# example Rydberg # Importations ``` python import jax from jax import numpy as jnp import matplotlib.pyplot as plt import pickle from matplotlib.colors import ListedColormap from matplotlib.colors import LinearSegmentedColormap import matplotlib.patches as patches cmap_blue = LinearSegmentedColormap.from_list("custom_cmap", ['#001733', '#13678A', '#60C7BB', '#FFFDA8']) cmap_blue ``` # Rydberg dataset This notebook presents how to use QDisc to reproduce the results on the Rydberg atom dataset. ## dataset and training The data are experimental snapshots obtained from a programmable Rydberg atom array. For more information, see the experimental paper: https://www.nature.com/articles/s41586-021-03582-4 ``` python ## load the data (not provided here, this will give an error) ## with open('data_RydbergExpL13 (1).pkl', 'rb') as f: data = pickle.load(f) snapshots = data['dataset'] all_Rb = jnp.array(data['all_Rb']) all_delta = jnp.array(data['all_delta']) L = 13 N = L*L ``` ``` python ## cast them into a QDisc Dataset object ## from qdisc.dataset.core import Dataset dataset = Dataset(data=snapshots, thetas=[all_Rb, all_delta], data_type='discrete', local_dimension=2, local_states=jnp.array([0,1])) ``` ``` python ## create Transformer encoder and decoder ## from qdisc.nn.core import Transformer_encoder from qdisc.nn.core import Transformer_decoder from qdisc.vae.core import VAEmodel encoder = Transformer_encoder(latent_dim=5, d_model=8, num_heads=4, num_layers=3, data_type='discrete') decoder = Transformer_decoder(d_model=8, num_heads=4, num_layers=3, data_type='discrete') myvae = VAEmodel(encoder=encoder, decoder=decoder) ``` ``` python ## training and representation ## from qdisc.vae.core import VAETrainer myvaetrainer = VAETrainer(model=myvae, dataset=dataset) key = jax.random.PRNGKey(6473) num_epochs = 200 myvaetrainer.train(num_epochs=num_epochs, batch_size=3000, beta=0.45, gamma=0., key=key, printing_rate=10, re_shuffle=False) ``` Start training... epoch=0 step=0 loss=119.569095476654 recon=117.79883895165419 logvar=[ 1.07415317 0.77997939 0.72378845 -0.93821395 -0.10032408] epoch=10 step=0 loss=57.38552010440751 recon=56.24097025543754 logvar=[-0.07107329 -0.16698306 -0.34394546 -4.30383545 -0.06234075] epoch=20 step=0 loss=56.43766805834163 recon=55.290590279764864 logvar=[-5.25762906e-02 -2.62134787e-02 -9.96280070e-02 -4.49870754e+00 -2.65689430e-03] epoch=30 step=0 loss=55.400550071167956 recon=54.15427316108366 logvar=[-0.24448012 -0.03299001 -0.01156073 -4.82013867 -0.0088556 ] epoch=40 step=0 loss=54.83681362387403 recon=53.43010166759967 logvar=[-1.07542669 -0.01479255 -0.00726525 -4.86872305 0.01455821] epoch=50 step=0 loss=54.53603219957166 recon=53.091557566642486 logvar=[-1.23756027e+00 -1.23709528e-02 -5.47091283e-03 -4.94531256e+00 3.28441727e-03] epoch=60 step=0 loss=54.43254836895113 recon=52.95765146657946 logvar=[-1.31491583e+00 7.30443888e-03 -1.43214228e-03 -5.12444969e+00 -1.06741663e-02] epoch=70 step=0 loss=54.1974911817511 recon=52.71919527568474 logvar=[-1.32034127e+00 -2.37230897e-02 1.86053524e-02 -5.13348633e+00 -1.70989903e-03] epoch=80 step=0 loss=54.12660673668517 recon=52.647385917253445 logvar=[-1.3428631 0.02448355 -0.01474814 -4.96093612 -0.00953531] epoch=90 step=0 loss=53.92428452520894 recon=52.40086608436129 logvar=[-1.34732496e+00 -6.91766618e-03 3.85537124e-03 -5.43389109e+00 1.05323362e-02] epoch=100 step=0 loss=53.81375978950114 recon=52.284231036530535 logvar=[-1.24221828e+00 1.59933671e-03 1.10860366e-02 -5.40422867e+00 4.23926713e-03] epoch=110 step=0 loss=53.947662847043595 recon=52.389166815528945 logvar=[-1.22268732 -0.01024584 0.01105267 -5.60247201 0.00754143] epoch=120 step=0 loss=53.65327323397351 recon=52.11970498272226 logvar=[-1.14891570e+00 1.44860368e-03 -7.84471019e-03 -5.52474337e+00 -3.39629913e-03] epoch=130 step=0 loss=53.58965383275648 recon=52.03493555562553 logvar=[-1.21718777e+00 -1.95895586e-03 1.06011300e-02 -5.49717389e+00 -1.73662012e-04] epoch=140 step=0 loss=53.741270003328054 recon=52.17246465086842 logvar=[-1.20641303e+00 2.43419631e-03 -9.57970176e-03 -5.35377156e+00 2.04272250e-02] epoch=150 step=0 loss=53.504663767647344 recon=51.920817517337696 logvar=[-1.13939071e+00 -4.74558356e-02 -5.17827363e-03 -5.59386329e+00 -1.79535286e-02] epoch=160 step=0 loss=53.42087330008696 recon=51.82503184617968 logvar=[-1.20954355 -0.04384308 -0.06335316 -5.62403719 -0.01117719] epoch=170 step=0 loss=53.35845947168567 recon=51.74085816470033 logvar=[-1.22017346 -0.04838106 -0.15391571 -5.70929859 -0.0138193 ] epoch=180 step=0 loss=53.38771218351293 recon=51.759155963089704 logvar=[-1.18558398 -0.07025403 -0.13406768 -5.76573897 -0.01678924] epoch=190 step=0 loss=53.30599297063178 recon=51.645958555152056 logvar=[-1.25971673 -0.01837498 -0.14288555 -5.67526412 -0.03959246] Training finished. ``` python data = myvaetrainer.get_data() ``` ``` python myvaetrainer.plot_training(num_epochs = num_epochs) ``` ![](qdisc_Rydberg_files/figure-commonmark/cell-8-output-1.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-8-output-2.png) ``` python latvar = myvaetrainer.compute_repr2d(theta_pair=(0,1), return_latvar = True) myvaetrainer.plot_repr2d(theta_pair=(0,1),subplot=False) ``` ![](qdisc_Rydberg_files/figure-commonmark/cell-9-output-1.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-9-output-2.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-9-output-3.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-9-output-4.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-9-output-5.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-9-output-6.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-9-output-7.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-9-output-8.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-9-output-9.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-9-output-10.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-9-output-11.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-9-output-12.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-9-output-13.