The short answer is no, the currency would not have the same value over the long history of the Roman Empire. That being said, it would still be recognized that earlier coins had significant precious metal content, and as a result you could potentially see people exchanging their old coins for new ones based on the metal content.

In case you're interested, I'll also give you a brief rundown of the periods of Roman coinage, and the major changes these systems underwent.

The regular minting of coinage largely began around the time of the 2nd Punic War. There was a system that predated this, but honestly I'm not very familiar with it and so I'll leave any discussion of that for better informed commenters. Near the end of the 3rd century BC, the Romans introduced their monetary system based around the sestertius (a large bronze coin) and the denarius (a smaller silver coin). There were many, many more denominations than these, but most values would be reckoned in fractions or multiples of denarii and sestertii. These coins would largely bear images of gods, especially Roma and Mars, and typically the name of the Roman citizen who was responsible for the minting of the coins. The metallic content of these coins was remarkably stable, with the silver denarius maintaining very high purity from c. 200 BC- c.200 AD. Denominations would be phased in and out, silver content might vary by a few percentage points, but over this time period, the coins were largely unchanged, though the iconography changed during the imperial period to feature more portrait busts of emperors and their families rather than almost exclusively divinities.

Things began to change substantially under the Severan dynasty in the very late2nd-3rd century AD. For a number of reasons, including the exhaustion of mines in Western Europe, the lack of new conquests, and a trade deficit with the far east (money flowing from Rome to Asia and India in exchange for goods, but no goods flowing from Rome to the East in exchange for more precious metals), the Roman supply of gold and (especially) silver was dwindling. Furthermore, the Severans greatly increased soldier pay, making the expenses of the state skyrocket. For all of these reasons, emperor Caracalla introduced a new denomination that had a nominal value of 2 denarii, but only had a roughly 50% increase in silver content. By introducing this coinage, Caracalla was hoping to bring in more silver than he was paying out, and thereby solve the problem. However, once the citizens became aware of this imbalance--and the purity would also continue to drop over the course of the 3rd century--the currency system began to crumble. Prices rose, coinage became more and more devalued, and soon the "silver" coins produced by the Empire had, at best, a tiny percentage of silver in them.

In the very late 3rd century, and especially in the 4th, we see the empire try to correct these problems and introduce a new system of coinage. Silver was largely abandoned, its value being deemed to volatile, and instead small bronze coins became the basic unit of currency, as well as high-value gold pieces known as solidi. Throughout the 4th century, we see these bronze coins being minted in incredible numbers. Gradually, over the course of the 5th century, these bronze coins tended to become more and more crude, and often tinier and tinier. By the end of the 5th century, counting the coins had become largely impractical, and people reverted to simply weighing bags of the tiny coins to determine value (which is exactly what the establishment of a system of coinage is supposed to replace).

In response to this, emperor Anastasios reforms the currency once again, and creates a system that would remain in place until the Middle Byzantine period. The basic, tiny copper currency was known as the nummus, and denominations of 5, 10, 12, 16, 20, and 40 nummi circulated widely. Gold solidi continued to be minted, and very infrequently silver coins would be minted as well. By the 6th century, the lower denominations of nummi began to be abandoned, and the large 40 nummi piece transitioned into what came to be called the follis.

The follis, solidi, and silver miliarense remained the basic system if currency in Byzantium through the 11th century. Under the Komnenans, we see another coinage reform, comprised of copper pieces such as the tetarteron (not too different from the follis), new "scyphate" or cupped coins, often made with some proportion of silver, and a revised gold currency, as the gold coinage had largely fallen apart during the tumultuous 11th century. This remained the basic system until after the 4th crusade; in the 13th and 14th centuries, we start to see Byzantium imitate western coinage, especially that of Venice.

I've probably forgotten some things here, but I hope that this helps give you an overview of how Roman coinage developed. Feel free to ask any follow-up questions!

Edit: My first gold, thank you very much!