# Physics-Informed Neural Networks for Dynamic Process Operations with Limited Physical Knowledge and Data

arXiv: https://arxiv.org/abs/2406.01528 · HTML: https://mlsanity.com/paper/2406.01528/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 76 references that are in the map.
- First posted on arXiv 3 Jun 2024.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.11833/) (2023)
- [The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?](https://mlsanity.com/paper/2103.09655/) (2021)
- [Inverse-Dirichlet Weighting Enables Reliable Training of Physics Informed Neural Networks](https://mlsanity.com/paper/2107.00940/) (2021)
- [A Hybrid Science-Guided Machine Learning Approach for Modeling and Optimizing Chemical Processes](https://mlsanity.com/paper/2112.01475/) (2021)
- [Physics-Driven Regularization of Deep Neural Networks for Enhanced Engineering Design and Analysis](https://mlsanity.com/paper/1810.05547/) (2018)
- [Fleet Prognosis with Physics-informed Recurrent Neural Networks](https://mlsanity.com/paper/1901.05512/) (2019)
- [Introducing Hybrid Modeling with Time-series-Transformers: A Comparative Study of Series and Parallel Approach in Batch Crystallization](https://mlsanity.com/paper/2308.05749/) (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)

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

- [Federated Learning from Molecules to Processes: A Perspective](https://mlsanity.com/paper/2506.18525/) (2025)
- [Sample-Efficient Reinforcement Learning of Koopman eNMPC](https://mlsanity.com/paper/2503.18787/) (2025)
- [Tensor-Compressed and Fully-Quantized Training of Neural PDE Solvers](https://mlsanity.com/paper/2512.09202/) (2025)
- [Estimating Dense-Packed Zone Height in Liquid-Liquid Separation: A Physics-Informed Neural Network Approach](https://mlsanity.com/paper/2601.18399/) (2026)
- [Hybrid Neural Ordinary Differential Equations for Data-Efficient Polymerization Modeling with Incomplete Kinetics](https://mlsanity.com/paper/2606.02145/) (2026)
- [Atoms to Processes: The Role of Artificial Intelligence and Machine Learning in Chemical Engineering](https://mlsanity.com/paper/2610.02014/) (2026)

## Abstract (arXiv)

In chemical engineering, process data are expensive to acquire, and complex phenomena are difficult to fully model. We explore the use of physics-informed neural networks (PINNs) for modeling dynamic processes with incomplete mechanistic semi-explicit differential-algebraic equation systems and scarce process data. In particular, we focus on estimating states for which neither direct observational data nor constitutive equations are available. We propose an easy-to-apply heuristic to assess whether estimation of such states may be possible. As numerical examples, we consider a continuously stirred tank reactor and a liquid-liquid separator. We find that PINNs can infer immeasurable states with reasonable accuracy, even if respective constitutive equations are unknown. We thus show that PINNs are capable of modeling processes when relatively few experimental data and only partially known mechanistic descriptions are available, and conclude that they constitute a promising avenue that warrants further investigation.