Neural Polytopes
Based on: 1) “Neural polytopes” (ArXiv:2307.00721 [cs.LG]) accepted at ICML2023 workshop poster2) “Multi-body wave function of ground and low-lying excited states using unornamented
deep neural network” (ArXiv:2302.08965) Phys. Rev. Research 5 (2023) 033189

Koji Hashimoto (Kyoto U., phys. dept.)
Tomoya Naito (RIKEN iTHEMS)
Hisashi Naito (Nagoya U., math dept.)ML journey from QM to polytopes
1. Intro: NN quantum states

4 pages

2. Solving QM with DNN

5 pages

3. Multi-particle / interaction

9 pages

4. Discrete geometry

5 pages

5. Neural polytopes

6 pages

ArXiv:2302.08965 [physics.comp-ph]

ArXiv:2302.08965 [physics.comp-ph]

Unpublished (on-going work)

ArXiv:2307.00721 [cs.LG]

1. Neural Network Quantum States 1/4
Find ground state wave function ψ(s1, s2, ⋯, sN )Q : Minimize its energy E for a given Hamiltonian H,

A : Use ansatz and optimize parameters!
- Matrix product states
ψ(s1, s2, ⋯) = tr[A (s1) A (s2)⋯]- Tensor network states
ψ(s1, s2, ⋯) =∑m,n

Bmn Ams1s2 Ans3s4

1. Neural Network Quantum States 2/4
Neural network can be wave functions- Boltzmann machine states [Carleo, Troyer `17],[Nomura, Darmawan, Yamaji, Imada `17], ..

Ex) 2-d antiferromagnetic
Heisenberg model was
better-approximatedEnergy with
RBM states# of hidden units- Feedfoward network states[Saito `18], ..
<latexit sha1_base64="a9OADadzNBfJAU2xBHV2TtSSoK0=">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</latexit>

ψ(s1, ⋯, sN ) =f σ
W s + bi∑ i
∑ ij jij1. Neural Network Quantum States 3/4
PINN (Physics-informed neural networks)[Raissi, Perdikaris, Karniadakis `17], ..

xt(x, t)<latexit sha1_base64="YOJpYwcc16acN7f/+7XqnfsYwLQ=">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</latexit><latexit sha1_base64="vRtdbZWQQYJnWLowO7S0JvxbRKM=">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</latexit><latexit sha1_base64="zo6T5Etn2zQVA56kGqyRQQSQSHA=">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</latexit>

Loss L = Ldata + LEoM + LBC<latexit sha1_base64="GWge9CLCx8TXloK4QzABkR9RM+o=">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</latexit>

Physics observations(x = xn , t = ti )] =<latexit sha1_base64="qD6bIyLupNY7uBBPd658Fsuvves=">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</latexit>

Equations of motion F[ (x, t)] = 0<latexit sha1_base64="SymoCbjS0kb7gE7sboT/gSWxqG8=">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</latexit>

Boundary conditions

B[ (x = x0 , t)] = 0<latexit sha1_base64="f5qVRapwsH4HZzXNOURVMhc1UM4=">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</latexit>data1. Neural Network Quantum States 4/4
My motivation : continuum ó discrete
So far, most of the work are separated toeither “discrete” inputs or “continuous” inputs.
- Discretization effects in DNN approximationof continuous systems is small enough?
- Concepts in continuous systems – such as topology –will be modified or surviving?
- How the wave function “space” is generated by
DNN and machine learning?

9ML journey from QM to polytopes
1. Intro: NN quantum states

4 pages

2. Solving QM with DNN

5 pages

3. Multi-particle / interaction

9 pages

4. Discrete geometry

5 pages

5. Neural polytopes

6 pages

ArXiv:2302.08965 [physics.comp-ph]

ArXiv:2302.08965 [physics.comp-ph]

Unpublished (on-going work)

ArXiv:2307.00721 [cs.LG]

2. Solving QM with DNN

1/5

QM on a lattice, suitable for DNN
Question
For a given Hamiltonian, solve for ground and excited states.
1d or multi-d. 1-particle or many particle. Boson or fermion.
Strategy
1) Prepare your QM Hamiltonian on a lattice.
2) DNN input : spatial coordinate values (x,y,..) on a lattice
DNN output : wave function value f(x,y,..)loss function : Energy expectation value, for the whole lattice3) Train DNN and obtain the ground state wave function.
11

2. Solving QM with DNN

2/5

1d harmonic oscillator on a lattice
Strategy 1) Prepare your QM Hamiltonian on a lattice.

12

2. Solving QM with DNN

3/5

1 à 1 DNN is prepared
Strategy
2) DNN input : spatial coordinate values (x,y,..) on a lattice
DNN output : wave function value f(x,y,..)loss function : Energy expectation value, for the whole lattice

Activation function:13

2. Solving QM with DNN

4/5

Successful training with good accuracy
Strategy 3) Train DNN and obtain the ground state wave function.

