Is there an equivalent of computation of physical processes in nature? I was watching a waterfall in the Austrian Alps. There were thousands of water droplets falling down, splattering on the stones below. I thought - how does nature find out so quickly where each droplet of water should go?
To find out what happens to a falling droplet of water, one can use the laws of motion and calculate the trajectory. To calculate, one needs some amount of time, some machine (the brain, computer) and some energy to feed the machine.
Does it make sense to ask what is the equivalent of this computation in nature? How does nature find out so quickly how things should move? More generally, to find out how anything should happen?
Where is the "calculation" in nature? There's no room or energy for a machine in the particles that make up things.
 A: This is a tricky question.  On one hand, its an important philosophical question about science.  On the other hand, the question is mu.  This is a term the Zen Buddhist masters will use in response to a question that should be unasked, for all possible answers lead to suffering!

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*One major challenge here is defining "computation."  If you use the most prevalent concept I know if, which is the computation associated with Turing machines, then we have a distinct difference between nature and computation.  As best as we understand it, nature is a continuous process, operating over real numbers (if one can say anything about what it is operating on at all!).  Turing Machines operate over integers.  This means every analogy for computation is going to be pretty strained.

*You mention the computations being fast.  This only matters if your sense of time is not part of the system being computed.  As a real practical example, I have simulations of sensors which run at 1/20th "real time."  As far as the sensor and the algorithms behind it are aware, they are running 'at real time' because their sense of time is tied to the rate I compute the simulation.  If the universe is computing, we can't say anything about how "long" it is taking.  Indeed, it could be tricky to even come up with a meaningful unit to measure it in!

*When we compute how a raindrop will fall, we don't make a raindrop fall.  No raindrops fall in our computation.  We have numbers that are symmetric with how raindrops fall.  Nature is not obliged to operate on such terms.  It simply does.

That all being said, you may find Rule 110 quite fascinating.

Rule 110 is one of Stephen Wolfram's 256 elementary cellular automaton, which are numbered based on their rules.  None of these rules for computing the evolution of the system require anything more than looking at the current cell, the one to its left, and the one to its right.  They are numbered based on how they compute the next value, using a binary encoding.
Rule 110 also happens to be Turing Complete [Complex Systems 15 (2004) 1-40].  It can compute the result of any computer algorithm in existence.
Now we are yet to find any instances of rule 110 in nature.  But cellular automata do exist in nature, such as the Rule 30 pattern of the conus textile snail

So while we may not be able to speak to how nature computes what it does, we can speak to how she might compute.
A: In addition to Oбжорoв's answer, I think two aspects are worth to add:

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*I understand your question such that a calculation needed to be necessary for the individual pieces of the world to function properly. This is in fact just exactly the opposite to what science is. Science is a way for humans to try to understand observed natural processes and, in a next step, to predict the natural flow of events into the future. In this context, the calculations are a well-suited way to apply our understanding of natural phenomena, which we captured in theories, to a certain set of start conditions (e.g. where is the water droplet at the beginning, how large is it etc.). The result for a given time is a set of numbers, which represent our best-guess prediction. But at each step, we have to make simplifications, because nature in itself does not work like a computer (for example, the uncertainty principle describes that we cannot even measure the complete set of conditions of some system for a single moment in time).


