🔳 l2hmc-qcd Example: 4D SU(3)

Sam Foreman Dec 5, 2023 12/05/23 updated: Aug 28, 2026 08/28/26 19 min read

 󰈤

Training and evaluating the L2HMC sampler on 4D SU(3) lattice gauge theory: HMC baseline, 100 training steps, and a plaquette comparison against HMC.

A minimal end-to-end run of l2hmc-qcd on four-dimensional $SU(3)$ lattice gauge theory. The model is a 27.9M-parameter sampler on a $4^4$ lattice with 8 chains, trained for 100 steps at $\beta = 6$ .

The arc is: run plain HMC as a baseline, train the learned sampler, evaluate it, then compare plaquette values between the two. The last figure is the payoff, the difference $|\delta U_{\mu\nu}|^{2}$ between evaluation and HMC.

Note

Console output below is shown as it was logged. The cell outputs come from a later re-run of the same notebook than the publication date suggests.

Load config + build Experiment
HMC
Training
Evaluation

In [1]:

# %matplotlib inline
import matplotlib_inline
matplotlib_inline.backend_inline.set_matplotlib_formats('svg')
import os
os.environ['COLORTERM'] = 'truecolor'
import lovely_tensors as lt
lt.monkey_patch()
lt.set_config(color=False)
# automatically detect and reload local changes to modules
%load_ext autoreload
%autoreload 2
import ezpz
import numpy as np
import matplotlib.pyplot as plt
from l2hmc.utils.plot_helpers import FigAxes
import ambivalent
plt.style.use(ambivalent.STYLES['ambivalent'])
#set_plot_style()

Out[1]:

Using device: cpu

In [2]:

import ezpz
from pathlib import Path
from typing import Optional
from rich import print

import lovely_tensors as lt
import matplotlib.pyplot as plt
import numpy as np
import torch
import yaml

# from l2hmc.utils.dist import setup_torch
seed = np.random.randint(2 ** 32)
print(f"seed: {seed}")

_ = ezpz.setup_torch(seed=seed)
torch.set_default_dtype(torch.float64)
# _ = setup_torch(precision='float64', backend='DDP', seed=seed, port='1234')

logger = ezpz.get_logger()

import l2hmc.group.su3.pytorch.group as g
# from l2hmc.utils.rich import get_console
from l2hmc.common import grab_tensor, print_dict
from l2hmc.configs import dict_to_list_of_overrides, get_experiment
from l2hmc.experiment.pytorch.experiment import Experiment, evaluate  # noqa

Out[2]:

seed: 979520535

Out[2]:

Manually specifying seed=979520535
Using [1 / 1] available "mps" devices !!

[2025-12-31 11:21:26,289] [INFO] [real_accelerator.py:239:get_accelerator] Setting ds_accelerator to mps (auto detect)

[rank0]:W1231 11:21:26.555000 12897 torch/distributed/elastic/multiprocessing/redirects.py:29] NOTE: Redirects are currently not supported in Windows or MacOs.

In [3]:

from l2hmc.utils.plot_helpers import (  # noqa
    plot_scalar,
    plot_chains,
    plot_leapfrogs
)

def savefig(fig: plt.Figure, fname: str, outdir: os.PathLike):
    pngfile = Path(outdir).joinpath(f"pngs/{fname}.png")
    svgfile = Path(outdir).joinpath(f"svgs/{fname}.svg")
    pngfile.parent.mkdir(exist_ok=True, parents=True)
    svgfile.parent.mkdir(exist_ok=True, parents=True)
    fig.savefig(svgfile, transparent=True, bbox_inches='tight')
    fig.savefig(pngfile, transparent=True, bbox_inches='tight', dpi=300)

def plot_metrics(metrics: dict, title: Optional[str] = None, **kwargs):
    outdir = Path(f"./plots-4dSU3/{title}")
    outdir.mkdir(exist_ok=True, parents=True)
    for key, val in metrics.items():
        fig, ax = plot_metric(val, name=key, **kwargs)
        if title is not None:
            ax.set_title(title)
        console.log(f"Saving {key} to {outdir}")
        savefig(fig, f"{key}", outdir=outdir)
        plt.show()

def plot_metric(
        metric: torch.Tensor,
        name: Optional[str] = None,
        **kwargs,
):
    assert len(metric) > 0
    if isinstance(metric[0], (int, float, bool, np.floating)):
        y = np.stack(metric)
        return plot_scalar(y, ylabel=name, **kwargs)
    element_shape = metric[0].shape
    if len(element_shape) == 2:
        y = grab_tensor(torch.stack(metric))
        return plot_leapfrogs(y, ylabel=name)
    if len(element_shape) == 1:
        y = grab_tensor(torch.stack(metric))
        return plot_chains(y, ylabel=name, **kwargs)
    if len(element_shape) == 0:
        y = grab_tensor(torch.stack(metric))
        return plot_scalar(y, ylabel=name, **kwargs)
    raise ValueError

Load config + build Experiment

In [4]:

from rich import print

from l2hmc.configs import CONF_DIR
su3conf = Path(f"{CONF_DIR}/su3test.yaml")
with su3conf.open('r') as stream:
    conf = dict(yaml.safe_load(stream))

In [5]:

import json
from rich import print_json
print_json(json.dumps(conf, indent=4, sort_keys=True))
overrides = dict_to_list_of_overrides(conf)

Out[5]:

