Machine learning, incomputably large data sets, and the string landscape

Transcription

1 Machine learning, incomputably large data sets, and the string landscape 2017 Workshop on Data Science and String Theory Northeastern University December 1, 2017 Washington (Wati) Taylor, MIT Based in part on arxiv: , , , written in collaboration with Y. Wang W. Taylor Machine learning and the string landscape 1 / 15

2 Outline 1. Comments on problems for machine learning and in the landscape 2. The skeleton of the F-theory landscape: a very large graph W. Taylor Machine learning and the string landscape 2 / 15

3 Some comments on problems for machine learning and in the landscape Some personal reflections 30 years ago I worked for a company named Thinking Machines. Goal: Build a machine that would be proud of us (D. Hillis) Company built a machine with 64k parallel processes; Gflops Richard Feynman designed the communication network/routing system W. Taylor Machine learning and the string landscape 3 / 15

4 As a side project, I worked on evolving neural network like systems; ultimate goal: play go, learning from scratch Difficult problem, not enough computer power; I went to grad school to learn string theory. Stopped following AI in any detail. AlphaGo Zero astonished me! 1. Computers are much faster, clearly But also, I suspect: 2. Humans are even less effective at playing go than I thought. A bit like physics research: we have some vague idea of our long term strategy, and some technical and analytic tools that we apply in a fairly limited human fashion... W. Taylor Machine learning and the string landscape 4 / 15

5 As a side project, I worked on evolving neural network like systems; ultimate goal: play go, learning from scratch Difficult problem, not enough computer power; I went to grad school to learn string theory. Stopped following AI in any detail. AlphaGo Zero astonished me! 1. Computers are much faster, clearly But also, I suspect: 2. Humans are even less effective at playing go than I thought. A bit like physics research: we have some vague idea of our long term strategy, and some technical and analytic tools that we apply in a fairly limited human fashion... W. Taylor Machine learning and the string landscape 4 / 15

6 As a side project, I worked on evolving neural network like systems; ultimate goal: play go, learning from scratch Difficult problem, not enough computer power; I went to grad school to learn string theory. Stopped following AI in any detail. AlphaGo Zero astonished me! 1. Computers are much faster, clearly But also, I suspect: 2. Humans are even less effective at playing go than I thought. A bit like physics research: we have some vague idea of our long term strategy, and some technical and analytic tools that we apply in a fairly limited human fashion... W. Taylor Machine learning and the string landscape 4 / 15

7 What kinds of problems is machine learning currently best at? Classification problems: image/face recognition, speech recognition,... Generally large data field, categorization into finite classes Optimization problems: pretty good solutions in high-dimensional spaces with reasonably smooth local structure W. Taylor Machine learning and the string landscape 5 / 15

8 Problems relevant for the string landscape Types of problems with issues that go beyond current machine learning 1. We don t know what we re doing, don t have a framework Fundamental background-independent formulation of ST/QG Nonperturbative definition of F-theory 2. We lack mathematical frameworks, even for semi-understood problems Classify non-geometric flux vacua Describe G 2 manifolds with nonabelian symmetries 3. Many things we don t know how to compute Classify Calabi-Yau threefolds (can t even prove finite number) Compute superpotential and low-energy EFT for F-theory (nonpert.) 4. We don t know what physics we are looking for SUSY/SUSY breaking? GUT SU(5)? SO(10)? E 6, E 8? non-higgsable SU(3) x SU(2) x U(1)? I believe machine learning will not solve any of these problems anytime soon. W. Taylor Machine learning and the string landscape 6 / 15

9 Problems relevant for the string landscape Types of problems with issues that go beyond current machine learning 1. We don t know what we re doing, don t have a framework Fundamental background-independent formulation of ST/QG Nonperturbative definition of F-theory 2. We lack mathematical frameworks, even for semi-understood problems Classify non-geometric flux vacua Describe G 2 manifolds with nonabelian symmetries 3. Many things we don t know how to compute Classify Calabi-Yau threefolds (can t even prove finite number) Compute superpotential and low-energy EFT for F-theory (nonpert.) 4. We don t know what physics we are looking for SUSY/SUSY breaking? GUT SU(5)? SO(10)? E 6, E 8? non-higgsable SU(3) x SU(2) x U(1)? I believe machine learning will not solve any of these problems anytime soon. W. Taylor Machine learning and the string landscape 6 / 15

