  Data-driven modeling can suffer from a constant demand for data, leading to
reduced accuracy and impractical for engineering applications due to the high
cost and scarcity of information. To address this challenge, we propose a
progressive reduced order modeling framework that minimizes data cravings and
enhances data-driven modeling's practicality. Our approach selectively
transfers knowledge from previously trained models through gates, similar to
how humans selectively use valuable knowledge while ignoring unuseful
information. By filtering relevant information from previous models, we can
create a surrogate model with minimal turnaround time and a smaller training
set that can still achieve high accuracy. We have tested our framework in
several cases, including transport in porous media, gravity-driven flow, and
finite deformation in hyperelastic materials. Our results illustrate that
retaining information from previous models and utilizing a valuable portion of
that knowledge can significantly improve the accuracy of the current model. We
have demonstrated the importance of progressive knowledge transfer and its
impact on model accuracy with reduced training samples. For instance, our
framework with four parent models outperforms the no-parent counterpart trained
on data nine times larger. Our research unlocks data-driven modeling's
potential for practical engineering applications by mitigating the data
scarcity issue. Our proposed framework is a significant step toward more
efficient and cost-effective data-driven modeling, fostering advancements
across various fields.
