Use computers to discover the Theory of Everything.
(I am not a physicist, so what I say here is probably wrong or confused, but I am saying it anyway, so at least someone could explain me where exactly am I wrong. Or maybe someone can improve the idea to make it workable.)
As far as I know, (1) we assume that the laws of the universe are simple, (2) we already have equations for relativity, and (3) we already have equations for quantum physics. However, we don’t yet have equations for relativistic quantum physics. We also have (4) data about chemical properties of atoms, that is, about electron orbitals. I assume that for large enough atoms, relativistic effects influence the chemical properties of the atoms.
The plan is the following: Let the computer explore different sets of equations that are supposed to represent laws of physics. That is, take a set of equations, calculate what would be the chemical properties of atoms according to these equations, and compare with known data. Output those sets of equations that seem to fit. Create a smart generator for sets of equations, that would generate random simple equations, or iterate through the equation space starting with the simplest ones. Then apply a lot of computing power and see what happens.
Thanks! I will savor the warm feeling that I generated an idea in a field I didn’t study that the people who study the field also consider hopeful. :D
Okay, if someone understands the topic, could you please tell me what exactly is the problem; why this wasn’t already solved? -- Is the space of realistically simple equations still too large? Is it a mathematical problem to predict the chemical properties from the equations? Are we missing sufficiently precise data about the chemical properties of large atoms? Are the relativistic effects even for large atoms too small? Is there so much noise that you can actually generate too many different sets of equations fitting the data, with no quick way to filter out the more hopeful ones? All of the above? Something else?
Noise is certainly a problem, but the biggest problem for any sort of atomic modelling is that you quickly run into an n-body problem. Each one of of n electrons in an atom interacts with every other electron in that atom and so to describe the behavior of each electron you end up with a set of 70 something coupled differential equations. As a consequence, even if you just want a good approximation of the wavefunction, you have to search through a 3n dimensional Hilbert space and even with a preponderance of good experimental data there’s not really a good way to get around the curse of dimensionality.
Am I understanding the relevance of the curse of dimensionality to this correctly: Generally, our goal is to find a simple pattern in some high-dimensional data. However, due to the high dimensionality there are exponentially many possible data points and, practically, we can only observe a very small fraction of that, so curse is that we are often left with an immense list of candidates for the true pattern. All we can do is to limit this list of candidates with certain heuristic priors, for example that the true pattern is a smooth, compact manifold (that worked well e.g. for relativity and machine learning, but for example quantum mechanics looks more like that the true pattern is not smooth but consists of individual particles).
The true pattern (i.e. the many-particle wavefunction) is smooth. The issue is that the pattern depends on the positions of every electron in the atom. The variational principle gives us a measure of the goodness of the wavefunction, but it doesn’t give us a way to find consistent sets of positions. We have to rely on numerical methods to find self-consistent solutions for the set of differential equations, but it’s ludicrously expensive to try to sample the solution space given the dimensionality of that space.
It’s really difficult to solve large systems of coupled differential equations. You run into different issues depending on how you attempt to solve them. For most machine-learning type approaches, those issues manifest themselves via the curse of dimensionality.
I fail to see how this would be qualitatively different from how physics has always been done. We’ve always been using computers to generate new laws to fit observations, except in the past those computers have been our brains, and in the past half-century they’ve increasingly been our brains augmented with artificial computing machines.
Our current lack of progress in physics doesn’t stem from lack of ideas, or even lack of ability to come up with theoretical predictions. We have plenty of ideas. Our lack of progress stems from lack of experimental data. We have a large number of competing explanations and they all work in the same in the infrared limit (physicist-speak for ‘everyday low-energy conditions’) but they have subtle differences in the high-energy limit. Our two main routes to physical evidence have been particle physics measurements and cosmological data. We are not yet able to probe to high enough energies in particle physics to sort out the various theories, and we have far too many uncertainties in cosmological data to substantially help us out.
Maybe better AI in the future will help us with this, but it would have to be incredibly powerful AI.
We have a large number of competing explanations and they all work in the same in the infrared limit (physicist-speak for ‘everyday low-energy conditions’) but they have subtle differences in the high-energy limit.
What are you talking about? I don’t think that’s true at all.
Added: I suppose the parameters of the standard model are subtle difference in the high energy domain, but I don’t think that’s what you mean.
I really like this topic, and I’m really glad you brought it up; it probably even deserves its own post.
There are definitely some people who are trying this, or similar approaches. I’m pretty sure it’s one of the end goals of Stephen Wolfram’s “New Kind of Science” and the idea of high-throughput searching of data for latent mathematical structure is definitely in vogue in several sub-branches of physics.