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-9-output-14.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-9-output-15.png) ``` python data.keys() ``` dict_keys(['params', 'history_loss', 'history_recon', 'history_logvar', 'latvar']) ``` python ## saving the data ## #data = myvaetrainer.get_data() with open('RydbergExpL13_data_cpVAE2_QDisc.pkl', 'wb') as f: pickle.dump(data, f) ``` ``` python ## example how to load the data ## with open('RydbergExpL13_data_cpVAE2_QDisc.pkl', 'rb') as f: data = pickle.load(f) VAE_params = data['params'] myvaetrainer = VAETrainer(model=myvae, dataset=dataset) key = jax.random.PRNGKey(0) myvaetrainer.init_state(key, dataset.data[0,0]) myvaetrainer.state = myvaetrainer.state.replace(params=VAE_params) ``` ## Exploring conditional probabilities with the decoder In this section, we demonstrate how the **decoder of the cpVAE** can be used to gain further insight into the ordering of each phase by examining the **conditional probabilities**. The procedure is as follows: 1. Specify the values of the latent variables corresponding to a region of interest in the latent space. 2. Identify the dataset samples associated with these latent values. 3. Input these samples into the decoder to compute the conditional probabilities of the spin configurations. ``` python latvar = myvaetrainer.latvar latvar.keys() ``` dict_keys(['id_lat', 'theta_pair', 'mu0', 'mu0_abs', 'logvar0', 'mu1', 'mu1_abs', 'logvar1', 'mu2', 'mu2_abs', 'logvar2', 'mu3', 'mu3_abs', 'logvar3', 'mu4', 'mu4_abs', 'logvar4']) ``` python ## a snake ordering is used, need it to map back to 2d for plotting ## snake_idx = jnp.zeros((N,2)) for i in range(N): #position of each atoms using the snake numbering yp = i//L xp = (yp%2==0)*(i%L) + (yp%2==1)*(L-1-i%L) snake_idx = snake_idx.at[i].set(jnp.array([yp, xp])) snake_idx = snake_idx.astype(jnp.int64) def plot_cp(p_add_cluster): plt.rcParams['font.size'] = 16 plt.figure(figsize=(5,5),dpi=75) p = jnp.mean(p_add_cluster,axis=0)[:,1] p2D = jnp.zeros((L,L)) p2D = p2D.at[snake_idx[:,0], snake_idx[:,1]].set(p) plt.imshow(p2D, cmap=cmap_blue) cbar = plt.colorbar(orientation="horizontal", pad=0.03, location="top") cbar.set_label('conditional probabilities') plt.xlabel(r'$x$ position on the lattice') plt.ylabel(r'$y$ position on the lattice') x_tick_positions = [i for i in range(0,13,2)] x_tick_labels = [str(i) for i in range(1,14,2)] plt.xticks(x_tick_positions, x_tick_labels) y_tick_positions = x_tick_positions y_tick_labels = [str(14-i) for i in range(1,14,2)] plt.yticks(y_tick_positions, y_tick_labels) plt.show() latvar = myvaetrainer.latvar mu1 = latvar['mu1'] mu0 = latvar['mu0'] data_cond_prob = {} ``` ``` python ## Additional cluter ## mu1 = latvar['mu1'] idx_add_cluster = jnp.argwhere(mu1>1.25) N = dataset.data.shape[-1] data_add_cluster = dataset.data[idx_add_cluster[:,0],idx_add_cluster[:,1],:,:].reshape(-1,N)[:2000] m = jnp.zeros((13,31)) for i in idx_add_cluster: m = m.at[i[0],i[1]].add(1) plt.rcParams['font.size'] = 16 plt.figure(figsize=(5,4),dpi=75) plt.imshow(jnp.flipud(m), aspect='auto') plt.show() input_samples = dataset.data[idx_add_cluster[:,0],idx_add_cluster[:,1],:,:].reshape(-1,N) key = jax.random.PRNGKey(0) cp = myvaetrainer.get_cp(input_samples) p_add_cluster = jnp.exp(cp) plot_cp(p_add_cluster) data_cond_prob['idx_add_cluster'] = idx_add_cluster data_cond_prob['cp_add_cluster'] = cp ``` ![](qdisc_Rydberg_files/figure-commonmark/cell-15-output-1.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-15-output-2.png) ``` python ## Striated phase ## id_somewhere_else = jnp.argwhere(mu1<-0.95) data_somewhere_else= dataset.data[id_somewhere_else[:,0],id_somewhere_else[:,1],:,:].reshape(-1,N)[:2000] m = jnp.zeros((13,31)) for i in id_somewhere_else: m = m.at[i[0],i[1]].add(1) plt.rcParams['font.size'] = 16 plt.figure(figsize=(5,4),dpi=75) plt.imshow(jnp.flipud(m), aspect='auto') plt.show() cp = myvaetrainer.get_cp(data_somewhere_else) p_somewhere_else = jnp.exp(cp) plot_cp(p_somewhere_else) idx_striated = jnp.median(jnp.array(id_somewhere_else), axis=0) data_cond_prob['idx_striated'] = idx_striated data_cond_prob['cp_striated'] = cp ``` ![](qdisc_Rydberg_files/figure-commonmark/cell-16-output-1.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-16-output-2.png) ``` python ## Checkerboard phase ## mu0 = latvar['mu0'] id_somewhere_else = jnp.argwhere(((mu0<-1.45)*1 + (mu0>-1.55)*1)==2) #just take the ones starting with 1 to not have degeneracies effect data_somewhere_else = dataset.data[id_somewhere_else[:,0],id_somewhere_else[:,1],:,:].reshape(-1,N)[:2000]#[jnp.argwhere(data_somewhere_else[:,0]==1)[:,0]] #data_somewhere_else = data_somewhere_else[jnp.argwhere(data_somewhere_else[:,0]==1)[:,0]] m = jnp.zeros((13,31)) for i in id_somewhere_else: m = m.at[i[0],i[1]].add(1) plt.rcParams['font.size'] = 16 plt.figure(figsize=(5,4),dpi=75) plt.imshow(jnp.flipud(m), aspect='auto') cbar = plt.colorbar(orientation="horizontal", pad=0.03, location="top") cbar.set_label(r'class', fontsize=20, labelpad=10) plt.show() cp = myvaetrainer.get_cp(data_somewhere_else) p_somewhere_else = jnp.exp(cp) plot_cp(p_somewhere_else) idx_check = jnp.median(jnp.array(id_somewhere_else), axis=0) data_cond_prob['idx_check'] = idx_check data_cond_prob['cp_check'] = cp ``` ![](qdisc_Rydberg_files/figure-commonmark/cell-17-output-1.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-17-output-2.png) ``` python ## Fully loaded lattice ## id_somewhere_else = jnp.argwhere(mu0<-2.2) data_somewhere_else= dataset.data[id_somewhere_else[:,0],id_somewhere_else[:,1],:,:].reshape(-1,N)[:2000] m = jnp.zeros((13,31)) for i in id_somewhere_else: m = m.at[i[0],i[1]].add(1) plt.rcParams['font.size'] = 16 plt.figure(figsize=(5,4),dpi=75) plt.imshow(jnp.flipud(m), aspect='auto') plt.show() cp = myvaetrainer.get_cp(data_somewhere_else) p_somewhere_else = jnp.exp(cp) p = jnp.mean(p_somewhere_else,axis=0)[:,1] plot_cp(p_somewhere_else) idx_full = jnp.median(jnp.array(id_somewhere_else), axis=0) data_cond_prob['idx_full'] = idx_full data_cond_prob['cp_full'] = cp ``` ![](qdisc_Rydberg_files/figure-commonmark/cell-18-output-1.