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The fast growth in practical applications of machine learning in a range of contexts has fueled a renewed interest in machine learning methods over recent years. Subsequently, scientific machine learning is an emerging discipline which merges scientific computing and machine learning. Whilst scientific computing focuses on large-scale models that are derived from scientific laws describing physical phenomena, machine learning focuses on developing data-driven models which require minimal knowledge and prior assumptions. With the contrast between these two approaches follows different advantages: scientific models are effective at extrapolation and can be fit with small data and few parameters whereas machine learning models require “big data” and a large number of parameters but are not biased by the validity of prior assumptions. Scientific machine learning endeavours to combine the two disciplines in order to develop models that retain the advantages from their respective disciplines. Specifically, it works to develop explainable models that are data-driven but require less data than traditional machine learning methods through the utilization of centuries of scientific literature. The resulting model therefore possesses knowledge that prevents overfitting, reduces the number of parameters, and promotes extrapolatability of the model while still utilizing machine learning techniques to learn the terms that are unexplainable by prior assumptions. We call these hybrid data-driven models as “model-constrained machine learning” (mcML) methods.
In the first part of the talk, we introduced PDE-constrained deterministic/statistical inverse problems their challenges, and present scalable methods using traditional applied mathematics approach. In the second part, we present a few efforts in the mcML direction: 1) ROM-ML approach, and 2) model-constrained deep neural networks, and 3) model-constrained autoencoders methods. Theoretical and numerical results for various PDE-constrained inverse problems will be presented to demonstrate the validity of the proposed approaches.