With that being said, while the idea has caught people’s interest, it’s far from obvious that it will work. There are a number of difficulties and open questions, both with the general method and the specific instance you outline.
As far as I know, (1) we assume that the laws of the universe are simple
It’s not clear that this is a good assumption, and it’s not totally clear what exactly it means. There are a couple of difficulties:
a.) We know that the universe exhibits regular structure on some length and time scales, but that’s almost certainly a necessary condition for the evolution of complex life, and the anthropic principle makes that very weak evidence that the universe exhibits similar regular structure on all length/time/energy scales. While clever arguments based on the anthropic principle are typically profoundly unsatisfying, the larger point is that we don’t know that the universe is entirely regular/mathematical/computable and it’s not clear that we have strong evidence to believe it is. As an example, we know that a vanishingly small percentage of real numbers are computable; since there is no known mechanism restricting physical constants to computable numbers, it seems eminently possible that the values taken by physical constants such as the gravitational constant are not computable.
b.) It’s also not really clear what it means to say the laws of physics are simple. Simplicity is a somewhat complicated concept. We typically talk about simplicity in terms of Occam’s razor and/or various mathematical descriptions of it such as message length and Kolmorogov complexity. We typically say that complexity is related to how long it takes to explain something, but the length of an explanation depends strongly on the language used for that explanation. While the mathematics that we’ve developed can be used to state physical laws relatively concisely, that doesn’t tell us very much about the complexity of the laws of physics, since mathematics was often created for just that purpose. Even assuming that all of physics can be concisely described by the language of mathematics, I’m not sure that mathematics itself is “simple”.
c.) Simple laws don’t necessarily lead to simple results. If I have a set of 3 objects interacting with each other via a 1/r^2 force like gravity there is no general closed form solution for the positions of those objects at some time t in the future. I can simulate their behavior numerically, but numerical simulations are often computationally expensive, the
numeric results may depend on the initial conditions in unpredictable ways, and small deviations in the initial set up or rounding errors early in the simulation may result in wildly different outcomes. This difficulty strongly affects our ability to model the chemical properties of atoms. Since each electron orbiting the nucleus interacts with each other electron via the coulomb force, there is currently no way to exactly describe the behavior of the electrons even for a single isolated many-electron atom.
d.) A simple set of equations is insufficient to specify a physical system. Most physical laws are concerned with the time evolution of physical systems, and they typically rely on the initial state of the system as a set of input parameters. For many of the systems physics is still trying to understand, it isn’t possible to accurately determining what the correct input parameters are. Because of the potentially strong dependence on the initial conditions outlined in c.), it’s difficult to know whether a negative result for a given set of equations/parameters implies needing a new set of laws, or just slightly different initial conditions.
In short, your proposal is difficult to enact for similar reasons that Solomonoff induction is difficult. In general there is a vast hypothesis space that varies over both a potentially infinite set of equations and a large number of initial conditions. The computational cost of evaluating a given hypothesis is unknown and potentially very expensive. It has the added difficulty that even given an infinite set of initial hypotheses, the correct hypothesis may not be among them.
No, it is not believed that gravity has a measurable effect on chemistry. People have pretty much no idea what kind of experiments would be relevant to quantum gravity. Moreover, the predictions that QFT makes about chemistry are too hard. I don’t think it is possible with current computers to compute the spectrum of helium, let alone lithium. A quantum computer could do this, though.
I’m by no means a physicist, but isn’t special relativity, which is related to gravity/spacetime, able to cause magnetism? Couldn’t that account for a chemical effect?
The program described, Eureqa, is available online and had a free trial. I have spent a lot of time playing with it and trying it on different problems and datasets.
I don’t think it could learn a theory of everything with out a lot of human help to reduce some parts of the problem. E.g. “here are a few numbers as inputs, here is a number as an output. Fit it out only using basic mathematical functions.” But its a start.
Nitpick: we have equations for (special) relativistic quantum physics. Dirac was one of the pioneers, and the Standard Model for instance is a relativistic quantum field theory. I presume you mean general relativity (gravity) and quantum mechanics that is the problem.
(Douglas_Knight) Moreover, the predictions that QFT makes about chemistry are too hard. I don’t think it is possible with current computers to compute the spectrum of helium, let alone lithium. A quantum computer could do this, though.
In the spirit of what Viliam suggested, maybe you could do computational searches for tractable approximations to QFT for chemistry i.e. automatically find things like density functional theory. A problem there might be that you do not get any insight from the result, and you might end up overfitting.
Just happened across this article summary today about people using atomic spectra to look for evidence of dark matter. I don’t know that they’ve found anything yet, but it’s sort of neat how closely related your proposal here is to their research.
Use computers to discover the Theory of Everything.