{
  "annealing_schedule": {
    "beta_final": 6.0,
    "beta_init": 6.0
  },
  "backend": "DDP",
  "conv": "none",
  "dynamics": {
    "eps": 0.01,
    "eps_fixed": false,
    "group": "SU3",
    "latvolume": [
      4,
      4,
      4,
      4
    ],
    "merge_directions": true,
    "nchains": 8,
    "nleapfrog": 4,
    "use_separate_networks": false,
    "use_split_xnets": false,
    "verbose": true
  },
  "framework": "pytorch",
  "init_aim": false,
  "init_wandb": false,
  "learning_rate": {
    "clip_norm": 1.0,
    "lr_init": "1e-04"
  },
  "loss": {
    "aux_weight": 0.0,
    "charge_weight": 0.0,
    "plaq_weight": 0.1,
    "rmse_weight": 0.1,
    "use_mixed_loss": true
  },
  "net_weights": {
    "v": {
      "q": 1.0,
      "s": 1.0,
      "t": 1.0
    },
    "x": {
      "q": 1.0,
      "s": 0.0,
      "t": 1.0
    }
  },
  "network": {
    "activation_fn": "tanh",
    "dropout_prob": 0.0,
    "units": [
      256
    ],
    "use_batch_norm": false
  },
  "restore": false,
  "save": false,
  "steps": {
    "log": 1,
    "nepoch": 10,
    "nera": 1,
    "print": 1,
    "test": 50
  },
  "use_tb": false,
  "use_wandb": false
}

In [6]:

ptExpSU3 = get_experiment(overrides=[*overrides], build_networks=True)

# console.print(ptExpSU3.config)
state = ptExpSU3.trainer.dynamics.random_state(6.0)
logger.info(f"checkSU(state.x): {g.checkSU(state.x)}")
logger.info(f"checkSU(state.x): {g.checkSU(g.projectSU(state.x))}")
assert isinstance(state.x, torch.Tensor)
assert isinstance(state.beta, torch.Tensor)
assert isinstance(ptExpSU3, Experiment)

Out[6]:

[2025-12-31 11:21:31,796887][I][utils/dist:229:setup_torch_DDP] Caught MASTER_PORT:54505 from environment!
[2025-12-31 11:21:31,830180][I][utils/dist:229:setup_torch_DDP] Caught MASTER_PORT:54505 from environment!
[2025-12-31 11:21:31,831212][W][pytorch/trainer:470:warning] Using torch.float32 on cpu!
[2025-12-31 11:21:32,041943][W][pytorch/trainer:470:warning] Using `torch.optim.Adam` optimizer
[2025-12-31 11:21:32,043077][I][pytorch/trainer:308:count_parameters] num_params in model: 27880456
[2025-12-31 11:21:32,109548][W][pytorch/trainer:274:__init__] logging with freq 1 for wandb.watch
[2025-12-31 11:21:32,123708][I][ipykernel_12897/1455121896:5:<module>] checkSU(state.x): (tensor[8] f64 x∈[1.555e-14, 1.280e-13] μ=5.128e-14 σ=4.647e-14 [1.555e-14, 2.123e-14, 1.571e-14, 4.069e-14, 1.280e-13, 1.997e-14, 4.937e-14, 1.197e-13], tensor[8] f64 x∈[3.730e-13, 4.090e-12] μ=1.462e-12 σ=1.414e-12 [3.730e-13, 5.266e-13, 4.105e-13, 1.261e-12, 4.090e-12, 4.025e-13, 1.513e-12, 3.120e-12])
[2025-12-31 11:21:32,131367][I][ipykernel_12897/1455121896:6:<module>] checkSU(state.x): (tensor[8] f64 x∈[2.760e-16, 2.865e-16] μ=2.803e-16 σ=3.214e-18 [2.760e-16, 2.804e-16, 2.812e-16, 2.865e-16, 2.797e-16, 2.783e-16, 2.781e-16, 2.825e-16], tensor[8] f64 x∈[8.924e-16, 9.900e-16] μ=9.342e-16 σ=3.144e-17 [9.298e-16, 9.900e-16, 9.658e-16, 9.112e-16, 8.924e-16, 9.120e-16, 9.338e-16, 9.386e-16])

In [7]:

# from l2hmc.utils.plot_helpers import set_plot_style
# set_plot_style()

from l2hmc.common import get_timestamp
TSTAMP = get_timestamp()
OUTPUT_DIR = Path(f"./outputs/pt4dSU3/{TSTAMP}")
HMC_DIR = OUTPUT_DIR.joinpath('hmc')
EVAL_DIR = OUTPUT_DIR.joinpath('eval')
TRAIN_DIR = OUTPUT_DIR.joinpath('train')
HMC_DIR.mkdir(exist_ok=True, parents=True)
EVAL_DIR.mkdir(exist_ok=True, parents=True)
TRAIN_DIR.mkdir(exist_ok=True, parents=True)

In [8]:

ptExpSU3.trainer.print_grads_and_weights()
logger.info(ptExpSU3.config)
#console.print(ptExpSU3.config)

Out[8]:

