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They are sent off as a suicide mission toward Vega or somewhere similar, and the AI is supposed to learn from them whatever they discover and return it somehow. They aren't told the truth, naturally. They figure out the nature of the mission, and also rapidly evolve new abilities. Their scientific understanding grows to the point where they figure out how to do FTL travel, and return to Earth, sending a bombardment of "heavy particles" before them that render all atomic fission and explosives inert. They are fairly pissed off at the people who sent them.

"The Gold at the Starbow's End", a novella by Frederik Pohl, which was expanded into a novel Starburst. We've had questions about this before.

Wikipedia summary:

The story is told with two narrative devices—reports from members of the crew of the U.S. Starship Constitution alternating with a traditional third-person narration of the activities back on Earth. The main protagonist of the activities on Earth is Dr. Dieter von Knefhausen, the scientist in charge of the U.S. space program.

In the first report from the starship, the reader learns that the ship is approximately one month into a multi-year journey to the Alpha Centauri star system, where the crew will begin colonization of the planet Alpha-Aleph. Already, the crew is finding they have too much free time and have begun filling that time by studying various problems in mathematics. In the first narration of the action on Earth, the reader learns that society has become dystopian. The possibility of colonizing Alpha-Aleph is a source of hope for a better future.

As the story progresses, the reader is told that the existence of the planet Alpha-Aleph is a hoax, perpetrated not only on the American people but also on the crew of the starship. The true purpose of the mission is to place the crew in a position where they will have nothing to do other than study mathematics. The hoax was the idea of Knefhausen, who believes that, if deprived of any other means of recreation, the crew will succeed in making scientific breakthroughs that will then be broadcast back to Earth. Knefhausen's theory proves true, but he learns that the crew quickly becomes bored with technological applications of their new-found mathematical prowess. Instead, they become increasingly interested in using it to develop their understanding of art and philosophy. These new understandings give the crew an unusual control over the physical universe and, by the end of the story, they have achieved god-like powers.

Two recurring mathematical themes in the story are Carnap-Ramsey sentences and Godel encoding.

The word "starbow" in the story's title is a word coined by one of the characters on the starship. It refers to the rainbow-like effect seen when stars are undergoing a relativistic Doppler effect.

In the novel the crew of the Constitution exact revenge on Earth by sending a beam of "strange" particles back to the home planet, causing all nuclear materials to melt down harmlessly, reducing the population to poverty in the absence of nuclear power. The last section relates the events surrounding the return of the crew's children to Earth.

Excerpt:

For it was clear to him that his success was already proved. What could be surer evidence of it than what had happened ten years ago? The "incident of next week" was as dramatic and complete as anyone could wish. If its details were still indecipherable, largely because of the demolition of the existing technology structure it had brought about, its main features were obvious. The shower of heavy particles—baryon? perhaps even quarks?—had drenched the Earth. The source had been traced to a point in the heavens identical with that plotted for the Constitution.

Also there were the messages received; taken together, there was no doubt that the astronauts had developed knowledge so far in advance of anything on Earth that, from two light-years out, they could impose their will on the human race. They had done it. In one downpour of particles, the entire military-industrial complex of the planet was put out of action.

How? How? Ah, thought Knefhausen, with envy and pride, that was the question. One could not know. All that was known was that every nuclear device—bomb, power plant, hospital radiation source or stockpile—had simultaneously soaked up the stream of particles and at that moment ceased to exist as a source of nuclear energy. It was not rapid and catastrophic, like a bomb. It was slow and long-lasting. The uranium and the plutonium simply melted, in the long, continuous reaction that was still bubbling away in the seething lava lakes where the silo had stood and the nuclear power plants had generated electricity. Little radiation was released, but a good deal of heat.