Simple:
* The scientific principles underlying the invention are complex. (meaning the principles form the basis of the invention)
* A deep-seated fear of failure underlies his reluctance to try new things. (meaning the fear is the reason for his reluctance)
More complex:
* The political tensions underlying the conflict are rooted in historical grievances. (showing how the conflict is a result of tensions)
* A strong work ethic underlies her success in business. (meaning her work ethic is the foundation of her success)
* The artist's use of bold colors and abstract forms underlies the powerful emotions conveyed in the painting. (emphasizing the connection between technique and effect)
Figurative:
* The entire argument was built on a misunderstanding that underlay the entire discussion. (using "underlie" to describe something foundational and hidden)
The word "underlie" implies a deeper, underlying cause or foundation. It helps to show that something is the reason for something else, even if it's not directly visible.