(I am not a physicist, so what I say here is probably wrong or confused, but I am saying it anyway, so at least someone could explain me where exactly am I wrong. Or maybe someone can improve the idea to make it workable.)
As far as I know, (1) we assume that the laws of the universe are simple, (2) we already have equations for relativity, and (3) we already have equations for quantum physics. However, we don’t yet have equations for relativistic quantum physics. We also have (4) data about chemical properties of atoms, that is, about electron orbitals. I assume that for large enough atoms, relativistic effects influence the chemical properties of the atoms.
The plan is the following: Let the computer explore different sets of equations that are supposed to represent laws of physics. That is, take a set of equations, calculate what would be the chemical properties of atoms according to these equations, and compare with known data. Output those sets of equations that seem to fit. Create a smart generator for sets of equations, that would generate random simple equations, or iterate through the equation space starting with the simplest ones. Then apply a lot of computing power and see what happens.
(Inspired by: Einstein’s Speed, That Alien Message.)
People are already attempting that since 2009 or so:
https://scholar.google.com/scholar?cluster=11583184257062107912
https://scholar.google.com/scholar?cluster=4202198002835248331
(Click /Cited by \d+/ to go down the rabbit hole.)
Thanks! I will savor the warm feeling that I generated an idea in a field I didn’t study that the people who study the field also consider hopeful. :D
Okay, if someone understands the topic, could you please tell me what exactly is the problem; why this wasn’t already solved? -- Is the space of realistically simple equations still too large? Is it a mathematical problem to predict the chemical properties from the equations? Are we missing sufficiently precise data about the chemical properties of large atoms? Are the relativistic effects even for large atoms too small? Is there so much noise that you can actually generate too many different sets of equations fitting the data, with no quick way to filter out the more hopeful ones? All of the above? Something else?
Noise is certainly a problem, but the biggest problem for any sort of atomic modelling is that you quickly run into an n-body problem. Each one of of n electrons in an atom interacts with every other electron in that atom and so to describe the behavior of each electron you end up with a set of 70 something coupled differential equations. As a consequence, even if you just want a good approximation of the wavefunction, you have to search through a 3n dimensional Hilbert space and even with a preponderance of good experimental data there’s not really a good way to get around the curse of dimensionality.
Am I understanding the relevance of the curse of dimensionality to this correctly: Generally, our goal is to find a simple pattern in some high-dimensional data. However, due to the high dimensionality there are exponentially many possible data points and, practically, we can only observe a very small fraction of that, so curse is that we are often left with an immense list of candidates for the true pattern. All we can do is to limit this list of candidates with certain heuristic priors, for example that the true pattern is a smooth, compact manifold (that worked well e.g. for relativity and machine learning, but for example quantum mechanics looks more like that the true pattern is not smooth but consists of individual particles).
The true pattern (i.e. the many-particle wavefunction) is smooth. The issue is that the pattern depends on the positions of every electron in the atom. The variational principle gives us a measure of the goodness of the wavefunction, but it doesn’t give us a way to find consistent sets of positions. We have to rely on numerical methods to find self-consistent solutions for the set of differential equations, but it’s ludicrously expensive to try to sample the solution space given the dimensionality of that space.
It’s really difficult to solve large systems of coupled differential equations. You run into different issues depending on how you attempt to solve them. For most machine-learning type approaches, those issues manifest themselves via the curse of dimensionality.
I fail to see how this would be qualitatively different from how physics has always been done. We’ve always been using computers to generate new laws to fit observations, except in the past those computers have been our brains, and in the past half-century they’ve increasingly been our brains augmented with artificial computing machines.
Our current lack of progress in physics doesn’t stem from lack of ideas, or even lack of ability to come up with theoretical predictions. We have plenty of ideas. Our lack of progress stems from lack of experimental data. We have a large number of competing explanations and they all work in the same in the infrared limit (physicist-speak for ‘everyday low-energy conditions’) but they have subtle differences in the high-energy limit. Our two main routes to physical evidence have been particle physics measurements and cosmological data. We are not yet able to probe to high enough energies in particle physics to sort out the various theories, and we have far too many uncertainties in cosmological data to substantially help us out.
Maybe better AI in the future will help us with this, but it would have to be incredibly powerful AI.
What are you talking about? I don’t think that’s true at all.
Added: I suppose the parameters of the standard model are subtle difference in the high energy domain, but I don’t think that’s what you mean.
I really like this topic, and I’m really glad you brought it up; it probably even deserves its own post.
There are definitely some people who are trying this, or similar approaches. I’m pretty sure it’s one of the end goals of Stephen Wolfram’s “New Kind of Science” and the idea of high-throughput searching of data for latent mathematical structure is definitely in vogue in several sub-branches of physics.
