Believe? No. Know? Yes.
I've experienced many miracles. And I'm not talking about the "common" type of "miracles" such as beautiful sunsets, birth of a baby, or surviving a car crash. Each of those are "ordinary" miracles that are easily explained by science, to a greater or lesser degree.
To me, "miracle" refers to something "extraordinary." Something, like you said, cannot be explained by science. But there's more. Not only can science not explain it; science will never explain it. Why?
Science deals with the realm of continuity -- space-time and energy-matter. Every effect or state that science studies is built upon continuity. Science would not exist without continuity. The predictability of events and effects, as in a laboratory, come from the fact of that continuity. Continuity means that one thing depends upon another. Molecules interacting chemically depend upon their properties and their states.
Miracles, on the other hand, come from the realm of discontinuity. This is where creation, inspiration and forgiveness originate. These are grounded in the realm of spirit. Science can never touch these.
I've written of my experiences with miracles. One was very much like that of Moses parting the sea. Quite literally, it took my breath away. It also taught me that we are not only Homo sapiens bodies, for a physical body could never do such a thing. Science tells us that. So, the only conclusion left is that I am more than merely physical.
Do I believe Peter stepped out of his storm-tossed boat onto the unsettled Sea of Galilee to stand on the water before his master, Christ? I wasn't there, but I believe it happened exactly that way. I also believe that doubt set in moments after he stepped out, and he promptly sank.
What is significant about that is that science depends heavily upon doubt -- that found in the paradigm known as "skepticism." Science could do better with a paradigm that did not depend upon doubt. It can find a better basis in "restraint" and "humility."