Machine learning for lattice theories Real-world lattices 2 - - PowerPoint PPT Presentation

Machine learning for lattice theories1

Michael S. Albergo, Gurtej Kanwar, Phiala E. Shanahan

1 Albergo, GK, Shanahan [PRD 100 (2019) 034515]

Deep Learning and Physics Kyoto, Japan (November 1, 2019) Center for Theoretical Physics, MIT

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Real-world lattices

Machine learning for lattice theories

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Real-world lattices Quantum field theories

Machine learning for lattice theories

Lattices in the real world

• Many materials have degrees of freedom pinned to a lattice structure

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[Ella Maru Studio] [Mazurenko et al. 1612.08436]

Lattices in the real world

• Thermodynamics describes collective behavior of many degrees of freedom
• At some temperature T, Boltzmann distribution over microstates

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with

Lattices in the real world

• Thermodynamics describes collective behavior of many degrees of freedom
• At some temperature T, Boltzmann distribution over microstates

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[ "Ising Model and Metropolis Algorithm", MathWorks Physics Team ]

with

Ising model has spin s = {↑,↓} per site, with energy penalty for neighboring spins differing. Typical microstates have patches of the same spin at some scale.

Lattices in the real world

• Derive thermodynamic observables by averaging microstates

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Partition function Boltzmann distribution total energy Helmholtz free energy total energy correlation function

. . . . . .

Lattices for quantum field theories

• Quantum-mechanical properties also computed as statistical expectation

values via Path Integral

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similar to partition function

Lattice Quantum Chromodynamics

• Predictions relevant to interpret upcoming

high-energy expts

○ Electron-Ion Collider will investigate detailed nuclear structure ○ Deep Underground Neutrino Expt requires nuclear cross sections with neutrinos

• Pen-and-paper methods fail, numerical evaluation of path integral req'd

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bnl.gov/eic dunescience.org [D. Leinweber, Visual QCD Archive]

So far! Hong-Ye's talk for holography ideas

Computational approach to lattice theories

• Partition functions and path integrals are typically intractable analytically
• Numerical approximation by Monte Carlo sampling
• Markov Chain Monte Carlo converges to samples from p(𝜚)

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sample integral according to estimate observables

. . .

approximately distributed ~ p(𝜚)

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lattice theories

Real-world lattices Quantum field theories

Machine learning for lattice theories

12

lattice theories

Real-world lattices Quantum field theories

numerical methods

– Thermodynamics – Collective phenomena – Spectrum – ...

Machine learning for lattice theories

13

lattice theories

Real-world lattices Quantum field theories

numerical methods

– Thermodynamics – Collective phenomena – Spectrum – ...

✘

Machine learning for lattice theories

hard to reach continuum limit / critical point in some theories

Machine learning for lattice theories

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lattice theories

Real-world lattices Quantum field theories

numerical methods

– Thermodynamics – Collective phenomena – Spectrum – ...

✘

+ ML

hard to reach continuum limit / critical point in some theories

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lattice theories

Real-world lattices Quantum field theories

numerical methods

– Thermodynamics – Collective phenomena – Spectrum – ...

✘

+ ML

Machine learning for lattice theories

Sampling using ML 2 Critical slowing down 1 Toy model results 3

Difficulties with Markov Chain Monte Carlo

• Need to wait for "burn-in period"
• Configurations close to each other on the chain will be correlated, so must

take many steps before drawing independent samples

• Burn-in and correlations both related to Markov chain "autocorrelation time"

→ smaller autocorrelation time means less computational cost!

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. . .

burn-in ~ p(𝜚) ~ p(𝜚) correlated typically quantify with integrated autocorrelation time:

Critical slowing down

• As params defining distribution approach

criticality, for Markov chains using local updates, autocorrelation time diverges

• Fitting 𝜐int to power law behavior gives

dynamical critical exponents

• Smaller dynamical critical exponent =

cheaper, closer approach to criticality

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continuum limit

Critical slowing down

• As params defining distribution approach

criticality, for Markov chains using local updates, autocorrelation time diverges

• Fitting 𝜐int to power law behavior gives

dynamical critical exponents

• Smaller dynamical critical exponent =

cheaper, closer approach to criticality

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continuum limit CSD also affects more realistic, complex models: ○ CPN-1 ○ O(N) ○ QCD ○ ...

[ALPHA collaboration 1009.5228] [Frick, et al. PRL 63, 2613] [Flynn, et al. 1504.06292]

CSD in scalar theory used in this work:

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lattice theories

Real-world lattices Quantum field theories

numerical methods

– Thermodynamics – Collective phenomena – Spectrum – ...