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-18-output-2.png) ``` python ## Empty lattice ## id_somewhere_else = jnp.argwhere(mu0>1.2) data_somewhere_else= dataset.data[id_somewhere_else[:,0],id_somewhere_else[:,1],:,:].reshape(-1,N)[:2000,:] m = jnp.zeros((13,31)) for i in id_somewhere_else: m = m.at[i[0],i[1]].add(1) plt.rcParams['font.size'] = 16 plt.figure(figsize=(5,4),dpi=75) plt.imshow(jnp.flipud(m), aspect='auto') plt.show() cp = myvaetrainer.get_cp(data_somewhere_else) p_somewhere_else = jnp.exp(cp) p = jnp.mean(p_somewhere_else,axis=0)[:,1] plot_cp(p_somewhere_else) idx_empty = jnp.median(jnp.array(id_somewhere_else), axis=0) data_cond_prob['idx_empty'] = idx_empty data_cond_prob['cp_empty'] = cp ``` ![](qdisc_Rydberg_files/figure-commonmark/cell-19-output-1.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-19-output-2.png) ``` python ## Edge ordered phase ## id_somewhere_else = jnp.argwhere(((mu1<1)*1 + (mu0>-.5)*1 + (mu0<0)*1 + (mu1>0.4)*1)==4) data_somewhere_else= dataset.data[id_somewhere_else[:,0],id_somewhere_else[:,1],:,:].reshape(-1,N)[:2000,:] m = jnp.zeros((13,31)) for i in id_somewhere_else: m = m.at[i[0],i[1]].add(1) plt.rcParams['font.size'] = 16 plt.figure(figsize=(5,4),dpi=75) plt.imshow(jnp.flipud(m), aspect='auto') plt.show() cp = myvaetrainer.get_cp(data_somewhere_else) p_somewhere_else = jnp.exp(cp) p = jnp.mean(p_somewhere_else,axis=0)[:,1] plot_cp(p_somewhere_else) idx_edge = jnp.median(jnp.array(id_somewhere_else), axis=0) data_cond_prob['idx_edge'] = idx_edge data_cond_prob['cp_edge'] = cp ``` ![](qdisc_Rydberg_files/figure-commonmark/cell-20-output-1.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-20-output-2.png) ``` python data['latvar'] = latvar data['data_cond_prob'] = data_cond_prob with open('RydbergExpL13_data_cpVAE2_QDisc.pkl', 'wb') as f: pickle.dump(data, f) ``` ## Symbolic regression In this section, we use `QDisc.SR.SymbolicRegression` with the `'genetic'` search space and the SR1 objective to find a symbolic function that characterizes the additional cluster. ``` python #import pysr import numpy as np from pysr import PySRRegressor #this take around 7min import sympy as sp import pickle from jax import numpy as jnp import jax from matplotlib import pyplot as plt ``` [juliapkg] Found dependencies: /usr/local/lib/python3.12/dist-packages/juliapkg/juliapkg.json [juliapkg] Found dependencies: /usr/local/lib/python3.12/dist-packages/juliacall/juliapkg.json [juliapkg] Found dependencies: /usr/local/lib/python3.12/dist-packages/pysr/juliapkg.json [juliapkg] Locating Julia 1.10.3 - 1.11 [juliapkg] Using Julia 1.11.5 at /usr/local/bin/julia [juliapkg] Using Julia project at /root/.julia/environments/pyjuliapkg [juliapkg] Writing Project.toml: | [deps] | PythonCall = "6099a3de-0909-46bc-b1f4-468b9a2dfc0d" | OpenSSL_jll = "458c3c95-2e84-50aa-8efc-19380b2a3a95" | SymbolicRegression = "8254be44-1295-4e6a-a16d-46603ac705cb" | Serialization = "9e88b42a-f829-5b0c-bbe9-9e923198166b" | | [compat] | PythonCall = "=0.9.26" | OpenSSL_jll = "~3.0" | SymbolicRegression = "~1.11" | Serialization = "^1" [juliapkg] Installing packages: | import Pkg | Pkg.Registry.update() | Pkg.add([ | Pkg.PackageSpec(name="PythonCall", uuid="6099a3de-0909-46bc-b1f4-468b9a2dfc0d"), | Pkg.PackageSpec(name="OpenSSL_jll", uuid="458c3c95-2e84-50aa-8efc-19380b2a3a95"), | Pkg.PackageSpec(name="SymbolicRegression", uuid="8254be44-1295-4e6a-a16d-46603ac705cb"), | Pkg.PackageSpec(name="Serialization", uuid="9e88b42a-f829-5b0c-bbe9-9e923198166b"), | ]) | Pkg.resolve() | Pkg.precompile() Detected IPython. Loading juliacall extension. See https://juliapy.github.io/PythonCall.jl/stable/compat/#IPython ``` python from matplotlib.colors import ListedColormap from matplotlib.colors import LinearSegmentedColormap import matplotlib.patches as patches custom_palette = ['#001733', '#13678A', '#60C7BB', '#FFFDA8'] cmap_blue = LinearSegmentedColormap.from_list("custom_cmap", custom_palette) custom_palette = ['#221226','#633D55','#A3648B','#F7FAFF'] cmap_purple = LinearSegmentedColormap.from_list("custom_cmap", custom_palette) custom_palette = ['#0D090D','#365925','#DAF2B6'] cmap_green3 = LinearSegmentedColormap.from_list("custom_cmap", custom_palette) ``` ``` python id_add_cluster = jnp.argwhere(mu1>1.2) id_add_cluster ``` Array([[ 0, 11], [ 0, 12], [ 0, 13], [ 1, 11], [ 1, 12], [ 2, 12], [ 2, 13]], dtype=int64) ``` python with open('data_RydbergExpL13.pkl', 'rb') as f: data = pickle.load(f) snapshots = data['dataset'] all_Rb = jnp.array(data['all_Rb']) all_delta = jnp.array(data['all_delta']) L = 13 N = L*L idx_add_cluster = jnp.array([[ 0, 11], [ 0, 12], [ 0, 13], [ 1, 11], [ 1, 12], [ 2, 12], [ 2, 13]]) ``` ``` python ## specify the indexe of the data labelled as outide and visualize ## #id_add_cluster = jnp.argwhere(mu1<-1.2) classes = 3*jnp.ones((13,31)) for i in idx_add_cluster: classes = classes.at[i[0],i[1]].set(1) classes = classes.at[7:,12:17].set(2) plt.rcParams['font.size'] = 16 plt.figure(figsize=(5,4),dpi=100) cmap_2bins = ListedColormap([cmap_blue(0.75), cmap_purple(0.65), cmap_blue(0.)]) im = plt.imshow(jnp.flipud(classes), aspect='auto', cmap=cmap_2bins) cbar = plt.colorbar(im, orientation="horizontal", pad=0.03, location="top", ticks=[1.33, 2, 2.66]) cbar.set_ticklabels(['0', '1', 'Not in']) cbar.set_label(r'Dataset SR1') plt.xlabel(r'$\Delta/\Omega$') plt.ylabel(r'$R_b/a$') x_tick_positions = [1, 10, 19, 27] # Positions for the ticks x_tick_labels = ['-2', '0', '2', '4'] # Labels for the ticks plt.xticks(x_tick_positions, x_tick_labels) y_tick_positions = [0,6,12] # Positions for the ticks y_tick_labels = ['2.0', '1.5', '1.0'] # Labels for the ticks plt.yticks(y_tick_positions, y_tick_labels) plt.show() data = snapshots all_corners = [[0,1,25,24], [12,13,11,14], [156,155,157,154], [168,167,143,144]] data_corners = jnp.concatenate([snapshots[..., corners] for corners in all_corners], axis=2) dataset_corners = Dataset(data=data_corners, thetas=[all_Rb, all_delta], data_type='discrete', local_dimension=2, local_states=jnp.array([0,1])) cluster_idx_in = jnp.argwhere(classes==1) cluster_idx_out = jnp.argwhere(classes==2) mySR = SymbolicRegression(dataset = dataset_corners, cluster_idx_in = cluster_idx_in, cluster_idx_out = cluster_idx_out, objective='SR1', search_space = "2_body_correlator" ) ``` ![](qdisc_Rydberg_files/figure-commonmark/cell-26-output-1.png) ``` python ## restrict the data to the corners of the lattice ## from qdisc.sr.core import SymbolicRegression data = snapshots all_corners = [[0,1,25,24], [12,13,11,14], [156,155,157,154], [168,167,143,144]] data_corners = jnp.concatenate([snapshots[..., corners] for corners in all_corners], axis=2) dataset_corners = Dataset(data=data_corners, thetas=[all_Rb, all_delta], data_type='discrete', local_dimension=2, local_states=jnp.array([0,1])) cluster_idx_in = jnp.argwhere(classes==1) cluster_idx_out = jnp.argwhere(classes==2) mySR = SymbolicRegression(dataset = dataset_corners, cluster_idx_in = cluster_idx_in, cluster_idx_out = cluster_idx_out, objective='SR1', search_space = "genetic", shift_data = False) ``` PySRRegressor imported ``` python ## search ## all_perf, all_eqs, all_terms = mySR.train( key = jax.random.PRNGKey(645), dataset_size = 5000, random_state = 123, # seed for reproductibility niterations = 20, # Number of iterations to search binary_operators = ["+", "*", "-"], # Allowed binary operations elementwise_loss = "loss(x,y) = -y*log(1/(1+exp(-x)))-(1-y)*log(1-1/(1+exp(-x)))", # sigmoid loss for SR1 maxsize = 20, # max complexity of the equations progress = True, # Show progress during training extra_sympy_mappings = {"C": "C"}, # Allow PySR to use constants batching = True, #batching, usually big dataset batch_size = 1000, turbo = True, deterministic = True, #for reproductibility parallelism = 'serial') ``` /usr/local/lib/python3.12/dist-packages/pysr/sr.py:2811: UserWarning: Note: it looks like you are running in Jupyter. The progress bar will be turned off. warnings.warn( Compiling Julia backend... INFO:pysr.sr:Compiling Julia backend... Resolving package versions... Installed ThreadingUtilities ─────────────── v0.5.5 Installed BitTwiddlingConvenienceFunctions ─ v0.1.6 Installed LayoutPointers ─────────────────── v0.1.17 Installed HostCPUFeatures ────────────────── v0.1.18 Installed ManualMemory ───────────────────── v0.1.8 Installed SIMDTypes ──────────────────────── v0.1.0 Installed VectorizationBase ──────────────── v0.21.72 Installed LoopVectorization ──────────────── v0.12.173 Installed SLEEFPirates ───────────────────── v0.6.43 Installed PolyesterWeave ─────────────────── v0.2.2 Installed UnPack ─────────────────────────── v1.0.2 Installed CloseOpenIntervals ─────────────── v0.1.13 Installed StaticArrayInterface ───────────── v1.9.0 Updating `~/.julia/environments/pyjuliapkg/Project.toml` [bdcacae8] + LoopVectorization v0.12.173 Updating `~/.julia/environments/pyjuliapkg/Manifest.toml` [62783981] + BitTwiddlingConvenienceFunctions v0.1.6 [2a0fbf3d] + CPUSummary v0.2.7 [fb6a15b2] + CloseOpenIntervals v0.1.13 [f70d9fcc] + CommonWorldInvalidations v1.0.0 [adafc99b] + CpuId v0.3.1 [3e5b6fbb] + HostCPUFeatures v0.1.18 [615f187c] + IfElse v0.1.1 [10f19ff3] + LayoutPointers v0.1.17 [bdcacae8] + LoopVectorization v0.12.173 [d125e4d3] + ManualMemory v0.1.8 [6fe1bfb0] + OffsetArrays v1.17.0 [1d0040c9] + PolyesterWeave v0.2.2 [94e857df] + SIMDTypes v0.1.0 [476501e8] + SLEEFPirates v0.6.43 [431bcebd] + SciMLPublic v1.0.1 [aedffcd0] + Static v1.3.1 [0d7ed370] + StaticArrayInterface v1.9.0 [8290d209] + ThreadingUtilities v0.5.5 [3a884ed6] + UnPack v1.0.2 [3d5dd08c] + VectorizationBase v0.21.72 Precompiling project... 1893.7 ms ✓ UnPack 1769.5 ms ✓ SIMDTypes 2015.6 ms ✓ ManualMemory 2236.4 ms ✓ BitTwiddlingConvenienceFunctions 3408.0 ms ✓ ThreadingUtilities 4904.3 ms ✓ StaticArrayInterface 1641.3 ms ✓ PolyesterWeave 3839.5 ms ✓ HostCPUFeatures 1658.1 ms ✓ StaticArrayInterface → StaticArrayInterfaceOffsetArraysExt 1264.0 ms ✓ CloseOpenIntervals 1058.3 ms ✓ LayoutPointers 13491.4 ms ✓ VectorizationBase 1244.7 ms ✓ SLEEFPirates 61586.3 ms ✓ LoopVectorization 2773.2 ms ✓ LoopVectorization → SpecialFunctionsExt 2875.6 ms ✓ LoopVectorization → ForwardDiffExt 4768.5 ms ✓ DynamicExpressions → DynamicExpressionsLoopVectorizationExt 17 dependencies successfully precompiled in 92 seconds. 139 already precompiled. No Changes to `~/.julia/environments/pyjuliapkg/Project.toml` No Changes to `~/.julia/environments/pyjuliapkg/Manifest.toml` ┌ Warning: #= /root/.julia/packages/DynamicExpressions/cYbpm/ext/DynamicExpressionsLoopVectorizationExt.jl:142 =#: │ `LoopVectorization.check_args` on your inputs failed; running fallback `@inbounds @fastmath` loop instead. │ Use `warn_check_args=false`, e.g. `@turbo warn_check_args=false ...`, to disable this warning. └ @ DynamicExpressionsLoopVectorizationExt ~/.julia/packages/LoopVectorization/GKxH5/src/condense_loopset.jl:1166 ┌ Warning: #= /root/.julia/packages/DynamicExpressions/cYbpm/ext/DynamicExpressionsLoopVectorizationExt.jl:152 =#: │ `LoopVectorization.check_args` on your inputs failed; running fallback `@inbounds @fastmath` loop instead. │ Use `warn_check_args=false`, e.g. `@turbo warn_check_args=false ...`, to disable this warning. └ @ DynamicExpressionsLoopVectorizationExt ~/.julia/packages/LoopVectorization/GKxH5/src/condense_loopset.jl:1166 ┌ Warning: #= /root/.julia/packages/DynamicExpressions/cYbpm/ext/DynamicExpressionsLoopVectorizationExt.jl:187 =#: │ `LoopVectorization.check_args` on your inputs failed; running fallback `@inbounds @fastmath` loop instead. │ Use `warn_check_args=false`, e.g. `@turbo warn_check_args=false ...`, to disable this warning. └ @ DynamicExpressionsLoopVectorizationExt ~/.julia/packages/LoopVectorization/GKxH5/src/condense_loopset.jl:1166 ┌ Warning: #= /root/.julia/packages/DynamicExpressions/cYbpm/ext/DynamicExpressionsLoopVectorizationExt.jl:161 =#: │ `LoopVectorization.check_args` on your inputs failed; running fallback `@inbounds @fastmath` loop instead. │ Use `warn_check_args=false`, e.g. `@turbo warn_check_args=false ...