10 Problems relevant for the string landscape Types of problems with issues that go beyond current machine learning 1. We don t know what we re doing, don t have a framework Fundamental background-independent formulation of ST/QG Nonperturbative definition of F-theory 2. We lack mathematical frameworks, even for semi-understood problems Classify non-geometric flux vacua Describe G 2 manifolds with nonabelian symmetries 3. Many things we don t know how to compute Classify Calabi-Yau threefolds (can t even prove finite number) Compute superpotential and low-energy EFT for F-theory (nonpert.) 4. We don t know what physics we are looking for SUSY/SUSY breaking? GUT SU(5)? SO(10)? E 6, E 8? non-higgsable SU(3) x SU(2) x U(1)? I believe machine learning will not solve any of these problems anytime soon. W. Taylor Machine learning and the string landscape 6 / 15

11 Problems relevant for the string landscape Types of problems with issues that go beyond current machine learning 1. We don t know what we re doing, don t have a framework Fundamental background-independent formulation of ST/QG Nonperturbative definition of F-theory 2. We lack mathematical frameworks, even for semi-understood problems Classify non-geometric flux vacua Describe G 2 manifolds with nonabelian symmetries 3. Many things we don t know how to compute Classify Calabi-Yau threefolds (can t even prove finite number) Compute superpotential and low-energy EFT for F-theory (nonpert.) 4. We don t know what physics we are looking for SUSY/SUSY breaking? GUT SU(5)? SO(10)? E 6, E 8? non-higgsable SU(3) x SU(2) x U(1)? I believe machine learning will not solve any of these problems anytime soon. W. Taylor Machine learning and the string landscape 6 / 15

12 Problems relevant for the string landscape Types of problems with issues that go beyond current machine learning 1. We don t know what we re doing, don t have a framework Fundamental background-independent formulation of ST/QG Nonperturbative definition of F-theory 2. We lack mathematical frameworks, even for semi-understood problems Classify non-geometric flux vacua Describe G 2 manifolds with nonabelian symmetries 3. Many things we don t know how to compute Classify Calabi-Yau threefolds (can t even prove finite number) Compute superpotential and low-energy EFT for F-theory (nonpert.) 4. We don t know what physics we are looking for SUSY/SUSY breaking? GUT SU(5)? SO(10)? E 6, E 8? non-higgsable SU(3) x SU(2) x U(1)? I believe machine learning will not solve any of these problems anytime soon. W. Taylor Machine learning and the string landscape 6 / 15

13 Nonetheless, many possible applications of large scale computation/ml Notwithstanding the preceding issues, we are getting a good enough handle on the landscape that we are beginning to have large datasets. Some (see below) are too large to enumerate (e.g. more elements than particles in the observable universe) Many difficult computational problems Need methods for dealing with large, poorly understood datasets (ML?) Identify patterns, suggest hypotheses for theoretical advances Look for elements combining desired features (e.g. SM gauge group, matter content, Yukawas, etc.; cf. Ruehle talk) Statistics and structure of BIG datasets W. Taylor Machine learning and the string landscape 7 / 15

14 Nonetheless, many possible applications of large scale computation/ml Notwithstanding the preceding issues, we are getting a good enough handle on the landscape that we are beginning to have large datasets. Some (see below) are too large to enumerate (e.g. more elements than particles in the observable universe) Many difficult computational problems Need methods for dealing with large, poorly understood datasets (ML?) Identify patterns, suggest hypotheses for theoretical advances Look for elements combining desired features (e.g. SM gauge group, matter content, Yukawas, etc.; cf. Ruehle talk) Statistics and structure of BIG datasets W. Taylor Machine learning and the string landscape 7 / 15

15 Some types of computationally challenging problems in the landscape: (examples of current projects) Large computations: 473,800,776 reflexive 4D polytopes Calabi-Yau 3-folds (KS database) e.g. Analyze elliptic fibration structure (w/ Y. Huang) Complex algebra: Many computations require solving large systems of algebraic equations easy to go beyond capacity of existing computational AG (e.g. Groebner basis) e.g. Analyze Weierstrass tunings (w/ N. Raghuram) Diophantine equations: Solving equations over integers analytically, computationally difficult e.g. Solve abelian 6D anomaly equations (w/ A. Turner) 6a b = q n q q 2 ; 3b b = q n q q 4 ; W. Taylor Machine learning and the string landscape 8 / 15

