Oh, I love this question!
I personally interpret Potterverse to possibly include scientific connections to magic.
A Time-turner, obviously, is related to time-travel, which I think touches on Einstein's Theory of Relativity:
The theory of relativity overturned the concept of motion from Newton's day, by positing that all motion is relative. Time was no longer uniform and absolute. Physics could no longer be understood as space by itself, and time by itself. Instead, an added dimension had to be taken into account with curved spacetime. Time now depended on velocity, and contraction became a fundamental consequence at appropriate speeds.
Wikipedia - Einstein's Theory of Relativity
Scientifically, I think we have to examine the Department of Mysteries very carefully. There are five known divisions in the Department of Mysteries that attend to love, time, thought, death, and space. As well, the Department of Mysteries houses Hall of Prophecies, although I do not know whether prophecies would fall under anything other than paranormal psychology.
The room housing the study of love, IIRC, is the room that is always locked and cannot be opened with any instrument (I think the knife that will open any lock or door that Sirius gave to Harry completely melted when the DA attempted to enter this room). Dumbledore had told Harry that this room contains a force so powerful that it would be completely overwhelming were it not contained. Love falls under several branches of psychology including, but not limited to: attachment psychology; biological psychology; forensic psychology; social psychology; and behavioral psychology. Per the Harry Potter Wiki, the love room may be used in attempts to understand and duplicate the magical protection self-sacrificing love creates, the only magic strong enough to repel the Killing Curse. Apparently there is said to be a large fountain of Amortentia in the room. What is the possible scientific explanation for love having the ability to block the killing curse?
The DoM room where thought is studied is the Brain Room and it is implied that when Ron touches the brains in the tanks there, and he is cut and injured on his forearms, it's by what is thought to be the physical manifestations of thought itself -- interpreted, feelings can hurt, or feel good, or sad, or happy, or angry, etc. Neurology, anyone? Neuropsychiatry? Thought disorders, learning diabilities, brain dysfunction, intelligence, mood, and a person's level of self-efficacy, and so so much more, are all related to the brain. Luna is forthcoming to the point of making others uncomfortable, yet is endearing beyond belief. Voldemort's thoughts are certainly disordered. Neville struggles to generally master magic and accept his placement in Gryffindor rather than Hufflepuff. Lockhart's brain is damaged beyond repair, as are Frank and Alice Longbottom's. Hermione's intellect is unusual and superior to most. Snape's moods are dark and angry and bitter. Harry demonstrates a high level of self-efficacy, despite the abuse and neglect he endured at the Dursleys.
The Veil occupies the room where death is studied and has been in the Department of Mysteries since the beginning of the Ministry of Magic (which is not quite as old as Hogwarts, but is many centuries old).
Thanatology is the scientific study of death. It investigates the mechanisms and forensic aspects of death, such as bodily changes that accompany death and the post-mortem period, as well as wider social aspects related to death. It is primarily an interdisciplinary study offered as a course of study at numerous colleges and universities.*
Wikipedia - Thanatology
Death is the theme of the Harry Potter series (that, and love). As far as the Veil room goes, the structure of the Veil itself implies that the death studies being done are not primarily pragmatic; my guess is that the research done on death in the DoM has more to do with the metaphysical, the soul, and the possibility of an afterlife (both Harry and Luna hear voices whispering from behind the veil). This likely falls more into social psychology than medical study. Perhaps there is study on how Avada Kedavra affects the body and why it leaves no trace. And wouldn't the study of Horcruxes -- were there to be such a study -- be fascinating in the context of thanatology? The physical manifestation of the preservation of life through a piece of torn soul. . . it's both highly scientific and deeply magical, IMO. Thestrals that appear only when dead has meaning and has "sunk in."
I already touched on time above regarding the Time-turner. During the Battle of the Department of Mysteries in Order of the Phoenix, we see all the Time-turners and clocks come smashing down to the floor, only to fly back up to their original storage spots and repair themselves, only to repeat the process endlessly. It's like a physical manifestation of the infinity of time. Even if our universe ceases to exist, time will continue. And Dumbledore values the gift of time when he tells Harry "It does not do to dwell on dreams and forget to live" when Harry is obsessed with the Mirror of Erised. (Speaking of the Mirror of Erised, hello telepathy!) Those who practice the art or study of measuring time are horologists (they are interested in the mechanics of clocks, watches, sundials, and, yes, presumably time-turners!) The entire book Prisoner of Azkaban deals with time, time-travel, and the timing of individual actions. Time touches so many things -- how quickly a spell is cast; how quickly Mandrakes grow to make the draught to revive the petrified victims of the Basilisk, how much time we get to spend with friends and loved ones (a big theme in HP as well -- family). Time is related to velocity, which relates to brooms and flying. Time is endless and it is fruitless to resist it (like Voldemort tried to), but life cycles infinitely. JK Rowling used the Celtic calendar to assign wand woods to each of her characters; calendars are representations of time. The prophecy was tied to "the end of July" 1980, a specific time frame.
During the Battle of the DoM, Luna finds herself in a room full of planets and stars -- space! Astronomy is a real science that is taught at Hogwarts, the study of the stars and constellations and the universe. Could Apparition be conceptually like warp speed? To instantaneously get from point A to point B when Apparating, do points A and B come together, bending the universe, deposit the Apparating traveler at point B, and then stretch out again, within seconds, realigning the universe? Or would Apparition be more like a wormhole? The way Portkeys are described, I would think they are more wormhole-like than possibly warp speedish, because of the way people seem to fall out of them to their final destination. Anyhow, magically, astronomy is applied by the centaurs to make their cryptic predictions ("Mars is bright tonight."). Could astronomy have something to do with the number seven being considered the most magical number, the number coveted by Tom Riddle/Voldemort?
The ghosts and poltergeist Peeves would fall under paranormal study; to some that is a true area of science (for others, not)
Dementors are JR Rowling's representation of depression, which would be categorized under psychology or psychiatry.
As @DVK first mentioned, there is Alchemy.
While the trio is on the run in Deathly Hallows it's noted that Hermione becomes skilled in picking edible roots and plants, perhaps even mushrooms? (Mycology)
Both the first and second Voldemort wars are fundamentally about Wizarding genetics. Purebloods are superior; half-bloods and Muggleborns less so. Squibs are anomalies that are not accepted in Wizarding society (Filch and Arabella Figg being exceptions) and are often hidden away or encouraged to join the Muggle world. This all relates to genetics and, in the case of the rabid purebloods, probably eugenics to some degree.
The Sorting Hat and Legilimency suggests telepathy; the Wiki states in the Harry Potter series by J. K. Rowling, telepathy is the magical skill known as Legilimency.
Ancient Runes -- Anthropology? Archaeology?