8. Gaussian process regression with MD data#
Lennard-Jones fluid#
Finally, we replace the mock MD data with actual simulation data. To generate a dataset, we run a similar active learning simulation than in the previous tutorial. The YAML input for this simulation looks like this:
options:
output: data/parabolic_lj_md
write_freq: 100
use_tstamp: True
grid:
Lx: 1470.
Ly: 1.
Nx: 200
Ny: 1
xE: ['D', 'N', 'N']
xW: ['D', 'N', 'N']
yS: ['P', 'P', 'P']
yN: ['P', 'P', 'P']
xE_D: 0.8
xW_D: 0.8
geometry:
type: parabolic
hmin: 12.
hmax: 60.
U: 0.12
V: 0.
numerics:
CFL: 0.5
adaptive: 1
tol: 1e-8
dt: 0.05
max_it: 50_000
properties:
shear: 2.15
bulk: 0.
EOS: DH
T: 1.0
rho0: 0.8
gp:
press:
fix_noise: True
atol: 1.5
rtol: 0.
obs_stddev: 2.e-2
max_steps: 10
shear:
fix_noise: True
atol: 1.5
rtol: 0.
obs_stddev: 4.e-3
max_steps: 10
db:
dtool: True
dtool_path: data/gapflow_training_lj
init_size: 5
init_method: rand
md:
system: lj # Lennard-Jones system
ncpu: 10 # Max. number of CPUs
atoms_per_cpu: 1000 # Min. number of atoms per CPU
infile: lmp/lj/in.lmp # Location of the LAMMPS input file
wallfile: lmp/lj/wall.lmp # Location of the LAMMPS file that contains the coordinates of the wall atoms
vWall: 0.12 # Sliding velocity of the lower wall (LJ units)
cutoff: 2.5 # Cutoff radius of LJ interactions
temp: 1.0 # Temperature
tsample: 100000 # Sampling time
Here, the md section of the YAML file configures the MD runs.
We ran the simulation from the command line using
mpirun -n 1 python GaPFlow -i parabolic_1d_lj_gp_lammps.yaml
Note that we start the run on a single processor (mpirun -n 1). The number of MPI processes for the LAMMPS simulations are configured in the YAML, and spawned from the parent process.
The active learning simulation generated 24 MD simulations over the course of a run which took approximately 3.5 hours.
We now want to use this dataset in a subsequent simulation with active learning turned off. The training dataset has been uploaded to Zenodo.
We now download the dataset to our local machine and test it there:
!wget -O- https://zenodo.org/records/18761223/files/gapflow_training_lj.tar.gz | tar -xz -C data
--2026-07-20 15:03:22-- https://zenodo.org/records/18761223/files/gapflow_training_lj.tar.gz
Resolving zenodo.org (zenodo.org)... 137.138.153.219, 188.184.98.114, 188.184.103.118, ...
Connecting to zenodo.org (zenodo.org)|137.138.153.219|:443...
connected.
HTTP request sent, awaiting response...