16 Some types of computationally challenging problems in the landscape: (examples of current projects) Large computations: 473,800,776 reflexive 4D polytopes Calabi-Yau 3-folds (KS database) e.g. Analyze elliptic fibration structure (w/ Y. Huang) Complex algebra: Many computations require solving large systems of algebraic equations easy to go beyond capacity of existing computational AG (e.g. Groebner basis) e.g. Analyze Weierstrass tunings (w/ N. Raghuram) Diophantine equations: Solving equations over integers analytically, computationally difficult e.g. Solve abelian 6D anomaly equations (w/ A. Turner) 6a b = q n q q 2 ; 3b b = q n q q 4 ; W. Taylor Machine learning and the string landscape 8 / 15

17 Some types of computationally challenging problems in the landscape: (examples of current projects) Large computations: 473,800,776 reflexive 4D polytopes Calabi-Yau 3-folds (KS database) e.g. Analyze elliptic fibration structure (w/ Y. Huang) Complex algebra: Many computations require solving large systems of algebraic equations easy to go beyond capacity of existing computational AG (e.g. Groebner basis) e.g. Analyze Weierstrass tunings (w/ N. Raghuram) Diophantine equations: Solving equations over integers analytically, computationally difficult e.g. Solve abelian 6D anomaly equations (w/ A. Turner) 6a b = q n q q 2 ; 3b b = q n q q 4 ; W. Taylor Machine learning and the string landscape 8 / 15

18 Some types of computationally challenging problems in the landscape: (examples of current projects) Large computations: 473,800,776 reflexive 4D polytopes Calabi-Yau 3-folds (KS database) e.g. Analyze elliptic fibration structure (w/ Y. Huang) Complex algebra: Many computations require solving large systems of algebraic equations easy to go beyond capacity of existing computational AG (e.g. Groebner basis) e.g. Analyze Weierstrass tunings (w/ N. Raghuram) Diophantine equations: Solving equations over integers analytically, computationally difficult e.g. Solve abelian 6D anomaly equations (w/ A. Turner) 6a b = q n q q 2 ; 3b b = q n q q 4 ; W. Taylor Machine learning and the string landscape 8 / 15

19 An example of a BIG dataset: The skeleton of the F-theory landscape This is a well defined graph, containing nodes and edges Contains > nodes Nodes = smooth toric threefold bases that support an elliptic CY4 Edges = transitions from blowing up toric points or curves Describes a class of geometries for 4D N = 1 F-theory vacua hypothesized to describe core/outline of F-theory landscape WT/Wang: Monte Carlo on set connected to P 3 : geometries (w/o codim. 2 (4, 6) curves) Halverson/Long/Sung: Systematic blow-ups of weak Fano 3-folds: (w/c2-46) WT/Wang: One-way MC on set w/c2-46: > geometries W. Taylor Machine learning and the string landscape 9 / 15

20 An example of a BIG dataset: The skeleton of the F-theory landscape This is a well defined graph, containing nodes and edges Contains > nodes Nodes = smooth toric threefold bases that support an elliptic CY4 Edges = transitions from blowing up toric points or curves Describes a class of geometries for 4D N = 1 F-theory vacua hypothesized to describe core/outline of F-theory landscape WT/Wang: Monte Carlo on set connected to P 3 : geometries (w/o codim. 2 (4, 6) curves) Halverson/Long/Sung: Systematic blow-ups of weak Fano 3-folds: (w/c2-46) WT/Wang: One-way MC on set w/c2-46: > geometries W. Taylor Machine learning and the string landscape 9 / 15

21 An example of a BIG dataset: The skeleton of the F-theory landscape This is a well defined graph, containing nodes and edges Contains > nodes Nodes = smooth toric threefold bases that support an elliptic CY4 Edges = transitions from blowing up toric points or curves Describes a class of geometries for 4D N = 1 F-theory vacua hypothesized to describe core/outline of F-theory landscape WT/Wang: Monte Carlo on set connected to P 3 : geometries (w/o codim. 2 (4, 6) curves) Halverson/Long/Sung: Systematic blow-ups of weak Fano 3-folds: (w/c2-46) WT/Wang: One-way MC on set w/c2-46: > geometries W. Taylor Machine learning and the string landscape 9 / 15