`, to disable this warning. └ @ DynamicExpressionsLoopVectorizationExt ~/.julia/packages/LoopVectorization/GKxH5/src/condense_loopset.jl:1166 ┌ Warning: #= /root/.julia/packages/DynamicExpressions/cYbpm/ext/DynamicExpressionsLoopVectorizationExt.jl:212 =#: │ `LoopVectorization.check_args` on your inputs failed; running fallback `@inbounds @fastmath` loop instead. │ Use `warn_check_args=false`, e.g. `@turbo warn_check_args=false ...`, to disable this warning. └ @ DynamicExpressionsLoopVectorizationExt ~/.julia/packages/LoopVectorization/GKxH5/src/condense_loopset.jl:1166 [ Info: Started! Expressions evaluated per second: 9.500e+02 Progress: 46 / 620 total iterations (7.419%) ════════════════════════════════════════════════════════════════════════════════════════════════════ ─────────────────────────────────────────────────────────────────────────────────────────────────── Complexity Loss Score Equation 1 6.940e-01 0.000e+00 y = -0.081661 3 6.346e-01 4.473e-02 y = x₁ + x₃ 5 6.208e-01 1.102e-02 y = (x₃ * 2.1711) + x₁ 7 6.018e-01 1.549e-02 y = (x₃ + (x₁ + -0.42624)) + x₃ 9 5.773e-01 2.080e-02 y = x₁ + (((x₃ - 0.21415) * 3.0877) + x₂) 11 5.772e-01 1.125e-04 y = (((x₃ + (x₁ + x₃)) + -0.60713) + x₂) * 1.4292 13 5.743e-01 2.526e-03 y = (x₃ * x₀) + (x₂ + ((-0.70466 + (x₃ + x₁)) + x₃)) 15 5.731e-01 1.021e-03 y = ((x₀ * (x₃ * 1.3817)) + (((x₂ + -0.70466) + x₃) + x₁))... + x₃ 17 5.726e-01 4.610e-04 y = ((x₀ * (x₃ * 0.91296)) + ((x₂ + -0.68221) + ((x₃ + x₁)... + x₃))) * 1.2419 19 5.710e-01 1.364e-03 y = (((x₃ + (x₁ - ((-0.0098262 - x₂) * (x₁ + 0.75249)))) -... -0.0073776) - (x₃ * -1.9871)) - 0.75249 ─────────────────────────────────────────────────────────────────────────────────────────────────── ════════════════════════════════════════════════════════════════════════════════════════════════════ Press 'q' and then to stop execution early. Expressions evaluated per second: 9.130e+02 Progress: 84 / 620 total iterations (13.548%) ════════════════════════════════════════════════════════════════════════════════════════════════════ ─────────────────────────────────────────────────────────────────────────────────────────────────── Complexity Loss Score Equation 1 6.932e-01 0.000e+00 y = -0.012652 3 6.346e-01 4.414e-02 y = x₁ + x₃ 5 6.208e-01 1.102e-02 y = (x₃ * 2.1711) + x₁ 7 6.016e-01 1.567e-02 y = x₁ + ((x₃ - 0.21415) * 3.0877) 9 5.773e-01 2.061e-02 y = x₁ + (((x₃ - 0.21415) * 3.0877) + x₂) 11 5.772e-01 1.125e-04 y = (((x₃ + (x₁ + x₃)) + -0.60713) + x₂) * 1.4292 13 5.739e-01 2.856e-03 y = (x₃ * x₀) + (((x₂ + (x₃ + x₁)) + -0.64617) + x₃) 15 5.731e-01 6.903e-04 y = ((x₀ * (x₃ * 1.3817)) + (((x₂ + -0.70466) + x₃) + x₁))... + x₃ 17 5.726e-01 4.610e-04 y = ((x₀ * (x₃ * 0.91296)) + ((x₂ + -0.68221) + ((x₃ + x₁)... + x₃))) * 1.2419 19 5.710e-01 1.364e-03 y = (((x₃ + (x₁ - ((-0.0098262 - x₂) * (x₁ + 0.75249)))) -... -0.0073776) - (x₃ * -1.9871)) - 0.75249 ─────────────────────────────────────────────────────────────────────────────────────────────────── ════════════════════════════════════════════════════════════════════════════════════════════════════ Press 'q' and then to stop execution early. Expressions evaluated per second: 9.440e+02 Progress: 125 / 620 total iterations (20.161%) ════════════════════════════════════════════════════════════════════════════════════════════════════ ─────────────────────────────────────────────────────────────────────────────────────────────────── Complexity Loss Score Equation 1 6.931e-01 0.000e+00 y = 0 3 6.346e-01 4.413e-02 y = x₁ + x₃ 5 6.208e-01 1.102e-02 y = (x₃ * 2.1711) + x₁ 7 6.016e-01 1.567e-02 y = x₁ + ((x₃ - 0.21415) * 3.0877) 9 5.773e-01 2.061e-02 y = x₁ + (((x₃ - 0.21415) * 3.0877) + x₂) 11 5.772e-01 1.125e-04 y = (((x₃ + (x₁ + x₃)) + -0.60713) + x₂) * 1.4292 13 5.730e-01 3.664e-03 y = (((x₃ + x₃) * (x₀ + x₃)) + x₁) + (x₂ + -0.67966) 15 5.701e-01 2.494e-03 y = (((x₃ + x₃) + x₃) + ((x₁ * x₂) + x₁)) + (x₂ + -0.76112... ) 17 5.691e-01 8.939e-04 y = x₃ + (x₃ + (x₃ + (((x₂ + (x₂ + x₁)) * x₁) + (x₂ + -0.5... 7925)))) ─────────────────────────────────────────────────────────────────────────────────────────────────── ════════════════════════════════════════════════════════════════════════════════════════════════════ Press 'q' and then to stop execution early. Expressions evaluated per second: 9.510e+02 Progress: 163 / 620 total iterations (26.290%) ════════════════════════════════════════════════════════════════════════════════════════════════════ ─────────────────────────────────────────────────────────────────────────────────────────────────── Complexity Loss Score Equation 1 6.931e-01 0.000e+00 y = 0 3 6.346e-01 4.413e-02 y = x₁ + x₃ 5 6.208e-01 1.102e-02 y = (x₃ * 2.1711) + x₁ 7 5.980e-01 1.866e-02 y = ((x₃ - 0.16037) * 2.7394) + x₁ 9 5.773e-01 1.762e-02 y = x₁ + (((x₃ - 0.21415) * 3.0877) + x₂) 11 5.755e-01 1.549e-03 y = ((x₃ * (x₀ + 1.7235)) + x₁) + (x₂ + -0.67966) 13 5.730e-01 2.228e-03 y = (((x₃ + x₃) * (x₀ + x₃)) + x₁) + (x₂ + -0.67966) 15 5.701e-01 2.494e-03 y = (((x₃ + x₃) + x₃) + ((x₁ * x₂) + x₁)) + (x₂ + -0.76112... ) 17 5.691e-01 8.939e-04 y = x₃ + (x₃ + (x₃ + (((x₂ + (x₂ + x₁)) * x₁) + (x₂ + -0.5... 7925)))) ─────────────────────────────────────────────────────────────────────────────────────────────────── ════════════════════════════════════════════════════════════════════════════════════════════════════ Press 'q' and then to stop execution early. Expressions evaluated per second: 9.660e+02 Progress: 205 / 620 total iterations (33.065%) ════════════════════════════════════════════════════════════════════════════════════════════════════ ─────────────────────────────────────────────────────────────────────────────────────────────────── Complexity Loss Score Equation 1 6.786e-01 0.000e+00 y = x₂ 3 6.346e-01 3.352e-02 y = x₁ + x₃ 5 6.207e-01 1.107e-02 y = (x₃ * 2.2239) + x₁ 7 5.979e-01 1.867e-02 y = ((x₃ - 0.1637) * 2.7958) + x₁ 9 5.773e-01 1.756e-02 y = x₁ + (((x₃ - 0.21415) * 3.0877) + x₂) 11 5.755e-01 1.549e-03 y = ((x₃ * (x₀ + 1.7235)) + x₁) + (x₂ + -0.67966) 13 5.730e-01 2.228e-03 y = (((x₃ + x₃) * (x₀ + x₃)) + x₁) + (x₂ + -0.67966) 15 5.701e-01 2.494e-03 y = (((x₃ + x₃) + x₃) + ((x₁ * x₂) + x₁)) + (x₂ + -0.76112... ) 17 5.691e-01 8.939e-04 y = x₃ + (x₃ + (x₃ + (((x₂ + (x₂ + x₁)) * x₁) + (x₂ + -0.5... 