*A calculation in itself is meaningless. Doing pure math with numbers does not tell anything about nature. A step towards a meaning are units, which are appended to the numbers and give information about the context in which the numbers describe something. If I give you a sheet of paper with a calculation and the result of "5", it would not mean anything, if it called "5 meters" you would at least grasp that there is some distance involved. However, the point is that even with units, any calculation is meaningless without someone to interpret its result. And nature itself is not a living entity which needs to express itself via some language of mathematics, but, as others have said here, it just "works". It is comparable to our idea of a mechanism, which is forced to follow a predetermined path over time. This deterministic understanding of nature has its limits as soon as you approach quantum mechanical effects, and caused some headaches amongst scientists.
This is one of the main pillars of modern science, not to involve a meta-physical being into physical processes.
A: Nobody knows, and perhaps nobody will ever know.
The only things that we can possibly know about the universe are the things that we can somehow observe, directly or indirectly. There's no way to gain information about how the universe operates other than observing its operation and thinking of rules that seem to explain what we have observed. As you know, these rules are called "laws of physics".
Nobody has ever observed anything that seems to provide any information about the computational mechanisms of the universe. It's likely that nobody ever will make any such observations. As a result, we don't know, and may never know, what those computational mechanisms are, or even whether or not there are any computational mechanisms at all.
A: Nature doesn’t need to find out where the droplets need to go. They just go.
This is the kind of question that has slowed down science from the Greek time to the Middle Ages, after which people slowly started to realise that there is no teleological purpose in nature. As Nike suggests, nature just does it !
A: That is, in my opinion, a good question. Re-phrased, "Does a physical process constitute a computation?"
The answer depends on the meaning of "computation".  Most often, computation is considered to be done by a physical system that represents the problem being solved or represents another system that is being simulated.
For example, one type of analog computer employs the fact that it is possible to construct an electronic circuit whose behavior precisely emulates an entirely different physical system, in that the same equations can be used to describe both the electronic circuit and the other physical system. So, the electronic circuit can be used to "calculate" the behavior of the other physical system.
Alternatively, the other physical system can be used to "calculate" the behavior of the electronic circuit.  But in either case, the electronic circuit or the physical system simply does what it must do.  That is, an electronic circuit that is a random assortment of components could be said to compute something, even if that something is never constructed or even visualized.
In my opinion (and it's just my opinion), an analog computer - or any physical system for that matter - "computes itself".  It is the perfect model of itself.  IMHO, to insist that "computation" requires one system to model another is an artificial and misleading distinction.
A: You feel awed when you see so many droplets falling together, but it really is the sum of all the individual interactions of the droplets between themselves. If you saw only two droplets falling, maybe you wouldn't be so surprised, as the interaction is , relatively, simpler . It's the same as millions of microchips in computers working together to show you this answer.
Also the properties of objects that are more than the sum of it's parts is called emergence.
A: I'm far from the best computer scientist to answer this, but I know that this is an important question (sidenote: I'm not sure about the "where does this computation take place" aspect of the question, but certainly the "how does nature compute" in the sense of "what do we know about the kinds of computations nature performs"):

*

*A physical system, by evolving, processes and reveals information.

*There are fundamental physical laws governing how information can be processed and revealed.

*These laws may include computational complexity limitations on nature itself (so to speak).

So far, this program has succeeded in the area of (quantum) information theory: we have a number of laws, such as no faster-than-light communication or the Bekenstein bound or even the second law of thermodynamics, that limit nature's ability to store, transmit, or process information. (Important: for us to accept these laws, we do not need to anthropomorphize nature as "computing" solutions using a slide rule and notebook or something; we just say that, however nature works, she seems to have information-processing limitations.)

Scott Aaaronson's article NP-complete Problems and Physical Reality reviews the natural line of thinking:

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*Things that happen in the world (like water droplets falling) seem like they would be very difficult for us to simulate computationally.


*Therefore, perhaps by setting up initial conditions, we can trick nature into "computing" the answers to hard problems.
So often people think of this and get excited about soap bubbles, or slime mold for a biology example. But we have failed to find evidence that this is possible. In other words, we have failed to falsify the hypothesis that natural processes cannot "efficiently" compute answers to "difficult" problems, for certain formalization of these terms.
Aaronson writes,

Can NP-complete problems be solved in polynomial time using the resources of the physical universe? I will argue that studying this question can yield new insights, not just about computer science but about physics as well.  More controversially, I will also argue that a negative answer might
eventually attain the same status as (say) the Second Law of Thermodynamics, or the impossibility of superluminal signalling.  In other words, while experiment will always be the last appeal, the presumed intractability of NP-complete problems might be taken as a useful constraint in the search for new physical theories.

