I note that there is a lack of notable, if at all, natural philosophers during that time, as opposed to Ancient Greek or medieval Islamic world where you can throw a stick and most likely hit a natural philosopher. Most Roman philosophers seem to focus on Stoicism or political philosophy. Was there no patronage? Or did the political and social structure of the time not support a dedication of time to study something which has little "impact" (why do we care that the earth revolves the sun, etc.) to everyday lives?
Although the Roman Empire certainly produced fewer "household name" natural philosophers, scientific inquiry did not cease. At the risk of oversimplification, however, Roman-era works of astronomy, biology, geology, etc. tended to be characterized more by codification and refinement than by dramatic theoretical advances.
First, a few Roman natural philosophers:
Claudius Ptolemy (flourished mid-second century CE) produced a series of brilliant and innovative works in mathematics, astronomy, and geography. His most famous work, the Almagest, used careful observations and nuanced theory to create the definitive model of the geocentric universe.
Galen (c. 129-216 CE) produced detailed studies of anatomy based on animal dissection.
Though not an original scholar, Pliny the Elder (23-79 CE) composed the compendious Natural History, a massive collection of knowledge about the natural world (and much else) that demonstrates, if nothing else, Roman curiosity on the topic.
It is true, however, that the Roman Empire offers few examples of dazzling theoretical work to match that of earlier periods. There are, I think, several reasons.
First, theoretical inquiry was never prevalent in the ancient world - Archaic Miletus, Classical Athens, and Hellenistic Alexandria are among the very few "centers" of natural philosophy, and each was characterized by special circumstances (usually, exceptionally concentrated wealth and/or royal patronage) that simply didn't obtain in most places and periods. Although the Library of Alexandria remained open through at least the first three centuries CE, the Romans never funded it in the way the Ptolemies had.
Second, the Romans - though seldom actually hostile to scientific advances or inquiry - did not really encourage it. There is a(n apocryphal) story in Petronius' Satyricon that nicely illustrates this attitude:
"But there was once a workman who made a glass cup that was unbreakable. So he was given an audience of the Emperor with his invention; he made Caesar give it back to him and then threw it on the floor. Caesar was as frightened as could be. But the man picked up his cup from the ground: it was dinted like a bronze bowl; then he took a little hammer out of his pocket and made the cup quite sound again without any trouble. After doing this he thought he had himself seated on the throne of Jupiter, especially when Caesar said to him: 'Does anyone else know how to blow glass like this?' Just see what happened. He said not, and then Caesar had him beheaded. Why? Because if his invention were generally known we should treat gold like dirt." (51)
This never happened - but the story illuminates the Roman tendency not to meddle with the status quo.
Finally, and I think most importantly, Greco-Roman culture in the imperial era was characterized by great veneration for the past, and especially for literary classics. It was more socially prestigious to comment on, compile, and refine the works of respected authorities than to innovate.