7925)))) ─────────────────────────────────────────────────────────────────────────────────────────────────── ════════════════════════════════════════════════════════════════════════════════════════════════════ Press 'q' and then to stop execution early. Expressions evaluated per second: 9.800e+02 Progress: 245 / 620 total iterations (39.516%) ════════════════════════════════════════════════════════════════════════════════════════════════════ ─────────────────────────────────────────────────────────────────────────────────────────────────── Complexity Loss Score Equation 1 6.786e-01 0.000e+00 y = x₂ 3 6.346e-01 3.352e-02 y = x₁ + x₃ 5 6.207e-01 1.108e-02 y = x₁ + (x₃ * 2.3544) 7 5.979e-01 1.867e-02 y = (x₃ * 2.8277) + (x₁ + -0.48327) 9 5.772e-01 1.762e-02 y = (x₃ * 3.0223) + ((x₁ + x₂) + -0.67966) 11 5.755e-01 1.490e-03 y = ((x₃ * (x₀ + 1.7235)) + x₁) + (x₂ + -0.67966) 13 5.722e-01 2.898e-03 y = ((x₂ + x₁) * x₂) + (x₁ + (-0.53616 + (x₃ * 2.8811))) 15 5.701e-01 1.824e-03 y = (((x₃ + x₃) + x₃) + ((x₁ * x₂) + x₁)) + (x₂ + -0.76112... ) 17 5.673e-01 2.459e-03 y = (x₃ + (((x₃ + x₃) + ((x₂ + (x₁ + x₂)) * x₁)) + x₂)) + ... -0.69276 19 5.673e-01 4.792e-05 y = x₃ + (((((x₁ + -0.61634) + x₂) * (x₂ + x₂)) + x₁) + (x... ₃ + (x₃ - 0.71109))) ─────────────────────────────────────────────────────────────────────────────────────────────────── ════════════════════════════════════════════════════════════════════════════════════════════════════ Press 'q' and then to stop execution early. Expressions evaluated per second: 9.530e+02 Progress: 282 / 620 total iterations (45.484%) ════════════════════════════════════════════════════════════════════════════════════════════════════ ─────────────────────────────────────────────────────────────────────────────────────────────────── Complexity Loss Score Equation 1 6.786e-01 0.000e+00 y = x₂ 3 6.346e-01 3.352e-02 y = x₁ + x₃ 5 6.207e-01 1.108e-02 y = x₁ + (x₃ * 2.3544) 7 5.979e-01 1.867e-02 y = (x₃ * 2.8277) + (x₁ + -0.48327) 9 5.772e-01 1.762e-02 y = (x₃ * 3.0223) + ((x₁ + x₂) + -0.67966) 11 5.755e-01 1.490e-03 y = ((x₃ * (x₀ + 1.7235)) + x₁) + (x₂ + -0.67966) 13 5.722e-01 2.898e-03 y = ((x₂ + x₁) * x₂) + (x₁ + (-0.53616 + (x₃ * 2.8811))) 15 5.673e-01 4.283e-03 y = (x₃ + (x₃ + x₃)) + (((x₂ + x₁) * (x₁ + x₂)) + -0.69276... ) 17 5.673e-01 4.426e-05 y = (x₃ + (x₃ + (((x₂ + (x₂ + x₁)) * x₁) + (x₃ + x₂)))) + ... -0.72049 19 5.655e-01 1.575e-03 y = (x₀ * x₃) + (x₃ + (x₃ + ((((x₂ + (x₁ + x₂)) * x₁) + x₂... ) + -0.57925))) ─────────────────────────────────────────────────────────────────────────────────────────────────── ════════════════════════════════════════════════════════════════════════════════════════════════════ Press 'q' and then to stop execution early. Expressions evaluated per second: 1.000e+03 Progress: 334 / 620 total iterations (53.871%) ════════════════════════════════════════════════════════════════════════════════════════════════════ ─────────────────────────────────────────────────────────────────────────────────────────────────── Complexity Loss Score Equation 1 6.512e-01 0.000e+00 y = x₃ 3 6.346e-01 1.288e-02 y = x₁ + x₃ 5 6.207e-01 1.108e-02 y = x₁ + (x₃ * 2.3544) 7 5.979e-01 1.867e-02 y = (x₃ * 2.8277) + (x₁ + -0.48327) 9 5.772e-01 1.762e-02 y = (x₃ * 3.0223) + ((x₁ + x₂) + -0.67966) 11 5.755e-01 1.490e-03 y = ((x₃ * (x₀ + 1.7235)) + x₁) + (x₂ + -0.67966) 13 5.673e-01 7.157e-03 y = (x₃ * 2.8811) + (((x₂ + x₁) * (x₁ + x₂)) + -0.72377) 15 5.637e-01 3.256e-03 y = ((x₃ * (2.3752 + x₀)) + ((x₁ + x₂) * (x₁ + x₂))) + -0.... 72377 ─────────────────────────────────────────────────────────────────────────────────────────────────── ════════════════════════════════════════════════════════════════════════════════════════════════════ Press 'q' and then to stop execution early. Expressions evaluated per second: 9.500e+02 Progress: 367 / 620 total iterations (59.194%) ════════════════════════════════════════════════════════════════════════════════════════════════════ ─────────────────────────────────────────────────────────────────────────────────────────────────── Complexity Loss Score Equation 1 6.512e-01 0.000e+00 y = x₃ 3 6.346e-01 1.288e-02 y = x₁ + x₃ 5 6.207e-01 1.108e-02 y = x₁ + (x₃ * 2.3544) 7 5.979e-01 1.867e-02 y = (x₃ * 2.8277) + (x₁ + -0.48327) 9 5.772e-01 1.762e-02 y = (x₃ * 3.0223) + ((x₁ + x₂) + -0.67966) 11 5.755e-01 1.490e-03 y = ((x₃ * (x₀ + 1.7235)) + x₁) + (x₂ + -0.67966) 13 5.673e-01 7.157e-03 y = (x₃ * 2.8811) + (((x₂ + x₁) * (x₁ + x₂)) + -0.72377) 15 5.637e-01 3.256e-03 y = ((x₃ * (2.3752 + x₀)) + ((x₁ + x₂) * (x₁ + x₂))) + -0.... 72377 ─────────────────────────────────────────────────────────────────────────────────────────────────── ════════════════════════════════════════════════════════════════════════════════════════════════════ Press 'q' and then to stop execution early. Expressions evaluated per second: 9.520e+02 Progress: 408 / 620 total iterations (65.806%) ════════════════════════════════════════════════════════════════════════════════════════════════════ ─────────────────────────────────────────────────────────────────────────────────────────────────── Complexity Loss Score Equation 1 6.512e-01 0.000e+00 y = x₃ 3 6.346e-01 1.288e-02 y = x₁ + x₃ 5 6.207e-01 1.108e-02 y = x₁ + (x₃ * 2.3544) 7 5.979e-01 1.867e-02 y = (x₃ * 2.8277) + (x₁ + -0.48327) 9 5.772e-01 1.762e-02 y = (x₃ * 3.0223) + ((x₁ + x₂) + -0.67966) 11 5.755e-01 1.490e-03 y = ((x₃ * (x₀ + 1.7235)) + x₁) + (x₂ + -0.67966) 13 5.673e-01 7.157e-03 y = (x₃ * 2.8811) + (((x₂ + x₁) * (x₁ + x₂)) + -0.72377) 15 5.637e-01 3.256e-03 y = ((x₃ * (2.3752 + x₀)) + ((x₁ + x₂) * (x₁ + x₂))) + -0.... 