A: I'd like to elaborate on and discuss the core idea of Cort Ammon's answer.
Nature really resembles a cellular automaton, with a few bells and whistles: In principle each (vaguely defined, fuzzy) space time "voxel" holds a state vector and only interacts with its immediate neighbors. In particular, modern physics did away with the implicit idea of immediate action at a distance: There is no Fernwirkung (action at a distance)2. These neighborly interactions are comparatively simple. The macroscopic "physical laws" we derive — like F = m*a or the conservation of momentum and energy — are abstract descriptions of phenomena emerging from these interactions in large collectives.1 They follow immediately from the underlying local interactions, much like embryo development or plant growth is mostly the result of local "decisions", which can be very nicely procedurally emulated.
A caveat: Given that computers are the reigning paradigm of our times, it is perhaps not so surprising that we see nature as a large computer simulation. When clocks were hip and fancy, people invariably thought of the universe as a clockwork. When gods and magic were hip, people invariably thought that deities and magic were running the place. Elon Musk's idea that we are living in a simulation is therefore probably as informed by our times as the caveman's notion of a Thunder God was informed by his (which I find equally consoling and disconcerting). Plato rules!

1 There is a rebellious group of physicists arguing that the substrate from which the observed phenomena emerge is of secondary importance, and that the ongoing, obsessive attempts to penetrate smaller and smaller structures is a dead end, partly because the results would be irrelevant (much like particular fluid molecules do not change fluid dynamics). See Robert B. Laughlin's A Different Universe (a somewhat unfavorable review can be found at https://physicstoday.scitation.org/doi/full/10.1063/1.2138425).

2 The "spooky" one does not really qualify. Which action?
A: Most answers so far rest on significant misconceptions about the nature of computation. I'll try to address these from a mathematical / computational perspective, and then we can come back to physics at the end.
Misconception: computation implies purpose or agency
Several answers include a line like "things happen because they happen", which I read as an abbreviation for "computation implies purpose or agency, which the universe does not have".
Computation, as studied by computer scientists and mathematicians, implies nothing of the sort. Computation is simply the processing of information according to determined (but not necessarily deterministic) rules. For example, the cellular automata mentioned in Cort Ammon's answer are computational processes, which are fully characterized by their mathematical definitions, independently of any electronic or other physical implementation.
Misconception: computation cannot handle infinite or continuum-sized structures
People who have worked with computer algebra systems may remark that (physical) computers can handle exact symbolic manipulations involving $e$, $\pi$ etc. But there's a far more literal sense in which computation using real numbers and their relatives is meaningful.
Think of computation on bit streams: you're provided a stream of bits as input, and asked to produce a stream of bits as output, where each new bit of output is determined by some computation on the input bits that you've received so far. Since at each stage you only need a finite amount of input to produce a finite amount of output, you can do this using "ordinary" finite computation. But the result looks like you're computing with an infinite amount of information: for example, you can add, multiply etc. arbitrary real numbers in this manner. This approach is called Type Two Effectivity, and it's been an active area of research for decades now.
The same idea applies to all the structures and processes that we care about "in applications": differential equations, stochastic processes etc. are all essentially computational in nature, which is what allows us to solve them in the first place. Even apparently discontinuous phenomena such as measures and distributions are more properly defined in terms of continuous maps between "hyper"spaces of open, compact, etc. subspaces, and can therefore be handled computationally.
A common philosophical objection here is that any finite computation can only handle finite approximations to these infinite objects. But I think most people working in this area would respond that if every infinite object we care about can be presented as an endless, converging sequence of finite approximations, and our finite algorithms can transform these into other sequences which converge to the correct output, then whether we're "actually" computing with infinite objects becomes a matter of mere semantics. For all practical purposes, we are.
Conclusion
The scope of "computational processes" is far broader than is often imagined. If a process is well-defined, such that its behaviour could be tested against an appropriate model (even if that model is not known) then it can be interpreted as a computational process. This includes processes with parameters, and even entire families of processes where the parameters take on every possible value in some domain.
It does not include so-called "supertasks", such as "toggling a binary switch infinitely many times then checking its state at the end", precisely because their behaviour is not well-defined. But it does include "continuum-sized" processes such as differential equations, as long as their behaviour is not so irregular that it simulates a supertask. In particular, it includes any process that can be modelled and tested / predicted in principle, even if in practice the appropriate model or tests cannot be found or carried out.
A: You are supposing that nature it's like a "rational agent" that "does something", but this can be just a projection of our human rationality to the outside world, by ourself generated on a base that we don't know (see Maya, concept discussed by Schopenhauer among the others)
In a way, nothing is a priori calculated (see how the time is relevant?) and quantum mechanics seems to say also that nothing is written. So things happen because they happen and trying to catch their reason to be it's like trying to hold their substantiality, but this is thin to handle, hard to catch like the wind. Physics doesn't answer this kind of questions, simply because it can't, but at the same time their existence is the reason why physics exists; a good circularity!
Maybe just from the chaos of the fields that constitute the space and time emerges something (see emergence), the nature of which is and will always be unthinkable.
Maybe everything that is (what does it mean to be?), is just a manifestation of these fields inside themselves, but this moves the problem before, because what the nature of these fields is, that's an unkown.
The question is also strictly related to the nature of time itself, this perpetual entity, always moving, that regulates the order in which phenomena seems to appear.
These questions are old as the man is and they are the reason why physics exists. At the same time is funny to think they will never get an answer, because are wrong from the start; they led us here, in this epoch in which the machine that you cited exists and in which quantum field theory is present.
But about understanding what actually is what are we talking about, we are at the same place of thousands of years ago. Maybe even more far, if we have the arrogance to think to have really understood something.
If you are interested in the question all the great thinkers have thought about that; take the ancient greeks or Nietzsche, Heidegger (he wrote "Being and Time - Sein und Zeit") or all the oriental thinkers; you can read "The Tao of Physics" by Fritjof Capra, about the relation between buddhism and quantum physics.
A: This is a remarkable question that many great thinkers have thought in one way or another. I think there are several ways you can view this question, one is regarding the continuum character of nature and the second one is viewing the universe as a computation rather than as a computer.