200 OK
Length: 70537865 (67M) [application/octet-stream]
Saving to: ‘STDOUT’
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The database has been generated with a legacy README schema. Before we can use it, we need to migrate to the new schema (0 -> 1), which adds also the wall velocities to the possible inputs X:
!gpf_migrate_readme data/gapflow_training_lj --U 0.12 --V 0.0
20260224_182647_lj-018 (75a567f8-044e-42b4-ad4d-03b9ecface92): schema_version 0 -> 1
20260224_155922_lj-001 (136f2017-079c-4fe3-954f-43df34d14a78): schema_version 0 -> 1
20260224_185038_lj-021 (76d4caaa-2631-438b-9f59-96788645c084): schema_version 0 -> 1
20260224_160819_lj-002 (6dc20674-c40c-4e80-baa7-10072d04ab10): schema_version 0 -> 1
20260224_184224_lj-020 (7d473ab6-d124-4053-9fa3-78956ba7047c): schema_version 0 -> 1
20260224_185838_lj-022 (e08cf49b-671a-407a-9a11-22c4257f9911): schema_version 0 -> 1
20260224_180027_lj-015 (f9886de7-bf1a-4207-90b5-ed8f555de8bd): schema_version 0 -> 1
20260224_170410_lj-009 (a16621f5-af5d-4a75-ad1b-65c603f30955): schema_version 0 -> 1
20260224_163858_lj-006 (e5ddd7d7-2b60-479a-b1f4-3d385580c2d0): schema_version 0 -> 1
20260224_171301_lj-010 (78fbb59f-b0ba-4f7c-b81e-52576dacd430): schema_version 0 -> 1
20260224_190623_lj-023 (8429bfe6-72e9-48ed-a59f-cae66f88171b): schema_version 0 -> 1
20260224_162410_lj-004 (3cb97a66-ff04-4c0a-87c2-c957aeba9bf8): schema_version 0 -> 1
20260224_180933_lj-016 (cd8a7e57-fadc-41b2-9b8b-76e0551b20eb): schema_version 0 -> 1
20260224_173203_lj-012 (261e68d4-e64b-4dc0-ae1c-cfbba8344d28): schema_version 0 -> 1
20260224_172232_lj-011 (72ffa906-4a7a-4d4b-8c4d-78856c000ab7): schema_version 0 -> 1
20260224_161608_lj-003 (efa4acb4-afb7-4d64-9362-0681462786ad): schema_version 0 -> 1
20260224_163134_lj-005 (36d17b6a-a918-4f26-a598-9ca1bf6571b5): schema_version 0 -> 1
20260224_174137_lj-013 (339d8f5c-6674-4d38-9442-0142412347c2): schema_version 0 -> 1
20260224_181817_lj-017 (f38ad65d-a362-4db6-acd8-6272a2d59345): schema_version 0 -> 1
20260224_165442_lj-008 (811e2b05-eff1-4054-acba-e0beff87ec3c): schema_version 0 -> 1
20260224_164826_lj-007 (2ccccd7a-04ab-4d78-be3d-cb3adbb186ac): schema_version 0 -> 1
20260224_175053_lj-014 (15152602-5c66-49f5-83fd-451124c4124d): schema_version 0 -> 1
20260224_191508_lj-024 (9efd81b7-69f6-4b25-997d-4014b1245d9f): schema_version 0 -> 1
20260224_183412_lj-019 (8cb5599d-a5db-4c53-bb8f-e819f05d89e1): schema_version 0 -> 1
Migrated 24 dataset(s), 0 already up to date.
The input file looks exactly the same, but we turn active learning off. We also have to make sure that GaPFlow looks for the training dataset in the right location.
lj_gp_input = """
options:
output: data/parabolic_lj_md
write_freq: 100
use_tstamp: True
grid:
Lx: 1470.
Ly: 1.
Nx: 200
Ny: 1
xE: ['D', 'N', 'N']
xW: ['D', 'N', 'N']
yS: ['P', 'P', 'P']
yN: ['P', 'P', 'P']
xE_D: 0.8
xW_D: 0.8
geometry:
type: parabolic
hmin: 12.
hmax: 60.
U: 0.12
V: 0.
numerics:
CFL: 0.5
adaptive: 1
tol: 1e-8
dt: 0.05
max_it: 50_000
properties:
shear: 2.15
bulk: 0.
EOS: DH
T: 1.0
rho0: 0.8
gp:
press:
fix_noise: True
atol: 1.5
rtol: 0.
max_steps: 10
active_learning: False # AL turned off, so no new MD data is generated
shear:
fix_noise: True
atol: 1.5
rtol: 0.