[2025-12-31 11:21:32,255438][I][pytorch/trainer:2003:print_grads_and_weights] --------------------------------------------------------------------------------
[2025-12-31 11:21:32,256420][I][pytorch/trainer:2004:print_grads_and_weights] GRADS:
[2025-12-31 11:21:32,257261][I][l2hmc/common:97:print_dict] networks.xnet.input_layer.xlayer.weight: None None
None
networks.xnet.input_layer.xlayer.bias: None None
None
networks.xnet.input_layer.vlayer.weight: None None
None
networks.xnet.input_layer.vlayer.bias: None None
None
networks.xnet.scale.coeff: None None
None

show the remaining 500 lines

networks.xnet.scale.layer.weight: None None
None
networks.xnet.scale.layer.bias: None None
None
networks.xnet.transf.coeff: None None
None
networks.xnet.transf.layer.weight: None None
None
networks.xnet.transf.layer.bias: None None
None
networks.xnet.transl.weight: None None
None
networks.xnet.transl.bias: None None
None
networks.vnet.input_layer.xlayer.weight: None None
None
networks.vnet.input_layer.xlayer.bias: None None
None
networks.vnet.input_layer.vlayer.weight: None None
None
networks.vnet.input_layer.vlayer.bias: None None
None
networks.vnet.scale.coeff: None None
None
networks.vnet.scale.layer.weight: None None
None
networks.vnet.scale.layer.bias: None None
None
networks.vnet.transf.coeff: None None
None
networks.vnet.transf.layer.weight: None None
None
networks.vnet.transf.layer.bias: None None
None
networks.vnet.transl.weight: None None
None
networks.vnet.transl.bias: None None
None
xeps.0: None None
None
xeps.1: None None
None
xeps.2: None None
None
xeps.3: None None
None
veps.0: None None
None
veps.1: None None
None
veps.2: None None
None
veps.3: None None
None
[2025-12-31 11:21:32,260291][I][pytorch/trainer:2006:print_grads_and_weights] --------------------------------------------------------------------------------
[2025-12-31 11:21:32,260825][I][pytorch/trainer:2007:print_grads_and_weights] WEIGHTS:
[2025-12-31 11:21:32,266638][I][l2hmc/common:97:print_dict] networks.xnet.input_layer.xlayer.weight: torch.Size([256, 18432]) torch.float64
[[ 5.34675944e-03  4.83692293e-03 -1.95424055e-03 ... -3.19398849e-03
  -5.18456944e-03 -4.53952094e-03]
 [-2.11885804e-03 -1.92370769e-03  2.95698048e-03 ... -5.08499583e-03
   5.75697440e-05  3.26761240e-03]
 [-6.75815378e-03  6.55236560e-03 -4.85125066e-03 ... -1.52208833e-03
  -7.36647905e-04 -2.70920219e-03]
 ...
 [ 3.86352724e-03 -5.44473292e-03  1.40281815e-03 ...  1.66045988e-03
  -4.66015396e-03  4.81116461e-03]
 [-2.21214706e-03  2.17645985e-03  5.16413433e-03 ...  2.66270520e-03
  -4.22490545e-03  1.29105152e-03]
 [ 3.40661623e-03 -6.68222571e-03  5.55093943e-03 ... -4.75621507e-03
   3.87005153e-03 -4.84463461e-03]]
networks.xnet.input_layer.xlayer.bias: torch.Size([256]) torch.float64
[-6.27517033e-03  6.95361315e-03  1.04622427e-03  7.10507091e-04
  5.23612152e-03  2.78631701e-03 -3.01457626e-03 -4.28143761e-03
  4.90115525e-03  5.15937100e-03 -4.35841365e-03 -8.57890304e-04
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 -6.01281254e-03 -9.80733306e-05 -2.83395712e-03  4.79620543e-03
  3.46183369e-03  5.38310159e-03 -5.16676674e-04 -4.46611003e-04
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 -5.75595100e-04  1.13273367e-03 -6.36804826e-03 -2.49421139e-03
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 -6.16607788e-03 -6.20185183e-04  5.16693342e-03  1.51241266e-03
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 -3.67713752e-03 -1.37108601e-03 -9.07445912e-04  4.72904453e-03
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 -5.20551542e-03 -4.37575844e-03  2.24417449e-03 -3.34303622e-03
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 -5.83262488e-03 -4.01208448e-03  3.18851118e-03  4.23947203e-03
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  6.42690689e-03  4.67194886e-03  7.16780595e-03  3.11963181e-03
 -2.64557081e-03  7.04843643e-03 -3.62632783e-03 -1.27351965e-03
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 -1.60449482e-03 -1.50436324e-03 -2.07081132e-03 -2.39192222e-03
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 -6.78312494e-04  5.16051779e-03  1.54800285e-03  6.42429157e-03
  5.16674219e-03  1.74328908e-03  1.74515415e-03 -7.45714487e-04