14

2. Solving QM with DNN

5/5

DNN wave function matches15ML journey from QM to polytopes
1. Intro: NN quantum states

4 pages

2. Solving QM with DNN

5 pages

3. Multi-particle / interaction

9 pages

4. Discrete geometry

5 pages

5. Neural polytopes

6 pages

ArXiv:2302.08965 [physics.comp-ph]

ArXiv:2302.08965 [physics.comp-ph]

Unpublished (on-going work)

ArXiv:2307.00721 [cs.LG]3. Multi-particle / interaction

1/9

2 particles, 1d harmonic oscillator(x,y) : location of the first and second particle
Introduction of interactionxy(x, y)<latexit sha1_base64="YOJpYwcc16acN7f/+7XqnfsYwLQ=">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</latexit><latexit sha1_base64="ig+CjE2hUswgtpcAxyIMMj4TiKM=">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</latexit><latexit sha1_base64="CMNOZ39NmxVDGrONffuwPl4ZIBQ=">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</latexit>

Simplicity: Symmetrization (boson) is imposed afterwards!!
17

3. Multi-particle / interaction

2/9

2 particles, 1d harmonic oscillator
Compare:For exact sol. at = 0<latexit sha1_base64="eDaArWmgJsN3yz7tDsUfcjohMSc=">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</latexit>

Boson :

Fermion :18

3. Multi-particle / interaction

3/9

2 particles, 1d harmonic oscillator

193. Multi-particle / interaction

4/9

2 particles, 1d harmonic oscillator

20

3. Multi-particle / interaction

5/9

2 particles, 1d harmonic oscillator
Generalization?!
Look inside: Raw wave function before (anti-)symmetrized21

3. Multi-particle / interaction

6/9

3 particles, 1d harmonic oscillator
Training result :<latexit sha1_base64="eDaArWmgJsN3yz7tDsUfcjohMSc=">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</latexit>=0Compare: For exact sol. at<latexit sha1_base64="eDaArWmgJsN3yz7tDsUfcjohMSc=">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</latexit>=0

Boson :
Fermion :22

3. Multi-particle / interaction

7/9

3 particles, 1d harmonic oscillator

Boson :

Fermion :23

3. Multi-particle / interaction

8/9

Excited states (1 particles, 1d harmonic oscillator)
Training result :DNN output

24

3. Multi-particle / interaction

9/9

Excited states (2 particles, 1d harmonic oscillator)

25

ML journey from QM to polytopes
1. Intro: NN quantum states

4 pages

2. Solving QM with DNN

5 pages

3. Multi-particle / interaction

9 pages

4. Discrete geometry

5 pages

5. Neural polytopes

6 pages

ArXiv:2302.08965 [physics.comp-ph]

ArXiv:2302.08965 [physics.comp-ph]

Unpublished (on-going work)

ArXiv:2307.00721 [cs.LG]4. Discrete geometry

1/5

Interpretation of the DNN : discrete geometry27

4. Discrete geometry

2/5

Region-wise linear approximation

2-particle 1d harmonic, DNN

DNN with ReLU replacement28

4. Discrete geometry

2/5

Region-wise linear approximation

2-particle 1d harmonic, DNN

DNN with ReLU replacement29

4. Discrete geometry

3/5

Working as a discrete geometry method
Scherk surface
h cos y ilog
cos x<latexit sha1_base64="SPLrBWU6CHdT87UJEUE/PJK5Vm4=">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</latexit>

DNN 2-4-4-4-1

DNN 2-8-8-8-1

DNN 2-20-20-20-1
30

4. Discrete geometry
High curvature region automatically designed

DNN 2-8-8-8-1

DNN 2-20-20-20-1

DNN 2-8-8-8-1

DNN 2-20-20-20-1

4/5

4. Discrete geometry

5/5

DNN giving optimal lattice, then?
Benefit
1. Computation-easy way to get discrete geometry
2. Works even when no smooth surface is given
3. Interpolates the smooth and the discrete
Quantum gravity!
1. Any space can be embedded as a surface in higher dim
2. Loss function can be Einstein equation
3. Weight quantization à Quantization of space
4. No spin network, thus different from AdS/DL

32

ML journey from QM to polytopes
1. Intro: NN quantum states

4 pages

2. Solving QM with DNN

5 pages

3. Multi-particle / interaction

9 pages

4. Discrete geometry

5 pages

5. Neural polytopes

6 pages

ArXiv:2302.08965 [physics.comp-ph]

ArXiv:2302.08965 [physics.comp-ph]

Unpublished (on-going work)

ArXiv:2307.00721 [cs.LG]

Quantum gravity and discrete surfaces :How can we sum all possible spacetimes?

Regge calculus

Dynamical triangulation

Fixed lattice architecture,
variable lengths

Randomly generated
lattice architecture,
fixed lengths[Regge `61]

[Ambjorn, Loll `98]

Approximating a sphere by neural network

Supervised data set :points on a sphere,
Architecture :
feedforward NN,
fully connected, no bias
Activation function :Generalized ReLU
Neural d-polytopes of typetraining||
Cross section defined by

[Result 1] Generative polytopes :successful discrete geometry by machine learning

Neural 2-polytopes of type (n ; 1) for n = 2, 3, 4, 5, 6, 7.

[Result 1] Generative polytopes :successful discrete geometry by machine learning

Neural 3-polytopes of type (n ; 1) for n = 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

[Result 1] Generative polytopes :successful discrete geometry by machine learning

Neural 5-polytopes of type (8 ; 1) sliced at codimension-2 plane rotated gradually.

[Result 2] Neural polytopes :
sphere approximation with various activation functions

[Result 2] Neural polytopes :
sphere approximation with various activation functions

Dual polytopes
p=1

p=∞

ML journey from QM to polytopes
1. Intro: NN quantum states

4 pages

2. Solving QM with DNN

5 pages

3. Multi-particle / interaction

9 pages

4. Discrete geometry

5 pages

5. Neural polytopes

6 pages

ArXiv:2302.08965 [physics.comp-ph]

ArXiv:2302.08965 [physics.comp-ph]

Unpublished (on-going work)

ArXiv:2307.00721 [cs.LG]