✘

+ ML

Machine learning for lattice theories

Sampling using ML 2 Critical slowing down 1 Toy model results 3

Sampling lattice configs

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likely likely unlikely

(log prob = 22) (log prob = 5) (log prob = -6107)

likely

(log prob = 25)

Sampling lattice configs ≅ generating images

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likely likely unlikely likely likely unlikely

[Karras, Lane, Aila / NVIDIA 1812.04948]

Unique features of the lattice sampling problem

✓

Probability density computable (up to normalization)

✓

Many symmetries in physics

○ Lattice symmetries like translation, rotation, and reflection ○ Per-site symmetries like negation

✘

High-dimensional (109 to 1012) samples

✘

Few (~1000) samples available ahead of time (fewer than # vars!)

○ Hard to use training paradigms that rely on existing samples from distribution

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Image generation via ML

1. Likelihood free methods:

E.g. Generative Adversarial Networks (GANs) ✘ Needs many real samples ✘ No associated likelihood for each produced sample

2. Autoencoding:

E.g. Variational Auto-Encoders (VAEs) ✔ Good for human interpretability ✘ Same issues as GANs

3. Normalizing flows:

Flow-based models learn a change-of-variables that transforms a known distribution to the desired distribution ✔ Exactly known likelihood for each sample ✔ Can be trained with samples from itself

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[Goodfellow et al. 1406.2661] [Kingma & Welling 1312.6114] [Rezende & Mohamed 1505.05770] [Shen & Liu 1612.05363]

Image generation via ML

1. Likelihood free methods:

E.g. Generative Adversarial Networks (GANs) ✘ Needs many real samples ✘ No associated likelihood for each produced sample

2. Autoencoding:

E.g. Variational Auto-Encoders (VAEs) ✔ Good for human interpretability ✘ Same issues as GANs

3. Normalizing flows:

Flow-based models learn a change-of-variables that transforms a known distribution to the desired distribution ✔ Exactly known likelihood for each sample ✔ Can be trained with samples from itself

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[Goodfellow et al. 1406.2661] [Kingma & Welling 1312.6114] [Rezende & Mohamed 1505.05770] [Shen & Liu 1612.05363]

Many related approaches

• Normalizing flows for many-body systems

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[Noé, Olsson, Köhler, Wu Science 365 (2019)

• Iss. 6457, 982]

[Liu, Qi, Meng, Fu 1610.03137] [Zhang, E, Wang 1809.10188]

• Continuous flows
• Self-Learning Monte Carlo

See talks by Junwei Liu, Lei Wang and Hong-Ye Hu

• Hamiltonian transforms

[Li, Dong, Zhang, Wang 1910.00024]

Using a change-of-variables, produce a distribution approximating what you want.

Flow-based generative models

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[Rezende & Mohamed 1505.05770]

Using a change-of-variables, produce a distribution approximating what you want.

Flow-based generative models

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Invertible & Tractable Jacobian Easily sampled Approximates desired dist.

[Rezende & Mohamed 1505.05770]

We chose real non-volume preserving (real NVP) flows for our work.

Flow-based generative models

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Easily sampled Approximates desired dist. Many simple layers composed to produce f

[Dinh et al. 1605.08803]

Invertible & Tractable Jacobian

We chose real non-volume preserving (real NVP) flows for our work.

Flow-based generative models

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Easily sampled Approximates desired dist.

[Dinh et al. 1605.08803]

Invertible & Tractable Jacobian

Real NVP coupling layer

1. Freeze 1/2 of the inputs, za 2. Feed frozen vars into neural networks s and t 3. Scale exp(-s) and offset -t applied to unfrozen, zb

• Simple inverse and Jacobian

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Application of gi

• 1

• Use known target probability density:
• For our application, train to minimize shifted KL divergence
• Can apply self-training: sampling model distribution p̃f(𝜚) to estimate loss

Loss function

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shift removes unknown normalization Z

Correcting for model error

• Known model and target densities, many options to correct for error
• We use MCMC with proposals from ML model (interoperable with standard

MC updates)

• Metropolis-Hastings step:

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ML model proposals Markov Chain

✘

model proposal, independent

• f previous sample

Overview of algorithm

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Parameterize flow using Real NVP coupling layers

Each layer contains arbitrary neural nets s and t

Overview of algorithm

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Parameterize flow using Real NVP coupling layers

Each layer contains arbitrary neural nets s and t

Training step

Draw samples from model Compute loss function Gradient descent

Desired accuracy?