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*The first view is perhaps the most straight forward, but it is very difficult nevertheless. And it is the fact that we regard nature as continuum. So how can we reconcile that we can subdivide space and matter infinitely, and the fact that it would take a computer infinite amount of time to perform such calculation. Thereby if we assume that nature has at most the power of a computer(Church-Turing Thesis), how is it that it can compute this results in a finite time ?


*The second view is that we better see nature as a computation, not as a computer. A computation, can be any simple rule(simple to describe), however its output can be extremely complicated. In fact it can be as complicated as any other computation(Turing Universality). In this sense nature is the data structure constantly updated by this running computation, but nature is not really the hardware where this computation is running. This misconception runs because of the contemporary idea that the universe is a simulation and that we are the program inside a computer of a supernatural entity. In this case, we would ask what is the clock-speed of this computer ?, or where is this data-structure stored ?. However this is not a scientific hypothesis, it is just a rephrase of the old tale of Plato's Cavern or the Jewish Golem.
Both of these points leave a lot of questions, but we don't really have a fundamental understanding on these kind of questions yet. However it is the case that nature is computing, and furthermore is constantly doing very difficult computations in various situations.
A: I'm reminded of what someone in Hollywood once said: "Everyone has at least one starring role in them: the one where they play themselves." I can make an entirely true-to-life movie of your life by following you around with a camera. Someone may ask, "How does David Meyer so accurately portray David Meyer in the movie?" In one sense, David Meyer cannot help but accurately portray David Meyer's life. What else could David Meyer do? [1]
If we define computation as the transformation of inputs at time $t_0$ to outputs at time $t_1 > t_0$ according to a finite list of rules in a finite time ($t_1 - t_0 < \infty$), then it is no surprise that a waterfall is good at computing the position of water drops that pass over a waterfall. The waterfall acts like a waterfall; how could it do anything else? The difficult trick is to get one system to compute the actions of another system. A computer is completely unlike a waterfall: the moisture content alone is a reliable distinguishing characteristic. So, it takes a lot of work to get a computer to simulate a waterfall by computing the positions of the water drops.
Another thing to consider: there is no "nature." At least, there does not seem to be any entity that encompasses all of existence and has knowledge of all the universes contents and directs their motion. Every particle, every infinitesimal piece of field, every bit of spacetime, acts independently and only responds to the immediately surrounding area. If you flick an electron, the effect on the electric field only propagates at the speed of light; it would take more than a second for anything on the Moon to notice that the electron has changed position. The technical name for this is "locality." Every individual thing in the universe only responds to its immediate surroundings. There is no need for anything to keep track of the overall picture. The universe is built up of local interactions. Every drop of water gets jostled by the other drops and the river bed (or lack thereof when it goes over the falls). The waterfall is the sum total of all these microscopic interactions.
[1] Malkovich, Malkovich, Malkovich ...