With that being said, while the idea has caught people’s interest, it’s far from obvious that it will work. There are a number of difficulties and open questions, both with the general method and the specific instance you outline.
It’s not clear that this is a good assumption, and it’s not totally clear what exactly it means. There are a couple of difficulties:
a.) We know that the universe exhibits regular structure on some length and time scales, but that’s almost certainly a necessary condition for the evolution of complex life, and the anthropic principle makes that very weak evidence that the universe exhibits similar regular structure on all length/time/energy scales. While clever arguments based on the anthropic principle are typically profoundly unsatisfying, the larger point is that we don’t know that the universe is entirely regular/mathematical/computable and it’s not clear that we have strong evidence to believe it is. As an example, we know that a vanishingly small percentage of real numbers are computable; since there is no known mechanism restricting physical constants to computable numbers, it seems eminently possible that the values taken by physical constants such as the gravitational constant are not computable.
b.) It’s also not really clear what it means to say the laws of physics are simple. Simplicity is a somewhat complicated concept. We typically talk about simplicity in terms of Occam’s razor and/or various mathematical descriptions of it such as message length and Kolmorogov complexity. We typically say that complexity is related to how long it takes to explain something, but the length of an explanation depends strongly on the language used for that explanation. While the mathematics that we’ve developed can be used to state physical laws relatively concisely, that doesn’t tell us very much about the complexity of the laws of physics, since mathematics was often created for just that purpose. Even assuming that all of physics can be concisely described by the language of mathematics, I’m not sure that mathematics itself is “simple”.
c.) Simple laws don’t necessarily lead to simple results. If I have a set of 3 objects interacting with each other via a 1/r^2 force like gravity there is no general closed form solution for the positions of those objects at some time t in the future. I can simulate their behavior numerically, but numerical simulations are often computationally expensive, the numeric results may depend on the initial conditions in unpredictable ways, and small deviations in the initial set up or rounding errors early in the simulation may result in wildly different outcomes. This difficulty strongly affects our ability to model the chemical properties of atoms. Since each electron orbiting the nucleus interacts with each other electron via the coulomb force, there is currently no way to exactly describe the behavior of the electrons even for a single isolated many-electron atom.
d.) A simple set of equations is insufficient to specify a physical system. Most physical laws are concerned with the time evolution of physical systems, and they typically rely on the initial state of the system as a set of input parameters. For many of the systems physics is still trying to understand, it isn’t possible to accurately determining what the correct input parameters are. Because of the potentially strong dependence on the initial conditions outlined in c.), it’s difficult to know whether a negative result for a given set of equations/parameters implies needing a new set of laws, or just slightly different initial conditions.
In short, your proposal is difficult to enact for similar reasons that Solomonoff induction is difficult. In general there is a vast hypothesis space that varies over both a potentially infinite set of equations and a large number of initial conditions. The computational cost of evaluating a given hypothesis is unknown and potentially very expensive. It has the added difficulty that even given an infinite set of initial hypotheses, the correct hypothesis may not be among them.
No, it is not believed that gravity has a measurable effect on chemistry. People have pretty much no idea what kind of experiments would be relevant to quantum gravity. Moreover, the predictions that QFT makes about chemistry are too hard. I don’t think it is possible with current computers to compute the spectrum of helium, let alone lithium. A quantum computer could do this, though.
I’m by no means a physicist, but isn’t special relativity, which is related to gravity/spacetime, able to cause magnetism? Couldn’t that account for a chemical effect?
Yes, special relativity is very important. Indeed, I was speaking of QED, a quantum mechanical model that incorporates special relativity.
You’ll probably be interested in reading this: http://www.wired.com/2009/04/newtonai/ and this: http://www.radiolab.org/story/91712-limits-of-science/
The program described, Eureqa, is available online and had a free trial. I have spent a lot of time playing with it and trying it on different problems and datasets.
I don’t think it could learn a theory of everything with out a lot of human help to reduce some parts of the problem. E.g. “here are a few numbers as inputs, here is a number as an output. Fit it out only using basic mathematical functions.” But its a start.
Nitpick: we have equations for (special) relativistic quantum physics. Dirac was one of the pioneers, and the Standard Model for instance is a relativistic quantum field theory. I presume you mean general relativity (gravity) and quantum mechanics that is the problem.
In the spirit of what Viliam suggested, maybe you could do computational searches for tractable approximations to QFT for chemistry i.e. automatically find things like density functional theory. A problem there might be that you do not get any insight from the result, and you might end up overfitting.
You’re leaving out dark matter, dark energy, and the possibility of discovering additional weird and surprising factors.
Just happened across this article summary today about people using atomic spectra to look for evidence of dark matter. I don’t know that they’ve found anything yet, but it’s sort of neat how closely related your proposal here is to their research.