 -7.09407018e-03 -2.28495549e-03 -6.81692804e-03  5.23045719e-03
  6.05730196e-03 -6.65441992e-03  4.86871512e-04  2.57458171e-03
 -6.80560892e-03  6.33736678e-03  2.74625043e-03  4.29353152e-04
  7.29708528e-03  6.67311357e-04  6.74459102e-03 -4.35460386e-03
  6.96138850e-03  1.83322686e-03 -5.98165282e-03 -5.23273300e-03
 -2.78394758e-03  4.86379841e-03  7.12929835e-03  6.31638649e-03
  8.46263464e-04  4.97016772e-03 -1.80937716e-03 -3.78782446e-03
 -6.20744782e-03  6.95460355e-03 -5.58908030e-03 -6.55041816e-03
  1.12782294e-03 -5.78133907e-03  6.95763098e-03 -2.85119940e-03
  2.84811140e-03 -5.93072944e-03  1.68583282e-03 -7.24992565e-03
  5.68267560e-03  6.84689886e-03  4.39475857e-03 -6.97825429e-03
 -8.91775789e-04 -6.67257741e-03 -1.61668491e-03 -3.75113533e-03]
networks.xnet.input_layer.vlayer.weight: torch.Size([256, 18432]) torch.float64
[[-7.12676676e-03 -7.19851121e-03  1.92262912e-03 ...  2.99740846e-03
  -6.84328585e-04 -4.82583241e-03]
 [-3.14961727e-03  3.82433539e-03 -3.95051939e-03 ... -5.88200204e-03
  -4.57862458e-03 -3.20982235e-03]
 [-4.21806104e-03 -1.25481908e-04  5.94817079e-03 ... -5.02316903e-03
   6.84295052e-03  2.60760743e-03]
 ...
 [-2.21680824e-03  1.66057820e-03 -3.77699599e-04 ... -6.31401523e-03
   5.11650858e-03  4.50395935e-03]
 [-5.39315008e-03  5.18607555e-03  1.48544059e-03 ...  7.04561156e-04
   1.51566670e-03  5.53408306e-05]
 [-3.43699633e-03  1.99222704e-03 -1.56282397e-03 ...  6.05750745e-03
  -3.89425094e-03  4.43250670e-03]]
networks.xnet.input_layer.vlayer.bias: torch.Size([256]) torch.float64
[-5.81100325e-03  1.49579324e-03  4.36190909e-03 -8.49009705e-04
  3.20082334e-03  3.33662172e-04  3.72210486e-03  3.36646725e-04
 -2.79477384e-03 -6.72874945e-04 -1.64154347e-03 -4.82888035e-03
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 -2.39514236e-03 -3.44068001e-03  3.37968481e-03 -3.55787200e-03
 -6.41927648e-03  5.97348240e-03  1.17699203e-03  1.10076478e-03
  3.95495686e-04 -8.75149301e-04  3.81136518e-03 -6.95569471e-03
 -1.76999892e-03  1.94536865e-03  3.12176239e-03  2.26232017e-03
  5.70797122e-03  6.74257010e-03  1.58784188e-03 -6.67101797e-03
  4.92471858e-03 -1.51666511e-03  5.14992317e-03 -5.77413524e-03
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  9.64015228e-03 -8.88051809e-03 -5.10172399e-04 -9.07101530e-03
 -1.06343967e-02  1.03739849e-02 -3.29581307e-03  2.30808303e-03
  6.19042182e-03 -1.01627108e-02 -5.90332461e-03  1.07983610e-02
  9.19950237e-03  1.23110613e-03 -3.21699028e-03  5.49590810e-03
  8.94747001e-03  3.44310787e-03 -9.38099103e-03  8.04877795e-03
 -4.02444667e-03  7.01264797e-03 -3.79652055e-03  9.48597155e-03
 -7.56496480e-03  1.00967279e-02 -4.17437588e-03  2.58722056e-03
  6.36675284e-03  9.32221153e-03  2.80300077e-03  5.41941176e-03
  3.34537557e-03 -5.73423342e-03 -6.67158697e-03  9.92357762e-03
 -6.93243096e-03 -2.01520317e-03  5.88531536e-03 -3.85475341e-03
  2.94204521e-03 -1.02806155e-03 -3.88829458e-03 -8.53240867e-03
  1.09355988e-02  8.25391223e-03  8.80964756e-03 -2.91565964e-03
  9.59728057e-03  7.75205289e-03 -7.38385104e-03 -3.29019614e-03
 -4.57266092e-03  3.43399890e-04  6.45364195e-03  1.39322244e-03
 -7.21065388e-03 -3.15960935e-03 -1.09171057e-02 -9.35888449e-03
 -6.55749088e-03  1.10386011e-02  3.43351264e-03 -8.20875171e-03
 -3.09228518e-03  1.76551474e-03  5.19667591e-03  1.03511356e-02
 -3.15706153e-03  1.07906224e-02 -4.48335762e-03 -8.08389191e-03
 -2.41927796e-03  1.09425138e-02  5.05842492e-04  4.76034417e-03
  8.71695420e-04 -1.03555491e-02 -1.96903837e-03 -5.88446821e-03
 -5.81500628e-03  7.83152856e-03 -1.01906154e-02  8.47155625e-03
 -3.04322853e-03 -1.51121378e-03 -8.13083116e-03 -7.11132059e-03
 -4.21085208e-03 -4.57075550e-04 -1.05307749e-05  6.18722644e-04]
networks.vnet.scale.coeff: torch.Size([1, 9216]) torch.float64
[[0. 0. 0. ... 0. 0. 0.]]
networks.vnet.scale.layer.weight: torch.Size([9216, 256]) torch.float64
[[-0.00761683  0.01649503 -0.03692442 ...  0.02964766 -0.00675128
  -0.04392627]
 [-0.00589     0.00963807  0.01736705 ... -0.00313597 -0.03191036
  -0.04455411]
 [ 0.04676324 -0.00047848 -0.03297401 ...  0.00322858 -0.04536244
   0.00209783]
 ...
 [-0.02007618 -0.03535818 -0.02410384 ... -0.05521103  0.01424467
