Firstly, science doesn’t “prove” anything. The scientific method is essentially a process of deductive reasoning, so it can “
dissprove” things though. So it can show what something is not.
Look into some of the modern saints and blessed. You’ll see that the miracles they need in order to be beatified or canonized are just about always answers to someones prayers.
These miracles have all been disproven to be anything else. Science has demonstrated that a miraculous cure is not due to medical treatment for example. That’s why we know it’s a miracle. The scientists and physicians determine that there is no plausible reason for a cure (most of the miracles are medical cures) . There is no Scientific explanation.
I’m emphasizing modern saints, so you know it’s modern science and medicine that can’t explain these miracles, not 13th century science which was obviously much different.
I’m sure there are good Catholic sources with much more details on each of these miracles, but here are some examples of miraculous answers to prayers from a few quick google searches. Proof prayer really works:
Mother Theresa:
bbc.com/news/world-asia-india-35129463
Pope John XXIII:
lastampa.it/2014/04/24/esteri/vatican-insider/en/a-miracle-has-been-attributed-to-john-xxiii-but-not-for-his-canonization-bxJaZWyhfuEaFNI5tLE3WO/pagina.html
Pope John Paul II:
abcnews.go.com/International/miracles-put-pope-john-paul-ii-fast-track/story?id=23467594
Andre Bessette:
catholicexchange.com/st-andre-bessettes-wondrous-miracles
But don’t forget, God answers many prayers that
are explicable with science and medicine as well. It’s not just the inexplicable that are answers to our prayers, it’s just that they’re just easiest to “prove”.