Box Dynamics Matching](https://mlsanity.com/paper/2602.17477/) (2026)
- [PILA: Physics-Informed Low Rank Augmentation for Interpretable Earth Observation](https://mlsanity.com/paper/2405.18953/) (2024)
- [Hybrid Generative Modeling for Incomplete Physics: Deep Grey-Box Meets Optimal Transport](https://mlsanity.com/paper/2506.22204/) (2025)
- [A Data-to-Product Multimodal Conceptual Framework to Achieve Automated Software Evolution for Context-rich Intelligent Applications](https://mlsanity.com/paper/2404.04821/) (2024)
- [Gray-Box Computed Torque Control for Differential-Drive Mobile Robot Tracking](https://mlsanity.com/paper/2509.00571/) (2025)
- [Sharpness-Aware Hybrid Model Learning for Architecture-Agnostic Parameter Estimation](https://mlsanity.com/paper/2602.06837/) (2026)
- [Bayesian Nonparametric Mixed-Effect ODEs with Gaussian Processes](https://mlsanity.com/paper/2605.13088/) (2026)

## Abstract (arXiv)

The combination of deep neural nets and theory-driven models, which we call deep grey-box modeling, can be inherently interpretable to some extent thanks to the theory backbone. Deep grey-box models are usually learned with a regularized risk minimization to prevent a theory-driven part from being overwritten and ignored by a deep neural net. However, an estimation of the theory-driven part obtained by uncritically optimizing a regularizer can hardly be trustworthy when we are not sure what regularizer is suitable for the given data, which may harm the interpretability. Toward a trustworthy estimation of the theory-driven part, we should analyze regularizers' behavior to compare different candidates and to justify a specific choice. In this paper, we present a framework that enables us to analyze a regularizer's behavior empirically with a slight change in the neural net's architecture and the training objective.