Save trained model

Overview of algorithm

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Parameterize flow using Real NVP coupling layers

Each layer contains arbitrary neural nets s and t

Training step

Draw samples from model Compute loss function Gradient descent

Markov chain using samples from model

Desired accuracy?

Save trained model

generating samples is "embarrassingly parallel"

36

lattice theories

Real-world lattices Quantum field theories

numerical methods

– Thermodynamics – Collective phenomena – Spectrum – ...

✘

+ ML

Machine learning for lattice theories

Sampling using ML 2 Critical slowing down 1 Toy model results 3

• One real number 𝜚(x) ∊ (-∞,∞) per lattice site x (2D lattice)
• Action: relativistic scalar with quartic coupling

Toy model: scalar 𝜚4 lattice field theory

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• One real number 𝜚(x) ∊ (-∞,∞) per lattice site x (2D lattice)
• Action: relativistic scalar with quartic coupling
• 5 lattice sizes L2 = {62, 82, 102, 122, 142} with bare parameters tuned to follow

a line of constant physics (symmetric phase)

• HMC and local Metropolis compared against our ML method

Toy model: scalar 𝜚4 lattice field theory

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Samples from ML model vs standard algorithms

By eye, ML model produces varied samples and correlations at the right scale

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Toy model: scalar 𝜚4 lattice field theory

• Measured observables:

○ Correlation functions, relating to masses in the (discretized) quantum field theory ○ Response of the vacuum to an impulse (two-point susceptibility) ○ Energy measurement relating to Ising model microstate energy in a particular limit of 𝜚4 theory

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Comparing observables (1)

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Ising energy and two-point susceptibility agree.

Comparing observables (2)

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correlation falls off with separation in both directions on periodic lattice

Correlation functions and pole masses agree.

No critical slowing down in ML approach

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HMC Local ML models by spending time training up-front, autocorrelations are fixed during sampling

Moving towards QCD

• Developing the method to apply to lattice QCD, tackling difficulties with

scaling volume, # dimensions, and to gauge theories

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David Murphy Dan Hackett Denis Boyda

Convolutional architectures

• Work in progress using convolutional networks for s and t, natural due to

locality of physical distributions

• Conv + checkerboard mask = translational invariance (up to parity)
• Convolutions also make scaling physical volume easy

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Transfer trained net + 10 mins retraining 3.5 days training

14 x 14 20 x 20

Hierarchical models

• Multiple flows, refining lattice between each
• Intuition: build correlations at multiple scales efficiently
• Computational gains allow scaling ML method

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flow flow flow flow hier flow for 32x32 𝜚4

[Dinh, Sohl-Dickstein, Bengio 1605.08803] [Li & Wang PRL 121 (2018) 260601]

Towards 3D and 4D lattices

• No theoretical obstacle to moving from two dimensions to 3D and 4D
• Work in progress to implement 4D convolutions efficiently in PyTorch

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8x8x8 model easily trained to 30% acceptance

Towards gauge theories

• Real NVP only directly works on fields taking real values 𝜚(x) ∊ (-∞,∞)
• Gauge theories (and O(N), CPN-1, ...) have compact domains
• Early success applying stereographic projection + Real NVP in collaboration

with DeepMind team

• Applications to discrete models (Ising, Potts, etc.)?

○ Some recent ideas emerging [Ziegler & Rush 1901.10548]

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[Gemici, Rezende, Mohammed 1611.02304]

Towards gauge theories

• Real NVP only directly works on fields taking real values 𝜚(x) ∊ (-∞,∞)
• Gauge theories (and O(N), CPN-1, ...) have compact domains
• Early success applying stereographic projection + Real NVP in collaboration

with DeepMind team

• Applications to discrete models (Ising, Potts, etc.)?

○ Some recent ideas emerging [Ziegler & Rush 1901.10548]

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[Gemici, Rezende, Mohammed 1611.02304]

Thanks! Questions?

Backup slides

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• Prior distribution chosen to be uncorrelated Gaussian,

i.e. for each site x,

• Real NVP model:

○ 8-12 Real NVP coupling layers ○ Alternating checkerboard pattern for variable split ○ 2-6 fully connected layers with 100-1024 hidden units

• Trained using shifted KL loss with Adam optimizer

○ Models trained until 50% and 70% acceptance rate in ML MCMC

ML method for scalar lattice field theory

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. . .

Autocorrelation time

• Well-behaved Markov chains have a "mixing time" determining how many

updates required to burn in / decorrelate samples

○ Hard to compute directly except for very special chains ○ Dominated by the slowest mixing mode

• Practically useful alternative: integrated autocorrelation time for an observable

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two-point correlation separated by tau Markov chain steps