72377 ─────────────────────────────────────────────────────────────────────────────────────────────────── ════════════════════════════════════════════════════════════════════════════════════════════════════ Press 'q' and then to stop execution early. Expressions evaluated per second: 9.710e+02 Progress: 441 / 620 total iterations (71.129%) ════════════════════════════════════════════════════════════════════════════════════════════════════ ─────────────────────────────────────────────────────────────────────────────────────────────────── Complexity Loss Score Equation 1 6.512e-01 0.000e+00 y = x₃ 3 6.346e-01 1.288e-02 y = x₁ + x₃ 5 6.207e-01 1.108e-02 y = x₁ + (x₃ * 2.3544) 7 5.979e-01 1.867e-02 y = (x₃ * 2.8277) + (x₁ + -0.48327) 9 5.772e-01 1.762e-02 y = (x₃ * 3.0223) + ((x₁ + x₂) + -0.67966) 11 5.755e-01 1.490e-03 y = ((x₃ * (x₀ + 1.7235)) + x₁) + (x₂ + -0.67966) 13 5.673e-01 7.157e-03 y = (x₃ * 2.8811) + (((x₂ + x₁) * (x₁ + x₂)) + -0.72377) 15 5.637e-01 3.256e-03 y = ((x₃ * (2.3752 + x₀)) + ((x₁ + x₂) * (x₁ + x₂))) + -0.... 72377 ─────────────────────────────────────────────────────────────────────────────────────────────────── ════════════════════════════════════════════════════════════════════════════════════════════════════ Press 'q' and then to stop execution early. Expressions evaluated per second: 8.780e+02 Progress: 474 / 620 total iterations (76.452%) ════════════════════════════════════════════════════════════════════════════════════════════════════ ─────────────────────────────────────────────────────────────────────────────────────────────────── Complexity Loss Score Equation 1 6.512e-01 0.000e+00 y = x₃ 3 6.346e-01 1.288e-02 y = x₁ + x₃ 5 6.207e-01 1.108e-02 y = x₁ + (x₃ * 2.3544) 7 5.979e-01 1.867e-02 y = (x₃ * 2.8277) + (x₁ + -0.48327) 9 5.772e-01 1.762e-02 y = (x₃ * 3.0223) + ((x₁ + x₂) + -0.67966) 11 5.755e-01 1.490e-03 y = ((x₃ * (x₀ + 1.7235)) + x₁) + (x₂ + -0.67966) 13 5.673e-01 7.157e-03 y = (x₃ * 2.8811) + (((x₂ + x₁) * (x₁ + x₂)) + -0.72377) 15 5.637e-01 3.256e-03 y = ((x₃ * (2.3752 + x₀)) + ((x₁ + x₂) * (x₁ + x₂))) + -0.... 72377 ─────────────────────────────────────────────────────────────────────────────────────────────────── ════════════════════════════════════════════════════════════════════════════════════════════════════ Press 'q' and then to stop execution early. Expressions evaluated per second: 8.880e+02 Progress: 518 / 620 total iterations (83.548%) ════════════════════════════════════════════════════════════════════════════════════════════════════ ─────────────────────────────────────────────────────────────────────────────────────────────────── Complexity Loss Score Equation 1 6.512e-01 0.000e+00 y = x₃ 3 6.346e-01 1.288e-02 y = x₁ + x₃ 5 6.207e-01 1.108e-02 y = x₁ + (x₃ * 2.3544) 7 5.979e-01 1.867e-02 y = (x₃ * 2.8277) + (x₁ + -0.48327) 9 5.772e-01 1.762e-02 y = (x₃ * 3.0223) + ((x₁ + x₂) + -0.67966) 11 5.755e-01 1.490e-03 y = ((x₃ * (x₀ + 1.7235)) + x₁) + (x₂ + -0.67966) 13 5.673e-01 7.157e-03 y = (x₃ * 2.8811) + (((x₂ + x₁) * (x₁ + x₂)) + -0.72377) 15 5.637e-01 3.256e-03 y = ((x₃ * (2.3752 + x₀)) + ((x₁ + x₂) * (x₁ + x₂))) + -0.... 72377 ─────────────────────────────────────────────────────────────────────────────────────────────────── ════════════════════════════════════════════════════════════════════════════════════════════════════ Press 'q' and then to stop execution early. Expressions evaluated per second: 8.760e+02 Progress: 551 / 620 total iterations (88.871%) ════════════════════════════════════════════════════════════════════════════════════════════════════ ─────────────────────────────────────────────────────────────────────────────────────────────────── Complexity Loss Score Equation 1 6.512e-01 0.000e+00 y = x₃ 3 6.346e-01 1.288e-02 y = x₁ + x₃ 5 6.207e-01 1.109e-02 y = (x₃ * 2.251) + x₁ 7 5.979e-01 1.866e-02 y = (x₃ * 2.8277) + (x₁ + -0.48327) 9 5.772e-01 1.762e-02 y = (x₃ * 3.0223) + ((x₁ + x₂) + -0.67966) 11 5.755e-01 1.490e-03 y = ((x₃ * (x₀ + 1.7235)) + x₁) + (x₂ + -0.67966) 13 5.673e-01 7.157e-03 y = (x₃ * 2.8811) + (((x₂ + x₁) * (x₁ + x₂)) + -0.72377) 15 5.637e-01 3.256e-03 y = ((x₃ * (2.3752 + x₀)) + ((x₁ + x₂) * (x₁ + x₂))) + -0.... 72377 ─────────────────────────────────────────────────────────────────────────────────────────────────── ════════════════════════════════════════════════════════════════════════════════════════════════════ Press 'q' and then to stop execution early. Expressions evaluated per second: 8.990e+02 Progress: 595 / 620 total iterations (95.968%) ════════════════════════════════════════════════════════════════════════════════════════════════════ ─────────────────────────────────────────────────────────────────────────────────────────────────── Complexity Loss Score Equation 1 6.512e-01 0.000e+00 y = x₃ 3 6.346e-01 1.288e-02 y = x₁ + x₃ 5 6.207e-01 1.109e-02 y = (x₃ * 2.251) + x₁ 7 5.979e-01 1.866e-02 y = (x₃ * 2.8277) + (x₁ + -0.48327) 9 5.772e-01 1.762e-02 y = (x₃ * 3.0223) + ((x₁ + x₂) + -0.67966) 11 5.755e-01 1.490e-03 y = ((x₃ * (x₀ + 1.7235)) + x₁) + (x₂ + -0.67966) 13 5.673e-01 7.157e-03 y = (x₃ * 2.8811) + (((x₂ + x₁) * (x₁ + x₂)) + -0.72377) 15 5.637e-01 3.256e-03 y = ((x₃ * (2.3752 + x₀)) + ((x₁ + x₂) * (x₁ + x₂))) + -0.... 72377 ─────────────────────────────────────────────────────────────────────────────────────────────────── ════════════════════════════════════════════════════════════════════════════════════════════════════ Press 'q' and then to stop execution early. ─────────────────────────────────────────────────────────────────────────────────────────────────── Complexity Loss Score Equation 1 6.512e-01 0.000e+00 y = x₃ 3 6.346e-01 1.288e-02 y = x₁ + x₃ 5 6.207e-01 1.109e-02 y = (x₃ * 2.251) + x₁ 7 5.979e-01 1.866e-02 y = (x₃ * 2.8277) + (x₁ + -0.48327) 9 5.772e-01 1.762e-02 y = (x₃ * 3.0223) + ((x₁ + x₂) + -0.67966) 11 5.755e-01 1.490e-03 y = ((x₃ * (x₀ + 1.7235)) + x₁) + (x₂ + -0.67966) 13 5.673e-01 7.157e-03 y = (x₃ * 2.8811) + (((x₂ + x₁) * (x₁ + x₂)) + -0.72377) 15 5.637e-01 3.256e-03 y = ((x₃ * (2.3752 + x₀)) + ((x₁ + x₂) * (x₁ + x₂))) + -0.... 