He is smarter than bats

for plot no..

but realistically he could sit and learn everything about human and alien science with his fast brain and already brilliant mind (his parents were genius scientists)

Batman would be useless so for plot Superman just doesnt learn how to do all those fancy martial arts and sciences

Superman is smarter than Batman in specific fields, however Superman can trump Batman in every field if he desired too. People bring Batman's intelligence onto a sort of high pedestal that it doesn't belong on. Batman is smart, don't get me wrong. But Superman can process information in nanoseconds, he can read entire libraries of information in mere seconds. He did it in the New 52 when Lois Lane needed surgery. He learned how to be a medical professional in mere seconds. He has stated before that he can modify Batman's gadgets far better than what they are now, and he can also create gadgets himself. He has X-ray vision, thermo vision, microscopic vision, he can see the entire electromagnetic spectrum. If he wanted to solve a crime, or learn thousands of battle strategies, or track down a criminal he could do it in a MUCH shorter time than Batman.

superman is in just every field better than batman. that is why bruce is the dark and superman the sun

Superman, when written correctly, is on a level of intelligence very few can match, and that includes Batman. His Super-Intelligence was there from the beginning, it is not an added power like many of his powers. He also has total recall. That doesn't render Batman useless, however, as in many ways Superman is TOO superior to the humans he protects, and he doesn't get human foibles and pettiness the way Batman does. If you read the best Superman stories, the challenges he meets are always a tough test for his powers and he has to think his way through them, either to use his powers correctly to stop the threat or to not overuse his powers and cause a lot of damage. Superman is a constant exercise in restraint. In fact, he purposely gimps himself a lot when on Earth so as to not break his code against killing.

Batman's areas of expertise- evidence analysis, psychology, deduction, etc, make his intellectual skillset very different from Superman's. While Superman might need to calculate just how fast to fly to counteract a tornado, Batman is concerned with piecing together very obscure bits of information and a knowledge of his foes to predict or influence their next moves. Like Superman, the best Batman stories also involve him using his mind to solve cases. Batman is an incredibly skilled martial artist, but you rarely see him jumping into 20 goons like Captain America does. That's because Cap is an action character and Batman is a stealth character.

Superman and Batman both take cues from one another pretty often, and they have much, much more in common than they have different. That's part of why having them be hostile to one another is a bad idea that just generates false conflict.