(549e) Learning Coarse-Scale ODEs/PDEs from Microscopic Data: What and How Can We Learn It from Data?

Traditionally, Ordinary Differential Equations (ODE) and Partial Differential Equations (PDE) have been widely used to understand complex physicochemical behaviors or dynamics in chemical engineering. However, identifying a proper ODE/PDE system for certain phenomena requires prior-knowledge about related physics, chemistry, and mathematics. Hence, the educators have tried to convey all these related concepts through various courses in classical chemical engineering curriculum.

Nowadays, thanks to the advances in machine learning techniques and data-driven modeling, we are able to effectively identify ODEs/PDEs from data directly without prior knowledge. To this end, the prerequisite for learning data-driven ODEs/PDEs will gradually evolve from the classical, mechanistic/physics based prerequisite courses in chemical engineering to include advanced statistics, machine learning techniques, and data mining. In this talk, we present some machine learning techniques of data-driven ODE/PDEs from microscopic data us (e.g. the use of ordinary neural network [1], Gaussian process [2], or ResNet [3]) Through these examples, we illustrate the concept of the black-box model and the (partially physics informed) gray box model to identify/explain model ODE/PDE. Moreover, we present a new challenge in data-driven ODE/PDE: (1) how to choose the right variables from data, (2) how to construct a proper data-driven model, and (3) what are the pros and cons for different approaches. Finally, this presentation will suggest a new direction for a future curriculum for data-driven modeling in chemical engineering.

[1]Chen, R.T., Rubanova, Y., Bettencourt, J. and Duvenaud, D., 2018. Neural ordinary differential equations. arXiv preprint arXiv:1806.07366.
[2] Lee, S., Kooshkbaghi, M., Spiliotis, K., Siettos, C.I. and Kevrekidis, I.G., 2020. Coarse-scale PDEs from fine-scale observations via machine learning. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(1), p.013141.
[3] Qin, T., Wu, K. and Xiu, D., 2019. Data driven governing equations approximation using deep neural networks. Journal of Computational Physics, 395, pp.620-635.