   0.05748986]
 [ 0.05310956 -0.00700633 -0.02713637 ...  0.04775035  0.02916621
   0.04902634]
 [-0.01296662 -0.04693715  0.01220499 ... -0.00491532  0.01583796
  -0.04115975]]
networks.vnet.scale.layer.bias: torch.Size([9216]) torch.float64
[-0.05532441  0.04261724  0.01327692 ... -0.02727258 -0.00564456
  0.05526191]
networks.vnet.transf.coeff: torch.Size([1, 9216]) torch.float64
[[0. 0. 0. ... 0. 0. 0.]]
networks.vnet.transf.layer.weight: torch.Size([9216, 256]) torch.float64
[[-8.06922849e-05 -4.86478073e-02 -1.74624561e-02 ...  5.14657732e-02
   1.11850013e-02 -2.96893353e-03]
 [ 5.42348063e-02 -6.03741696e-02  7.92496492e-04 ... -6.04141558e-02
   3.33679678e-02  4.60974095e-03]
 [ 3.08612215e-02  4.24380905e-02 -3.32743200e-02 ...  1.54626832e-02
   5.21812386e-02  3.67741292e-02]
 ...
 [-5.65227179e-02  8.79422558e-03  7.17933897e-03 ... -5.81218986e-02
   6.02176274e-02  6.21102000e-02]
 [ 6.00315605e-02 -3.43763009e-02 -1.19883634e-02 ... -3.51821714e-02
   3.69714337e-02 -5.36188244e-02]
 [ 1.06812561e-02  2.56660157e-02 -1.52740900e-02 ... -1.88780392e-02
   1.36668521e-02  2.64287697e-02]]
networks.vnet.transf.layer.bias: torch.Size([9216]) torch.float64
[-0.04448601  0.05444971  0.05047074 ... -0.02935933 -0.0370173
 -0.03672931]
networks.vnet.transl.weight: torch.Size([9216, 256]) torch.float64
[[ 0.03190299  0.03209705 -0.00548753 ... -0.01850558 -0.03685719
  -0.01781093]
 [ 0.05555422  0.01567202  0.05491162 ... -0.01233239 -0.01591408
   0.00792362]
 [ 0.03025957 -0.02511794 -0.05441319 ... -0.03709203 -0.00967987
  -0.05173925]
 ...
 [ 0.06073248  0.05392621 -0.04660168 ...  0.00607929 -0.03720785
   0.00577768]
 [ 0.00502593 -0.01161339  0.04961748 ...  0.02408661 -0.03853835
  -0.00770556]
 [-0.00473506  0.04545451 -0.05643493 ...  0.04181123 -0.03172008
  -0.05468202]]
networks.vnet.transl.bias: torch.Size([9216]) torch.float64
[ 0.02885058 -0.01754039  0.02482559 ... -0.014913   -0.02643838
 -0.02004313]
xeps.0: torch.Size([]) torch.float64
0.01
xeps.1: torch.Size([]) torch.float64
0.01
xeps.2: torch.Size([]) torch.float64
0.01
xeps.3: torch.Size([]) torch.float64
0.01
veps.0: torch.Size([]) torch.float64
0.01
veps.1: torch.Size([]) torch.float64
0.01
veps.2: torch.Size([]) torch.float64
0.01
veps.3: torch.Size([]) torch.float64
0.01
[2025-12-31 11:21:32,293773][I][pytorch/trainer:2009:print_grads_and_weights] --------------------------------------------------------------------------------
[2025-12-31 11:21:32,294670][I][ipykernel_12897/3178487732:2:<module>] ExperimentConfig(wandb={'setup': {'id': None, 'group': None, 'config': None, 'save_code': True, 'sync_tensorboard': True, 'mode': 'online', 'resume': 'allow', 'entity': 'l2hmc-qcd', 'project': 'l2hmc-qcd', 'settings': {'start_method': 'thread'}, 'tags': ['beta_init=6.0', 'beta_final=6.0']}}, steps=Steps(nera=1, nepoch=10, test=50, log=1, print=1, extend_last_era=1), framework='pytorch', loss=LossConfig(use_mixed_loss=True, charge_weight=0.0, rmse_weight=0.1, plaq_weight=0.1, aux_weight=0.0), network=NetworkConfig(units=[256], activation_fn='tanh', dropout_prob=0.0, use_batch_norm=False), conv=ConvolutionConfig(filters=[], sizes=[], pool=[]), net_weights=NetWeights(x=NetWeight(s=0.0, t=1.0, q=1.0), v=NetWeight(s=1.0, t=1.0, q=1.0)), dynamics=DynamicsConfig(nchains=8, group='SU3', latvolume=[4, 4, 4, 4], nleapfrog=4, eps=0.01, eps_hmc=0.25, use_ncp=True, verbose=True, eps_fixed=False, use_split_xnets=False, use_separate_networks=False, merge_directions=True), learning_rate=LearningRateConfig(lr_init=0.0001, mode='auto', monitor='loss', patience=5, cooldown=0, warmup=1000, verbose=True, min_lr=1e-06, factor=0.98, min_delta=0.0001, clip_norm=1.0), annealing_schedule=AnnealingSchedule(beta_init=6.0, beta_final=6.0, dynamic=False), gradient_accumulation_steps=1, restore=False, save=False, c1=0.0, port='2345', compile=True, profile=False, init_aim=False, init_wandb=False, use_wandb=False, use_tb=False, debug_mode=False, default_mode=True, print_config=True, precision='float32', ignore_warnings=True, backend='DDP', seed=9992, ds_config_path='/Users/samforeman/projects/saforem2/l2hmc-qcd/src/l2hmc/conf/ds_config.yaml', name=None, width=200, nchains=None, compression=False)

