[memetic status: stating directly despite it being a clear consequence of core AI risk knowledge because many people have "but nature will survive us" antibodies to other classes of doom and misapply them here.]
Unfortunately, no.[1]
Technically, “Nature”, meaning the fundamental physical laws, will continue. However, people usually mean forests, oceans, fungi, bacteria, and generally biological life when they say “nature”, and those would not have much chance competing against a misaligned superintelligence for resources like sunlight and atoms, which are useful to both biological and artificial systems.
There’s a thought that comforts many people when they imagine humanity going extinct due to a nuclear catastrophe or runaway global warming: Once the mushroom clouds or CO2 levels have settled, nature will reclaim the cities. Maybe mankind in our hubris will have wounded Mother Earth and paid the price ourselves, but she’ll recover in time, and she has all the time in the world.
AI is different. It would not simply destroy human civilization with brute force, leaving the flows of energy and other life-sustaining resources open for nature to make a resurgence. Instead, AI would still exist after wiping humans out, and feed on the same resources nature needs, but much more capably.
You can draw strong parallels to the way humanity has captured huge parts of the biosphere for ourselves. Except, in the case of AI, we’re the slow-moving process which is unable to keep up.
A misaligned superintelligence would have many cognitive superpowers, which include developing advanced technology. For almost any objective it might have, it would require basic physical resources, like atoms to construct things which further its goals, and energy (such as that from sunlight) to power those things. These resources are also essential to current life forms, and, just as humans drove so many species extinct by hunting or outcompeting them, AI could do the same to all life, and to the planet itself.
Planets are not a particularly efficient use of atoms for most goals, and many goals which an AI may arrive at can demand an unbounded amount of resources. For each square meter of usable surface, there are millions of tons of magma and other materials locked up. Rearranging these into a more efficient configuration could look like strip mining the entire planet and firing the extracted materials into space using self-replicating factories, and then using those materials to build megastructures in space to harness a large fraction of the sun’s output. Looking further out, the sun and other stars are themselves huge piles of resources spilling unused energy out into space, and no law of physics renders them invulnerable to sufficiently advanced technology.
Some time after a misaligned, optimizing AI wipes out humanity, it is likely that there will be no Earth and no biological life, but only a rapidly expanding sphere of darkness eating through the Milky Way as the AI reaches and extinguishes or envelops nearby stars.
This is generally considered a less comforting thought.
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It is not at all clear to me that most of the atoms in a planet could be harnessed for technological structures, or that doing so would be energy efficient. Most of the mass of an earthlike planet is iron, oxygen, silicon and magnesium, and while useful things can be made out of these elements, I would strongly worry that other elements that are needed also in those useful things will run out long before the planet has been disassembled. By historical precedent, I would think that an AI civilization on Earth will ultimately be able to use only a tiny fraction of the material in the planet, similarly to how only a very small fraction of a percent of the carbon in the planet is being used by the biosphere, in spite of biological evolution having optimized organisms for billions of years towards using all resources available for life.
The scenario of a swarm of intelligent drones eating up a galaxy and blotting out its stars I think can empirically be dismissed as very unlikely, because it would be visible over intergalactic distances. Unless we are the only civilization in the observable universe in the present epoch, we would see galaxies with dark spots or very strangely altered spectra somewhere. So this isn't happening anywhere.
There are probably some historical analogs for the scenario of a complete takeover, but they are very far in the past, and have had more complex outcomes than intelligent grey goo scenarios normally portray. One instance I can think of is the Great Oxygenation Event. I imagine an observer back then might have envisioned that the end result of the evolution of cyanobacteria doing oxygenic photosynthesis would be the oceans and lakes and rivers all being filled with green slime, with a toxic oxygen atmosphere killing off all other life. While indeed this prognosis would have been true to a first order approximation - green plants do dominate life on Earth today - the reality of what happened is infinitely more complex than this crude picture suggests. And even anaerobic organisms survive to this day in some niches.
The other historical precedent that comes to mind would be the evolution of organisms that use DNA to encode genetic information using the specific genetic code that is now universal to all life, in whatever pre-DNA world existed at the beginning of life. These seem to have indeed completely erased all other kinds of life (claims of a shadow biosphere of more primitive organisms are all dubious to my knowledge), but also have not resulted in a less complex world.