# Partial observations and conservation laws: Grey-box modeling in biotechnology and optogenetics

arXiv: https://arxiv.org/abs/1909.04234 · HTML: https://mlsanity.com/paper/1909.04234/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Sep 2019.

## Built on by (most cited first, 9 of 9)

- [A Framework for Machine Learning of Model Error in Dynamical Systems](https://mlsanity.com/paper/2107.06658/) (2021)
- [AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identification](https://mlsanity.com/paper/2310.01433/) (2023)
- [Constructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approach](https://mlsanity.com/paper/2201.13323/) (2022)
- [Data-driven Control of Agent-based Models: an Equation/Variable-free Machine Learning Approach](https://mlsanity.com/paper/2207.05779/) (2022)
- [Representation Meets Optimization: Training PINNs and PIKANs for Gray-Box Discovery in Systems Pharmacology](https://mlsanity.com/paper/2504.07379/) (2025)
- [Personalized Algorithm Generation: A Case Study in Learning ODE Integrators](https://mlsanity.com/paper/2105.01303/) (2021)
- [A Recursively Recurrent Neural Network (R2N2) Architecture for Learning Iterative Algorithms](https://mlsanity.com/paper/2211.12386/) (2022)
- [Implementation and (Inverse Modified) Error Analysis for implicitly-templated ODE-nets](https://mlsanity.com/paper/2303.17824/) (2023)
- [Some of the variables, some of the parameters, some of the times, with some physics known: Identification with partial information](https://mlsanity.com/paper/2304.14214/) (2023)

## Abstract (arXiv)

Developing accurate dynamical system models from physical insight or data can be impeded when only partial observations of the system state are available. Here, we combine conservation laws used in physics and engineering with artificial neural networks to construct "grey-box" system models that make accurate predictions even with limited information. These models use a time delay embedding (c.f., Takens embedding theorem) to reconstruct effect of the intrinsic states, and can be used for multiscale systems where macroscopic balance equations depend on unmeasured micro/meso scale phenomena. By incorporating physics knowledge into the neural network architecture, we regularize variables and may train the model more accurately on smaller data sets than black-box neural network models. We present numerical examples from biotechnology, including a continuous bioreactor actuated using light through optogenetics (an emerging technology in synthetic biology) where the effect of unmeasured intracellular information is recovered from the histories of the measured macroscopic variables.