Simon Trebst. Quantitative Modeling of Complex Systems

Transcription

1 Quantitative Modeling of Complex Systems

2 Machine learning In computer science, machine learning is concerned with algorithms that allow for data analytics, most prominently dimensional reduction and feature extraction. Many computer programs/apps have machine learning algorithms built-in already. spam filters face recognition voice recognition

3 Machine learning Applications of machine learning techniques are booming and poised to enter our daily lives. digital assistants self-driving cars

4 Machines at play Machine learning techniques can make computers play. A computer at play is probably one of the most striking realization of artificial intelligence. 1996: G. Kasparow vs. IBM s deep blue 2016: L. Sedol vs. Google s AlphaGo

5 How do machines learn? How do machines learn? What is it that they can learn? Can we control what they learn? How can we benefit from machine learning? x1 x2 x3 x4 x5 biological neural network artificial neural network

6 Time Title Speaker Institute 09:00 Intro Simon Trebst 09:15 Quantum Machine Learning Simon Trebst Theoretical Physics 09:45 kerndeepstacknet: An R package for tuning kernel deep stacking networks Thomas Welchowski and Matthias Schmid IMBIE, Bonn 10:15 coffee break 10:45 Applying Machine Learning to Text Mining Jürgen Hermes Department of Linguistics 11:15 Machine learning and the social sciences Jörn Grahl Digital Transformation and Value Creation 11:45 lunch break 13:15 Compressed Sensing for Sparse and Low-Rank Models David Gross Theoretical Physics 13:45 Sparse PCA and convex optimization Frank Vallentin Mathematical Institute 14:15 coffee break 14:45 Reconstruction of Biological Networks Achim Tresch Bioinformatics and Computational Biology 15:15 Illuminating Genetic Networks with Random Forest Andreas Beyer CECAD 15:45 End

7 Machine learning quantum phases of matter Quantum Machine Learning Machine Learning Day Cologne, April 2017 Simon Trebst Institute for Theoretical Physics April 2017 niversity of Cologne titute for Theoretical physics -of-the-art machine learning niques have recently attracted vid interest from the statistical cs community for their capacity to nguish different phases of quantum er in an automated way. This day mini-workshop will provide nformal setting for discussions gst numerical practitioners who recently started to apply machine ng techniques to quantum stical physics problems. Besides talks there will be ample time for actions amongst the attendees. Speakers Peter Bröcker Cologne Guiseppe Carleo Zurich Maciej Koch-Janusz Zurich Roger Melko Waterloo Titus Neupert Zurich Evert van Nieuwenb Caltech Yi (Frank) Zhang Cornell

8 artificial neural networks x1 x2 x3 x4 x5 Artificial neural networks mimic biological neural networks (albeit at a much smaller scale). They allow for an implicit knowledge representation, which is infused in supervised or unsupervised learning settings.

9 artificial neural networks artificial neurons Θ(z) 1 (binary) input x1 x2 x3 w2 w1 w3 b output 0 z 0 ( ~w ~x b) example Should I skip the first talk (or sleep in)? 9am really? +3 free coffee? -2 2 sleep in quantum stuff? +2

10 artificial neural networks Artificial neural networks are pretty powerful. x1 x x1 x output NAND-gate Like circuits of NAND gates artificial neural networks can encode arbitrarily complex logic functions, thus allowing for universal computation. But the power of neural networks really comes about by varying the weights such that one obtains some desired functionality.

11 neural network architectures x1 feedforward network x2 x3 x4 x5 input layer hidden layers output layer Neural networks with multiple hidden layers have been popularized as deep learning networks.

12 How to train a neural network? x1 x2 x3 x4 x5 quadratic cost function C( ~w, ~ b)= 1 2n X x y(x) a(x) 2 desired output actual output Small adjustments on the level of a single neuron should result in small changes of the cost function. perceptrons sigmoid neurons Θ(z) 1 Θ(z) z 0 0 z

13 How to train a neural network? x1 x2 x3 x4 x5 quadratic cost function C( ~w, ~ b)= 1 2n X x y(x) a(x) 2 desired output actual output back propagation algorithm Rumelhart, Hinton & Williams, Nature (1986) gradient descent extremely efficient way to calculate all needed for a gradient descent optimization.

14 pattern recognition x1 x2 x3 x4 x Some 60 lines of code (Python/Julia) will do this for you with >95% accuracy. Much higher accuracy possible for networks with additional convolutional layers. conv pool conv pool full dropout full matrix reductions conventional

15 convolutional neural networks Convolutional neural networks look for recurring patterns using small filters.

16 convolutional neural networks Convolutional neural networks look for recurring patterns using small filters.

17 convolutional neural networks Convolutional neural networks look for recurring patterns using small filters. filter element-wise matrix product activation map Slide filters across image and create new image based on how well they fit.

18 GPUs & open-source codes

19 Machine learning quantum phases of matter

20 Quantum matter superconductor water ice Bose-Einstein condensate

21 Model systems pair-wise interactions favor alignment of spins Ising model interacting many-body system

22 Model systems pair-wise interactions favor alignment of spins Ising model interacting many-body system

23 Model systems pair-wise interactions favor alignment of spins Ising model interacting many-body system paradigmatic model for a phase transition high temperature critical temperature low temperature

24 Supervised learning approach General setup Consider some system, which as a function of some parameter λ exhibits a phase transition between two phases. Supervised learning approach 1) train convolutional neural network on representative images deep within the two phases 2) apply trained network to images sampled elsewhere to predict phases + transition step 1 train here phase A phase transition phase B train here step 2 predict phases by applying neural network here What are the right images to feed into the neural network?

25 classical phases of matter Machine learning of thermal phase transition in the classical Ising model 6 Nature Physics (2017) L = Output layer T < Tc T > Tc T

26 quantum phases of matter Hubbard model on the honeycomb lattice semi-metal SDW

27 quantum phase transition Green s functions are ideal objects/images for machine learning based discrimination of quantum phases. Dirac SDW L = L = 9 prediction L = 15 L = 12 L = L = U = 1.0 interaction U U = 16.0

28 unsupervised learning

29 Unsupervised learning Employ ability to blindly distinguish phases to map out an entire phase diagram with no hitherto knowledge about the phases. Example: hardcore bosons / XXZ model on a square lattice H = X hi,ji S + i S j + S i S+ j + X hi,ji S z i S z j + h X i S z i 6 4 PRL 88, (2002) checkerboard solid ρ=1/2 hs + i S j i + hs i S+ j i V/t 2 Heisenberg point empty full ρ=0 superfluid ρ= (µ 2V)/t

30 Unsupervised learning Employ ability to blindly distinguish phases to map out an entire phase diagram with no hitherto knowledge about the phases. Example: hardcore bosons / XXZ model on a square lattice H = X hi,ji S + i S j + S i S+ j + X hi,ji S z i S z j + h X i S z i 6 4 PRL 88, (2002) checkerboard solid ρ=1/2 hs z i S z j i V/t 2 Heisenberg point empty full ρ=0 superfluid ρ= (µ 2V)/t

31 summary

32 Summary Monte Carlo + machine learning approach can be used to distinguish phases of interacting many-body systems in (quantum) statistical physics. The approach can be adapted to an unsupervised setting for quickly and semi-automatically mapping out phase diagrams of (quantum) many-body systems. Can we identify even some of the more subtle quantum mechanical properties such as long-range entanglement and (non-local) topological order?

33 Thanks!