I have pondered this issue for nearly 35 years. If the universe came into being by way of a Big Bang process we will probably never find the actual center at which it commenced:
REASONS A CENTER MAY NEVER BE FOUND
1. First, we should always keep this in mind: We did not see the beginning of our universe take place. There are no eyewitnesses who can tell us what actually happened. Therefore, it is a 13.7+ billions of years old COLD case. In other words everything we put forward regarding the beginning of the universe will always be only speculation.
We have no way to prove anything that we hypothesize regarding how our universe commenced (no matter how many theories are proposed regarding the beginning of the universe nor how good they and their corresponding math equations appear to be, there is no way to fully test them to prove out anything they make claims for).
In other words, even if the numbers do not add up perfectly you can always come up with another constant or sub-theory that makes it look more correct, none of which is provable in our real world predicament. This means it may never be possible to determine if a BB took place or if a center ever existed at all.
2. That said, even if the Big Bang is responsible for how the universe expanded (a kind of explosion of a plasma type soupy mix under pressure against gravity) to its current state, problems or barriers may prevent us from locating the real location of the center from which it all supposedly started.
3. The starting point, regarding the BB, was supposedly a state of singularity. What singularity actually would be is not really known. However, its state would be a point about the size of a marble in which presumably all the laws of physics as we know them are broken down, a kind of state of total annihilation of all atomic structure and its various particles. If the supposed early "super" inflation phase occurred it would be opaque because photons would not be present nor released during this phase. Therefore, our view of events up to this point would be obscured.
In other words, we can never see through this phase nor would there be any way to see this phase of inflation because we would need to see photons. Only where photons are released are we able to see even part of what was there (the so-called visible universe limit).
4. Because we can not see beyond the visible universe (which would include the parts of the electromagnetic spectrum that can not be seen by the naked eye, that can only be seen through equipment that can detect invisible wavelengths) we would have almost no chance of ever finding the real center of the BB.
5. According to one theory space exists and inflates separate from matter and energy. That is we are located within and are a part of this inflation. According to this proposal space (fabric of spacetime) is inflating in all directions all around us, that we are within the spot, the center. The result being that we can not detect a direction in which the BB started from. Supposedly, because we are located within the BB inflation, this precludes us from having a frame of reference that would allow us to have the ability to trace back and locate the real center or even the general location of the BB starting point.
6. The real world frame of reference for our universe is three dimensional plus time. The visible universe matter is not uniform in all directions. That is all the galaxies, etc. are not uniformly spaced along its x, y & z axis (unlike the two dimensional demonstration models with evenly spaced matter). Since, supposedly the Big Bang started with a very tiny singularity blob (at least for this theory), smaller than a marble, then there ought to be various vector trails (even if they are fragmented or somewhat askew, blurred or dissipated) of some sort going back to the marble size singularity state or at least into the outward edge area of the 'super' iflation phase (especially since matter is not uniformly spaced and would have to change direction and consistancy in order for large areas or gaps at irregular intervals to be present as they are now).
Even if no hole or hollow area exists there should be a huge clouded area or an area with different characteristics than most of the rest of the universe if a BB took place. However, so far we have not been able to locate any evidence that would suggest that the BB came from a particular direction which has also precluded us from locating a BB center.
7. Unfortunately, two dimensional demonstation models lack ability to show true three dimensional effect regarding starting at a tiny point through the present state of inflation or expansion of 13.7 billion light years. In other words, if we look across through our universe sphere (matter does not exist evenly spaced on two dimensional planes here) somewhere at least near part of the edge of our visible universe we should see at least some large area where galaxies are much closer together compared to say an eight or ten billion light years area around where our galaxy is located now. So far we have not found any evidence. Perhaps this means the BB did not occur.
Certainly if the universe had a tiny starting point there should be a visible change somewhere in an inflation or expansion that covers such a huge area of more than 13.7 billion light years (there would be some indication in which direction we should concentrate our effort to find something regarding at least the general direction of the BB's original location). So far such indication has not been found.
8. Also, there may be a force(s) outside our universe that is the driving force for the inflation or expansion of our universe, in which case it would most likely make it impossible to ever find evidence of a center if one ever existed.
9. If any kind of BB happened at all I seriously doubt that the universe could be 'flat' unless there was an amazing powerful set of unkown events that caused the universe to change shape. Generally a roundish center that inflated with tremendous outward force would retain a general roundish shape even for our current universe. Perhaps if unseen forces from outside the universe caused some huge obstruction to the shaping process that would have come by way of a BB, there could be a different shape now. However, I doubt that there will ever be any way to varify such a claim.