HMC

In [9]:

xhmc, history_hmc = evaluate(
    nsteps=50,
    exp=ptExpSU3,
    beta=6.0,
    x=state.x,
    eps=0.1,
    nleapfrog=8,
    job_type='hmc',
    nlog=1,
    nprint=50,
    grab=True
)

Out[9]:

[2025-12-31 11:21:32,364810][I][pytorch/experiment:117:evaluate] Running 50 steps of hmc at beta=6.0000
[2025-12-31 11:21:32,365632][I][pytorch/experiment:121:evaluate] STEP: 0
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[2025-12-31 11:21:33,232529][I][pytorch/experiment:121:evaluate] STEP: 3
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[2025-12-31 11:21:35,182088][I][pytorch/experiment:121:evaluate] STEP: 10

show the remaining 39 lines

[2025-12-31 11:21:35,435679][I][pytorch/experiment:121:evaluate] STEP: 11
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In [10]:

dataset_hmc = history_hmc.get_dataset()
_ = history_hmc.plot_all(title='HMC')

Out[10]:

In [11]:

xhmc = ptExpSU3.trainer.dynamics.unflatten(xhmc)
print(f"checkSU(x_eval): {g.checkSU(xhmc)}")
print(f"checkSU(x_eval): {g.checkSU(g.projectSU(xhmc))}")

Out[11]:

checkSU(x_eval): (tensor[8] f64 x∈[2.502e-16, 5.331e-16] μ=4.279e-16 σ=1.346e-16 [2.502e-16, 5.244e-16, 5.210e-16,
2.808e-16, 5.331e-16, 5.305e-16, 5.168e-16, 2.662e-16], tensor[8] f64 x∈[7.632e-16, 1.578e-15] μ=1.246e-15
σ=3.609e-16 [7.632e-16, 1.513e-15, 1.578e-15, 8.960e-16, 1.497e-15, 1.480e-15, 1.454e-15, 7.840e-16])

Out[11]:

checkSU(x_eval): (tensor[8] f64 x∈[2.436e-16, 3.161e-16] μ=2.897e-16 σ=3.147e-17 [2.436e-16, 3.149e-16, 3.161e-16,
2.702e-16, 3.107e-16, 3.075e-16, 3.091e-16, 2.455e-16], tensor[8] f64 x∈[7.665e-16, 9.669e-16] μ=8.918e-16
σ=8.773e-17 [7.839e-16, 9.658e-16, 9.669e-16, 7.665e-16, 9.371e-16, 9.338e-16, 9.663e-16, 8.140e-16])

Training

In [12]:

import time
from l2hmc.utils.history import BaseHistory, summarize_dict

history_train = BaseHistory()
x = state.x
for step in range(100):
    # log.info(f'HMC STEP: {step}')
    tic = time.perf_counter()
    x, metrics_ = ptExpSU3.trainer.train_step(
        (x, state.beta)
    )
    toc = time.perf_counter()
    metrics = {
        'train_step': step,
        'dt': toc - tic,
        **metrics_,
    }
    if step % 5 == 0:
        avgs = history_train.update(metrics)
        summary = summarize_dict(avgs)
        logger.info(summary)

Out[12]:

[2025-12-31 11:21:53,926183][I][ipykernel_12897/30352159:21:<module>] train_step=0 dt=2.476 energy=28.872 logprob=28.636 logdet=0.236 sldf=0.149 sldb=-0.154 sld=0.236 xeps=0.010 veps=0.010 acc=0.202 sumlogdet=-0.022 beta=6.000 acc_mask=0.250 loss=91.790 plaqs=-0.000 sinQ=0.001 intQ=0.019 dQint=0.006 dQsin=0.000
[2025-12-31 11:22:04,277222][I][ipykernel_12897/30352159:21:<module>] train_step=5 dt=2.144 energy=-320.934 logprob=-321.240 logdet=0.306 sldf=0.180 sldb=-0.160 sld=0.306 xeps=0.010 veps=0.010 acc=1.000 sumlogdet=0.081 beta=6.000 acc_mask=1.000 loss=-682.159 plaqs=0.038 sinQ=0.000 intQ=0.006 dQint=0.022 dQsin=0.001
[2025-12-31 11:22:15,931363][I][ipykernel_12897/30352159:21:<module>] train_step=10 dt=2.276 energy=-734.319 logprob=-734.432 logdet=0.113 sldf=0.082 sldb=-0.104 sld=0.113 xeps=0.011 veps=0.010 acc=1.000 sumlogdet=-0.078 beta=6.000 acc_mask=1.000 loss=-561.834 plaqs=0.084 sinQ=0.001 intQ=0.015 dQint=0.019 dQsin=0.001
[2025-12-31 11:22:26,985074][I][ipykernel_12897/30352159:21:<module>] train_step=15 dt=2.453 energy=-1211.909 logprob=-1211.984 logdet=0.075 sldf=0.041 sldb=-0.034 sld=0.075 xeps=0.011 veps=0.010 acc=1.000 sumlogdet=0.027 beta=6.000 acc_mask=1.000 loss=-583.813 plaqs=0.128 sinQ=0.001 intQ=0.009 dQint=0.016 dQsin=0.001
[2025-12-31 11:22:38,760535][I][ipykernel_12897/30352159:21:<module>] train_step=20 dt=2.289 energy=-1519.041 logprob=-1519.097 logdet=0.056 sldf=0.042 sldb=-0.048 sld=0.056 xeps=0.012 veps=0.010 acc=0.880 sumlogdet=-0.022 beta=6.000 acc_mask=0.875 loss=-361.916 plaqs=0.164 sinQ=-0.000 intQ=-0.001 dQint=0.024 dQsin=0.002
[2025-12-31 11:22:50,090777][I][ipykernel_12897/30352159:21:<module>] train_step=25 dt=2.657 energy=-1783.602 logprob=-1783.604 logdet=0.002 sldf=-0.001 sldb=0.002 sld=0.002 xeps=0.012 veps=0.011 acc=0.967 sumlogdet=0.014 beta=6.000 acc_mask=1.000 loss=-463.398 plaqs=0.196 sinQ=-0.001 intQ=-0.009 dQint=0.010 dQsin=0.001
[2025-12-31 11:23:00,910782][I][ipykernel_12897/30352159:21:<module>] train_step=30 dt=2.121 energy=-1992.581 logprob=-1992.687 logdet=0.106 sldf=0.070 sldb=-0.076 sld=0.106 xeps=0.013 veps=0.011 acc=0.907 sumlogdet=-0.029 beta=6.000 acc_mask=0.875 loss=-390.638 plaqs=0.215 sinQ=0.001 intQ=0.020 dQint=0.025 dQsin=0.002
[2025-12-31 11:23:12,800585][I][ipykernel_12897/30352159:21:<module>] train_step=35 dt=2.863 energy=-2166.294 logprob=-2166.477 logdet=0.183 sldf=0.119 sldb=-0.127 sld=0.183 xeps=0.013 veps=0.011 acc=0.983 sumlogdet=-0.030 beta=6.000 acc_mask=1.000 loss=-419.961 plaqs=0.239 sinQ=-0.001 intQ=-0.011 dQint=0.018 dQsin=0.001
[2025-12-31 11:23:25,106273][I][ipykernel_12897/30352159:21:<module>] train_step=40 dt=2.729 energy=-2411.163 logprob=-2411.124 logdet=-0.039 sldf=-0.044 sldb=0.084 sld=-0.039 xeps=0.013 veps=0.011 acc=0.467 sumlogdet=0.072 beta=6.000 acc_mask=0.500 loss=117.772 plaqs=0.261 sinQ=-0.001 intQ=-0.014 dQint=0.010 dQsin=0.001
[2025-12-31 11:23:36,199067][I][ipykernel_12897/30352159:21:<module>] train_step=45 dt=2.253 energy=-2521.896 logprob=-2521.849 logdet=-0.047 sldf=-0.027 sldb=0.023 sld=-0.047 xeps=0.014 veps=0.011 acc=0.875 sumlogdet=-0.015 beta=6.000 acc_mask=0.875 loss=-110.950 plaqs=0.275 sinQ=-0.000 intQ=-0.001 dQint=0.016 dQsin=0.001
[2025-12-31 11:23:46,263515][I][ipykernel_12897/30352159:21:<module>] train_step=50 dt=1.960 energy=-2705.348 logprob=-2705.224 logdet=-0.125 sldf=-0.090 sldb=0.114 sld=-0.125 xeps=0.014 veps=0.012 acc=0.955 sumlogdet=0.072 beta=6.000 acc_mask=0.875 loss=-304.184 plaqs=0.289 sinQ=0.000 intQ=0.000 dQint=0.012 dQsin=0.001
[2025-12-31 11:23:57,308554][I][ipykernel_12897/30352159:21:<module>] train_step=55 dt=2.079 energy=-2851.051 logprob=-2850.890 logdet=-0.161 sldf=-0.099 sldb=0.092 sld=-0.161 xeps=0.013 veps=0.012 acc=0.883 sumlogdet=-0.051 beta=6.000 acc_mask=0.875 loss=-286.513 plaqs=0.304 sinQ=0.000 intQ=0.006 dQint=0.019 dQsin=0.001

show the remaining 8 lines

[2025-12-31 11:24:07,402194][I][ipykernel_12897/30352159:21:<module>] train_step=60 dt=1.962 energy=-2979.375 logprob=-2979.152 logdet=-0.223 sldf=-0.142 sldb=0.146 sld=-0.223 xeps=0.013 veps=0.013 acc=1.000 sumlogdet=0.027 beta=6.000 acc_mask=1.000 loss=-456.457 plaqs=0.322 sinQ=0.001 intQ=0.012 dQint=0.007 dQsin=0.001
[2025-12-31 11:24:18,182804][I][ipykernel_12897/30352159:21:<module>] train_step=65 dt=1.809 energy=-3174.287 logprob=-3174.011 logdet=-0.276 sldf=-0.179 sldb=0.200 sld=-0.276 xeps=0.013 veps=0.013 acc=1.000 sumlogdet=0.063 beta=6.000 acc_mask=1.000 loss=-478.375 plaqs=0.344 sinQ=-0.000 intQ=-0.004 dQint=0.019 dQsin=0.001
[2025-12-31 11:24:28,457927][I][ipykernel_12897/30352159:21:<module>] train_step=70 dt=1.881 energy=-3395.198 logprob=-3394.849 logdet=-0.348 sldf=-0.214 sldb=0.212 sld=-0.348 xeps=0.013 veps=0.014 acc=1.000 sumlogdet=-0.039 beta=6.000 acc_mask=1.000 loss=-399.666 plaqs=0.362 sinQ=-0.000 intQ=-0.000 dQint=0.019 dQsin=0.001
[2025-12-31 11:24:38,221877][I][ipykernel_12897/30352159:21:<module>] train_step=75 dt=1.766 energy=-3494.564 logprob=-3494.095 logdet=-0.469 sldf=-0.292 sldb=0.290 sld=-0.469 xeps=0.013 veps=0.014 acc=1.000 sumlogdet=-0.012 beta=6.000 acc_mask=1.000 loss=-480.226 plaqs=0.381 sinQ=-0.001 intQ=-0.015 dQint=0.015 dQsin=0.001
[2025-12-31 11:24:47,896467][I][ipykernel_12897/30352159:21:<module>] train_step=80 dt=1.986 energy=-3711.719 logprob=-3710.920 logdet=-0.799 sldf=-0.481 sldb=0.447 sld=-0.799 xeps=0.013 veps=0.015 acc=1.000 sumlogdet=-0.128 beta=6.000 acc_mask=1.000 loss=-521.006 plaqs=0.402 sinQ=-0.000 intQ=-0.002 dQint=0.015 dQsin=0.001
[2025-12-31 11:24:58,094056][I][ipykernel_12897/30352159:21:<module>] train_step=85 dt=2.155 energy=-3878.745 logprob=-3877.647 logdet=-1.097 sldf=-0.698 sldb=0.720 sld=-1.097 xeps=0.012 veps=0.015 acc=1.000 sumlogdet=0.054 beta=6.000 acc_mask=1.000 loss=-624.681 plaqs=0.421 sinQ=-0.001 intQ=-0.011 dQint=0.013 dQsin=0.001
[2025-12-31 11:25:08,390086][I][ipykernel_12897/30352159:21:<module>] train_step=90 dt=2.014 energy=-4033.966 logprob=-4032.690 logdet=-1.276 sldf=-0.798 sldb=0.806 sld=-1.276 xeps=0.012 veps=0.016 acc=1.000 sumlogdet=0.006 beta=6.000 acc_mask=1.000 loss=-473.295 plaqs=0.440 sinQ=0.000 intQ=0.007 dQint=0.013 dQsin=0.001
[2025-12-31 11:25:19,868897][I][ipykernel_12897/30352159:21:<module>] train_step=95 dt=3.220 energy=-4188.433 logprob=-4186.851 logdet=-1.581 sldf=-0.977 sldb=0.967 sld=-1.581 xeps=0.012 veps=0.016 acc=1.000 sumlogdet=-0.075 beta=6.000 acc_mask=1.000 loss=-855.853 plaqs=0.457 sinQ=-0.002 intQ=-0.023 dQint=0.013 dQsin=0.001