22 Definition of F-theory skeleton graph: nodes Toric threefold defined by rays and cone structure in N = Z 3 Rays v i Z 3, Edges e ij = (v i, v j ), Cones σ ijk = (v i, v j, v k ) Edges and cones define a triangulation of projection on sphere S 2 Defines a complex threefold, (C ) 3 + toric divisors D i (from v i ), curves (from e ij ), points (from σ ijk ) Smooth if each σ ijk has unit volume Example: complex projective space CP 3 {v i } = {(1, 0, 0), (0, 1, 0), (0, 0, 1), ( 1, 1, 1)}; D i are vanishing loci of homogeneous coordinates [x, y, z, w] W. Taylor Machine learning and the string landscape 10 / 15

23 Definition of F-theory skeleton graph: nodes Toric threefold defined by rays and cone structure in N = Z 3 Rays v i Z 3, Edges e ij = (v i, v j ), Cones σ ijk = (v i, v j, v k ) Edges and cones define a triangulation of projection on sphere S 2 Defines a complex threefold, (C ) 3 + toric divisors D i (from v i ), curves (from e ij ), points (from σ ijk ) Smooth if each σ ijk has unit volume Example: complex projective space CP 3 {v i } = {(1, 0, 0), (0, 1, 0), (0, 0, 1), ( 1, 1, 1)}; D i are vanishing loci of homogeneous coordinates [x, y, z, w] W. Taylor Machine learning and the string landscape 10 / 15

24 Definition of F-theory skeleton graph: nodes Toric threefold defined by rays and cone structure in N = Z 3 Rays v i Z 3, Edges e ij = (v i, v j ), Cones σ ijk = (v i, v j, v k ) Edges and cones define a triangulation of projection on sphere S 2 Defines a complex threefold, (C ) 3 + toric divisors D i (from v i ), curves (from e ij ), points (from σ ijk ) Smooth if each σ ijk has unit volume Example: complex projective space CP 3 {v i } = {(1, 0, 0), (0, 1, 0), (0, 0, 1), ( 1, 1, 1)}; D i are vanishing loci of homogeneous coordinates [x, y, z, w] W. Taylor Machine learning and the string landscape 10 / 15

25 Definition of F-theory skeleton graph: edges Edges connect the nodes (toric threefolds) by blowing up points, curves. Blowing up a point σ ijk : add new vertex v = v i + v j + v k Blowing up a curve e ij : add new vertex v = v i + v j So far, defines an infinite connected graph, starting with e.g. P 3 W. Taylor Machine learning and the string landscape 11 / 15

26 Definition of F-theory skeleton graph: edges Edges connect the nodes (toric threefolds) by blowing up points, curves. Blowing up a point σ ijk : add new vertex v = v i + v j + v k Blowing up a curve e ij : add new vertex v = v i + v j So far, defines an infinite connected graph, starting with e.g. P 3 W. Taylor Machine learning and the string landscape 11 / 15

27 Definition of F-theory skeleton graph: edges Edges connect the nodes (toric threefolds) by blowing up points, curves. Blowing up a point σ ijk : add new vertex v = v i + v j + v k Blowing up a curve e ij : add new vertex v = v i + v j So far, defines an infinite connected graph, starting with e.g. P 3 W. Taylor Machine learning and the string landscape 11 / 15

28 Definition of F-theory skeleton graph: edges Edges connect the nodes (toric threefolds) by blowing up points, curves. Blowing up a point σ ijk : add new vertex v = v i + v j + v k Blowing up a curve e ij : add new vertex v = v i + v j So far, defines an infinite connected graph, starting with e.g. P 3 W. Taylor Machine learning and the string landscape 11 / 15

29 Gauge groups and upper bounds on threefold geometries Elliptic Calabi-Yau fourfold over a base B defined by Weierstrass model y 2 = x 3 + fx + g f, g sections of line bundles O( 4K), O( 6K) On toric threefold base, f, g combinations of monomials in M 4, M 6 M k = {m N : m, v i k, i}. Orders of vanishing of f, g on divisor D i ord Di f = min m M4 (4 + m, v i ), ord Di g = min m M6 (6 + m, v i ) ord f, g, = 4f g 2 generic (non-higgsable) gauge groups. ord (f, g) > (4, 6) on codimension one (divisor) no CY resolution So analyzing M 4, M 6 gives gauge group on generic elliptic fibration, limiting bound on set of nodes Number of nodes is finite, since number of elliptic CY4 s finite (Di Cerbo, Svaldi 17) W. Taylor Machine learning and the string landscape 12 / 15