max_steps: 10
active_learning: False # AL turned off, so no new MD data is generated
db:
dtool: True
dtool_path: data/gapflow_training_lj # downloaded from Zenodo
init_size: 5
init_method: rand
init_width: 0.01 # default (for density)
md:
system: lj
ncpu: 10
atoms_per_cpu: 1000
infile: lmp/lj/in.lmp
wallfile: lmp/lj/wall.lmp
vWall: 0.12
cutoff: 2.5
temp: 1.0
tsample: 100000
"""
from GaPFlow import Problem
lj_problem = Problem.from_string(lj_gp_input)
*************************************************************
* PROBLEM SETUP *
*************************************************************
- options:
- output : data/parabolic_lj_md
- write_freq : 100
- use_tstamp : True
- silent : False
- grid:
- Nx : 200
- Lx : 1470.0
- dx : 7.35
- Ny : 1
- Ly : 1.0
- dy : 1.0
- dim : 1
- bc_xE_P : [False, False, False]
- bc_xE_D : [True, False, False]
- bc_xE_N : [False, True, True]
- bc_xW_P : [False, False, False]
- bc_xW_D : [True, False, False]
- bc_xW_N : [False, True, True]
- bc_xE_D_val : 0.8
- bc_xW_D_val : 0.8
- bc_yS_P : [True, True, True]
- bc_yS_D : [False, False, False]
- bc_yS_N : [False, False, False]
- bc_yN_P : [True, True, True]
- bc_yN_D : [False, False, False]
- bc_yN_N : [False, False, False]
- geometry:
- U : 0.12
- V : 0.0
- type : parabolic
- flip : False
- hmin : 12.0
- hmax : 60.0
- numerics:
- tol : 1e-08
- max_it : 50000
- dt : 0.05
- adaptive : True
- CFL : 0.5
- MC_order : 1
- properties:
- shear : 2.15
- bulk : 0.0
- EOS : DH
- rho0 : 0.8
- P0 : 101325.0
- C1 : 35000000000.0
- C2 : 1.23
- elastic:
- enabled : False
- gp:
- press_gp : True
- shear_gp : True
- derived_features : []
- press:
- obs_stddev : 0.0
- fix_noise : True
- max_steps : 10
- pause_steps : 1000
- active_learning : False
- similarity_check : False
- allowed_skips : 0
- pause_on_high_residual : False
- tolerance_protocol : rtol_delta
- atol : 1.5
- rtol : 0.0
- atol_reduction_factor : 0.5
- tol_rmid : 1e-06
- tol_alpha : 2.0
- active_dims : [0, 3]
- shear:
- obs_stddev : 0.0
- fix_noise : True
- max_steps : 10
- pause_steps : 1000
- active_learning : False
- similarity_check : False
- allowed_skips : 0
- pause_on_high_residual : False
- tolerance_protocol : rtol_delta
- atol : 1.5
- rtol : 0.0
- atol_reduction_factor : 0.5
- tol_rmid : 1e-06
- tol_alpha : 2.0
- active_dims_x : [0, 1, 3]
- active_dims_y : [0, 2, 3]
- db:
- dtool_path : data/gapflow_training_lj
- init_size : 5
- init_method : rand
- init_halfwidth : [0.01, 0.5, 0.5]
- init_seed : 123
- normalizer_X : minmax
- normalizer_Y : standard
- md:
- system : lj
- ncpu : 10
- atoms_per_cpu : 1000
- infile : lmp/lj/in.lmp
- wallfile : lmp/lj/wall.lmp
- vWall : 0.12
- cutoff : 2.5
- temp : 1.0
- tsample : 100000
*************************************************************
* PROBLEM SETUP COMPLETED *
*************************************************************
- 75a567f8-044e-42b4-ad4d-03b9ecface92 (20260224_182647_lj-018)
- 136f2017-079c-4fe3-954f-43df34d14a78 (20260224_155922_lj-001)
- 76d4caaa-2631-438b-9f59-96788645c084 (20260224_185038_lj-021)