In [13]:

dataset_train = history_train.get_dataset()
_ = history_train.plot_all(
    title='Train',
    num_chains=x.shape[0],
)

Out[13]:

Evaluation

In [14]:

# state = ptExpSU3.trainer.dynamics.random_state(6.0)
xeval, history_eval = evaluate(
    nsteps=50,
    exp=ptExpSU3,
    beta=6.0,
    # x=state.x,
    job_type='eval',
    nlog=1,
    nprint=50,
    grab=True,
)

Out[14]:

[2025-12-31 11:25:33,856738][I][pytorch/experiment:117:evaluate] Running 50 steps of eval at beta=6.0000
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In [15]:

dataset_eval = history_eval.get_dataset()
_ = history_eval.plot_all(title='Eval')

Out[15]:

In [16]:

xeval = ptExpSU3.trainer.dynamics.unflatten(xeval)
logger.info(f"checkSU(x_eval): {g.checkSU(xeval)}")
logger.info(f"checkSU(x_eval): {g.checkSU(g.projectSU(xeval))}")

Out[16]:

[2025-12-31 11:26:09,856048][I][ipykernel_12897/2193937887:2:<module>] checkSU(x_eval): (tensor[8] f64 x∈[1.386e-16, 1.523e-16] μ=1.437e-16 σ=4.820e-18 [1.401e-16, 1.386e-16, 1.523e-16, 1.436e-16, 1.473e-16, 1.470e-16, 1.416e-16, 1.390e-16], tensor[8] f64 x∈[4.900e-16, 7.789e-16] μ=6.360e-16 σ=1.034e-16 [6.312e-16, 5.391e-16, 5.597e-16, 7.129e-16, 6.306e-16, 7.453e-16, 4.900e-16, 7.789e-16])
[2025-12-31 11:26:09,868908][I][ipykernel_12897/2193937887:3:<module>] checkSU(x_eval): (tensor[8] f64 x∈[1.351e-16, 1.500e-16] μ=1.409e-16 σ=4.762e-18 [1.420e-16, 1.388e-16, 1.392e-16, 1.500e-16, 1.351e-16, 1.446e-16, 1.413e-16, 1.365e-16], tensor[8] f64 x∈[4.802e-16, 7.133e-16] μ=6.128e-16 σ=1.057e-16 [7.131e-16, 5.905e-16, 4.802e-16, 7.133e-16, 4.960e-16, 7.133e-16, 5.094e-16, 6.869e-16])

In [17]:

import matplotlib.pyplot as plt
pdiff = dataset_eval.plaqs - dataset_hmc.plaqs
pdiff
import xarray as xr

fig, ax = plt.subplots(figsize=(12, 4))
(pdiff ** 2).plot(ax=ax)  #, robust=True)
ax.set_title(r"$\left|\delta U_{\mu\nu}\right|^{2}$ (HMC - Eval)")
outfile = Path(EVAL_DIR).joinpath('pdiff.svg')
#%xmode fig.savefig(outfile.as_posix(), dpi=400, bbox_inches='tight')
plt.show()

Out[17]:

Cite this post

@article{foreman2023-l2hmc-qcd-example,
    author = {Sam Foreman},
    title = {{l2hmc-qcd Example: 4D SU(3)}},
    year = {2023},
    month = dec,
    day = {05},
    journal = {Sam Foreman's Blog},
    howpublished = {https://samf.sh},
    url = {https://samf.sh/posts/2023/12/05/},
}

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posts/ 🎢 l2hmc-qcd Example: 2D U(1) [n]ext → posts/ 🎢 L2HMC for LQCD