30 Gauge groups and upper bounds on threefold geometries Elliptic Calabi-Yau fourfold over a base B defined by Weierstrass model y 2 = x 3 + fx + g f, g sections of line bundles O( 4K), O( 6K) On toric threefold base, f, g combinations of monomials in M 4, M 6 M k = {m N : m, v i k, i}. Orders of vanishing of f, g on divisor D i ord Di f = min m M4 (4 + m, v i ), ord Di g = min m M6 (6 + m, v i ) ord f, g, = 4f g 2 generic (non-higgsable) gauge groups. ord (f, g) > (4, 6) on codimension one (divisor) no CY resolution So analyzing M 4, M 6 gives gauge group on generic elliptic fibration, limiting bound on set of nodes Number of nodes is finite, since number of elliptic CY4 s finite (Di Cerbo, Svaldi 17) W. Taylor Machine learning and the string landscape 12 / 15

31 The skeleton graph and the landscape: 6D analogue Similar story in 6D, but bases are toric surfaces. 61,539 smooth toric base surfaces (w/o c2-46 s) [Morrison/WT] h h 11 (blue = generic ecy3 over toric base, gray = full KS database) Fill range of (known) Calabi-Yaus. For all elliptic CY s need: non-toric (WT/Wang 15, h 2,1 > 150) all tunings of Weierstrass model over each base (Johnson/WT, large h 2,1 ) 6D story fairly well under control W. Taylor Machine learning and the string landscape 13 / 15

32 The skeleton graph and the landscape: 4D Expect that the graph of smooth toric threefold bases captures rough global structure of the landscape. For a complete understanding, need: Non-toric bases Tunings of Weierstrass Superpotential, fluxes, brane DOF, etc. Clarify physics of codimension two (4, 6) curves, other singularities First step: understand scope and features of 4D skeleton graph Enumerate nodes with rare features? (e.g. no c2-46, minimal,... ) Find nodes with specific combinations of features? Exploring this graph: see Long, Wang talks Physics: Standard model? Cosmology? Need more detailed analysis of fluxes, superpotential etc. W. Taylor Machine learning and the string landscape 14 / 15

33 The skeleton graph and the landscape: 4D Expect that the graph of smooth toric threefold bases captures rough global structure of the landscape. For a complete understanding, need: Non-toric bases Tunings of Weierstrass Superpotential, fluxes, brane DOF, etc. Clarify physics of codimension two (4, 6) curves, other singularities First step: understand scope and features of 4D skeleton graph Enumerate nodes with rare features? (e.g. no c2-46, minimal,... ) Find nodes with specific combinations of features? Exploring this graph: see Long, Wang talks Physics: Standard model? Cosmology? Need more detailed analysis of fluxes, superpotential etc. W. Taylor Machine learning and the string landscape 14 / 15

34 The skeleton graph and the landscape: 4D Expect that the graph of smooth toric threefold bases captures rough global structure of the landscape. For a complete understanding, need: Non-toric bases Tunings of Weierstrass Superpotential, fluxes, brane DOF, etc. Clarify physics of codimension two (4, 6) curves, other singularities First step: understand scope and features of 4D skeleton graph Enumerate nodes with rare features? (e.g. no c2-46, minimal,... ) Find nodes with specific combinations of features? Exploring this graph: see Long, Wang talks Physics: Standard model? Cosmology? Need more detailed analysis of fluxes, superpotential etc. W. Taylor Machine learning and the string landscape 14 / 15

35 Conclusions 1. While there are many hard physics problems in the landscape that go beyond current machine learning methods, we have reached the point where significant large datasets may be amenable to big data/ml methods, and may give physically significant results. 2. The skeleton graph of toric threefold bases for elliptic Calabi-Yau fourfolds forms a prototype of a model of the 4D N = 1 landscape that offers many interesting questions for investigation. W. Taylor Machine learning and the string landscape 15 / 15