72377 ─────────────────────────────────────────────────────────────────────────────────────────────────── [ Info: Final population: [ Info: Results saved to: - outputs/20260213_091433_AHqlFf/hall_of_fame.csv ``` python ## look at the last equation ## i = 7 perf = all_perf[i] equation_sympy = all_eqs[i] terms = all_terms[i] print('equation {}: {}, terms: {}'.format(i,perf,terms)) print(equation_sympy) print('\n') ``` equation 7: 0.6969000101089478, terms: [1, x1**2, x2**2, x3, x0*x3, x1*x2] x0*x3 + x1**2 + 2*x1*x2 + x2**2 + 2.3751783*x3 - 0.72377163 ``` python import sympy as sp import jax import jax.numpy as jnp f_str = '2.0894327*x0*x3 + 2.0894327*x1*x2 + 0.857685211456824*x1 + x2 + 2.0894327*x3 - 0.776886377897628' f_str = 'x0*x3 + x1 + 2*x1*x2 + x2 + 2.3751783*x3 - 0.72377163' f_sympy = sp.sympify(f_str) x0, x1, x2, x3 = sp.symbols('x0 x1 x2 x3') f_callable = sp.lambdify([x0, x1, x2, x3], f_sympy, 'jax') all_corners = [[0,1,25,24], [12,13,11,14], [156,155,157,154], [168,167,143,144]] snapshots_coners = jnp.concatenate([snapshots[..., corners] for corners in all_corners], axis=2) all_predictions = f_callable(snapshots_coners[:,:,:,0], snapshots_coners[:,:,:,1], snapshots_coners[:,:,:,2], snapshots_coners[:,:,:,3]) plt.rcParams['font.size'] = 16 plt.figure(figsize=(5,4),dpi=100) plt.imshow(jnp.flipud(jnp.mean(all_predictions, axis=-1)), cmap=cmap_blue, aspect='auto') cbar = plt.colorbar(orientation="horizontal", pad=0.03, location="top") cbar.set_label(r'f(x)') plt.xlabel(r'$\Delta/\Omega$') plt.ylabel(r'$R_b/a$') x_tick_positions = [1, 10, 19, 27] # Positions for the ticks x_tick_labels = ['-2', '0', '2', '4'] # Labels for the ticks plt.xticks(x_tick_positions, x_tick_labels) y_tick_positions = [0,6,12] # Positions for the ticks y_tick_labels = ['2.0', '1.5', '1.0'] # Labels for the ticks plt.yticks(y_tick_positions, y_tick_labels) plt.show() plt.rcParams['font.size'] = 16 plt.figure(figsize=(5,4),dpi=100) plt.imshow(jnp.flipud(jnp.mean(all_predictions>0, axis=-1)), cmap=cmap_blue, aspect='auto') cbar = plt.colorbar(orientation="horizontal", pad=0.03, location="top") cbar.set_label(r'f(x)>0') plt.xlabel(r'$\Delta/\Omega$') plt.ylabel(r'$R_b/a$') x_tick_positions = [1, 10, 19, 27] # Positions for the ticks x_tick_labels = ['-2', '0', '2', '4'] # Labels for the ticks plt.xticks(x_tick_positions, x_tick_labels) y_tick_positions = [0,6,12] # Positions for the ticks y_tick_labels = ['2.0', '1.5', '1.0'] # Labels for the ticks plt.yticks(y_tick_positions, y_tick_labels) plt.show() plt.rcParams['font.size'] = 16 plt.figure(figsize=(5,4),dpi=100) plt.imshow(jnp.flipud(jnp.mean(all_predictions, axis=-1)>0), cmap=cmap_blue, aspect='auto') cbar = plt.colorbar(orientation="horizontal", pad=0.03, location="top") cbar.set_label(r'f(x)>0') plt.xlabel(r'$\Delta/\Omega$') plt.ylabel(r'$R_b/a$') x_tick_positions = [1, 10, 19, 27] # Positions for the ticks x_tick_labels = ['-2', '0', '2', '4'] # Labels for the ticks plt.xticks(x_tick_positions, x_tick_labels) y_tick_positions = [0,6,12] # Positions for the ticks y_tick_labels = ['2.0', '1.5', '1.0'] # Labels for the ticks plt.yticks(y_tick_positions, y_tick_labels) plt.show() ``` ![](qdisc_Rydberg_files/figure-commonmark/cell-30-output-1.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-30-output-2.png) ![](qdisc_Rydberg_files/figure-commonmark/cell-30-output-3.png) [ColabKernelApp] WARNING | Error caught during object inspection: Can't clean for JSON: Using our prior knowledge of the system, we can construct an order parameter by using the found correlator on the corner and subtracting the correlations on the edges and in the bulk. ``` python def compute_correlation_bulk_pos(x): """Computes the bulk correlation of a single configuration.""" c = [] for i in range(N): if i>L and i<(L*L-L) and i%L!=0 and (i+1)%L!=0: yi = i//L xi = (yi%2==0)*(i%L) + (yi%2==1)*(L-1-i%L) j = i+1 c.append(x[i]*x[j]) return jnp.mean(jnp.array(c),axis=-1)/2 def compute_correlation_edges_pos(x): """Computes the edge correlation of a single configuration.""" c = [] for i in range(L-1): c.append(x[i]*x[i+1]) for i in range(L*L-L,L*L-1): c.append(x[i]*x[i+1]) for i in range(L-1,L*L,L): c.append(x[i]*x[i+1]) for i in range(0,L*L,L): c.append(x[i]*x[i+25]) return jnp.mean(jnp.array(c),axis=-1) def compute_correlation_corner(x): """Computes the corner correlation of a single configuration.""" c = [] all_corners = [[0,1,25,24], [12,13,11,14], [156,155,157,154], [168,167,143,144]] x_corner = jnp.concatenate([x[None, corners] for corners in all_corners], axis=0) c.append(x_corner[:,0]*x_corner[:,3]) c.append(2*x_corner[:,1]*x_corner[:,2]) co = jnp.abs(jnp.mean(jnp.array(c).reshape(-1),axis=-1)) ed = jnp.abs(compute_correlation_edges_pos(x)) bu = jnp.abs(compute_correlation_bulk_pos(x)) return co - ed - bu compute_correlation_corner_vmap = jax.vmap(jax.vmap(jax.vmap(compute_correlation_corner))) plt.rcParams['font.size'] = 18 plt.figure(figsize=(3,3),dpi=200) #S = data_exact['S'] cor = jnp.mean(compute_correlation_corner_vmap(snapshots*2-1), axis=-1) plt.imshow(jnp.flipud(cor), cmap=cmap_green3, aspect='auto') cbar = plt.colorbar(orientation="horizontal", pad=0.03, location="top") cbar.set_label(r'f(x)', fontsize=20, labelpad=10) plt.xlabel(r'$\Delta/\Omega$') plt.ylabel(r'$R_b/a$') x_tick_positions = [1, 10, 19, 27] # Positions for the ticks x_tick_labels = ['-2', '0', '2', '4'] # Labels for the ticks plt.xticks(x_tick_positions, x_tick_labels) y_tick_positions = [0,6,12] # Positions for the ticks y_tick_labels = ['2.0', '1.5', '1.0'] # Labels for the ticks plt.yticks(y_tick_positions, y_tick_labels) #cbar.set_label(r'$\langle Z(r_j)Z(r_i) \rangle$ $(r_j,r_i)\in NN$') #plt.title('S') plt.show() ``` ![](qdisc_Rydberg_files/figure-commonmark/cell-31-output-1.png) ``` python plt.rcParams['font.size'] = 18 plt.figure(figsize=(3,3),dpi=200) plt.imshow(jnp.flipud(cor>0.2), cmap=cmap_green3, aspect='auto') cbar = plt.colorbar(orientation="horizontal", pad=0.03, location="top") cbar.set_label(r'f(x)', fontsize=20, labelpad=10) plt.xlabel(r'$\Delta/\Omega$') plt.ylabel(r'$R_b/a$') x_tick_positions = [1, 10, 19, 27] # Positions for the ticks x_tick_labels = ['-2', '0', '2', '4'] # Labels for the ticks plt.xticks(x_tick_positions, x_tick_labels) y_tick_positions = [0,6,12] # Positions for the ticks y_tick_labels = ['2.0', '1.5', '1.0'] # Labels for the ticks plt.yticks(y_tick_positions, y_tick_labels) #cbar.set_label(r'$\langle Z(r_j)Z(r_i) \rangle$ $(r_j,r_i)\in NN$') #plt.title('S') plt.show() ``` ![](qdisc_Rydberg_files/figure-commonmark/cell-32-output-1.png)