# Modern, Efficient, and Differentiable Transport Equation Models using JAX: Applications to Population Balance Equations

arXiv: https://arxiv.org/abs/2411.00742 · HTML: https://mlsanity.com/paper/2411.00742/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 38 references that are in the map.
- First posted on arXiv 1 Nov 2024.

## Builds on (most cited first, 9 of 9)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Enhancing Computational Fluid Dynamics with Machine Learning](https://mlsanity.com/paper/2110.02085/) (2021)
- [Physics-based Deep Learning](https://mlsanity.com/paper/2109.05237/) (2021)
- [The transformative potential of machine learning for experiments in fluid mechanics](https://mlsanity.com/paper/2303.15832/) (2023)
- [Differentiable Physics: A Position Piece](https://mlsanity.com/paper/2109.07573/) (2021)
- [Machine Learning-Assisted Discovery of Flow Reactor Designs](https://mlsanity.com/paper/2308.08841/) (2023)
- [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)

## Built on by (most cited first, 1 of 1)

- [Differentiable Hybrid Modelling for Learning and Optimising Chemical Transport Processes from Experimental Data](https://mlsanity.com/paper/2609.04011/) (2026)

## Abstract (arXiv)

Population balance equation (PBE) models have potential to automate many engineering processes with far-reaching implications. In the pharmaceutical sector, crystallization model-based design can contribute to shortening excessive drug development timelines. Even so, two major barriers, typical of most transport equations, not just PBEs, have limited this potential. Notably, the time taken to compute a solution to these models with representative accuracy is frequently limiting. Likewise, the model construction process is often tedious and wastes valuable time, owing to the reliance on human expertise to guess constituent models from empirical data. Hybrid models promise to overcome both barriers through tight integration of neural networks with physical PBE models. Towards eliminating experimental guesswork, hybrid models facilitate determining physical relationships from data, also known as 'discovering physics'. Here, we aim to prepare for planned Scientific Machine Learning (SciML) integration through a contemporary implementation of an existing PBE algorithm, one with computational efficiency and differentiability at the forefront. To accomplish this, we utilized JAX, a cutting-edge library for accelerated computing. We showcase the speed benefits of this modern take on PBE modelling by benchmarking our solver to others we prepared using older, more widespread software. Primarily among these software tools is the ubiquitous NumPy, where we show JAX achieves up to 300x relative acceleration in PBE simulations. Our solver is also fully differentiable, which we demonstrate is the only feasible option for integrating learnable data-driven models at scale. We show that differentiability can be 40x faster for optimizing larger models than conventional approaches, which represents the key to neural network integration for physics discovery in later work.