- 6dc20674-c40c-4e80-baa7-10072d04ab10 (20260224_160819_lj-002)
- 7d473ab6-d124-4053-9fa3-78956ba7047c (20260224_184224_lj-020)
- e08cf49b-671a-407a-9a11-22c4257f9911 (20260224_185838_lj-022)
- f9886de7-bf1a-4207-90b5-ed8f555de8bd (20260224_180027_lj-015)
- a16621f5-af5d-4a75-ad1b-65c603f30955 (20260224_170410_lj-009)
- e5ddd7d7-2b60-479a-b1f4-3d385580c2d0 (20260224_163858_lj-006)
- 78fbb59f-b0ba-4f7c-b81e-52576dacd430 (20260224_171301_lj-010)
- 8429bfe6-72e9-48ed-a59f-cae66f88171b (20260224_190623_lj-023)
- 3cb97a66-ff04-4c0a-87c2-c957aeba9bf8 (20260224_162410_lj-004)
- cd8a7e57-fadc-41b2-9b8b-76e0551b20eb (20260224_180933_lj-016)
- 261e68d4-e64b-4dc0-ae1c-cfbba8344d28 (20260224_173203_lj-012)
- 72ffa906-4a7a-4d4b-8c4d-78856c000ab7 (20260224_172232_lj-011)
- efa4acb4-afb7-4d64-9362-0681462786ad (20260224_161608_lj-003)
- 36d17b6a-a918-4f26-a598-9ca1bf6571b5 (20260224_163134_lj-005)
- 339d8f5c-6674-4d38-9442-0142412347c2 (20260224_174137_lj-013)
- f38ad65d-a362-4db6-acd8-6272a2d59345 (20260224_181817_lj-017)
- 811e2b05-eff1-4054-acba-e0beff87ec3c (20260224_165442_lj-008)
- 2ccccd7a-04ab-4d78-be3d-cb3adbb186ac (20260224_164826_lj-007)
- 15152602-5c66-49f5-83fd-451124c4124d (20260224_175053_lj-014)
- 9efd81b7-69f6-4b25-997d-4014b1245d9f (20260224_191508_lj-024)
- 8cb5599d-a5db-4c53-bb8f-e819f05d89e1 (20260224_183412_lj-019)
Loading 24 local datasets in 'data/gapflow_training_lj'.
Writing output into: data/2026-07-20_150420_parabolic_lj_md
We see that the downloaded training simulations have been recognized. Thus, we are ready to run the simulation.
lj_problem.run()
#-----------------GP TRAINING (ZZ)-----------------
# Timestep : 0
# Reason : DB
# Database size: 24
# Objective : -5.1033
# Hyperparam :
# - Log scale : 1.85187 1.74605
# - Log amp : 4.16391
# - Log noise : -4.28893
#--------------------------------------------------
#-----------------GP TRAINING (XZ)-----------------
# Timestep : 0
# Reason : DB
# Database size: 24
# Objective : 43.765
# Hyperparam :
# - Log scale : 3.24342 -0.24961 0.37362
# - Log amp : 0.98870
# - Log noise : -1.38213
#--------------------------------------------------
-------------------------------------------------------------
Step Timestep Time CFL Residual
-------------------------------------------------------------
0 1.1084e-01 0.0000e+00 5.0000e-01 1.0000e+00
100 1.1046e-01 1.1064e+01 5.0000e-01 1.4385e-03
200 1.1026e-01 2.2099e+01 5.0000e-01 6.0513e-04
300 1.1021e-01 3.3122e+01 5.0000e-01 1.6131e-04
400 1.1023e-01 4.4144e+01 5.0000e-01 1.9536e-04
500 1.1029e-01 5.5170e+01 5.0000e-01 4.8111e-04
600 1.1035e-01 6.6202e+01 5.0000e-01 6.8646e-04
700 1.1040e-01 7.7239e+01 5.0000e-01 8.0962e-04
800 1.1044e-01 8.8281e+01 5.0000e-01 8.5502e-04
900 1.1046e-01 9.9327e+01 5.0000e-01 8.3209e-04
1000 1.1044e-01 1.1037e+02 5.0000e-01 7.5378e-04
1100 1.1041e-01 1.2141e+02 5.0000e-01 6.3232e-04
1200 1.1039e-01 1.3245e+02 5.0000e-01 4.8060e-04
1300 1.1036e-01 1.4349e+02 5.0000e-01 3.1817e-04
1400 1.1034e-01 1.5453e+02 5.0000e-01 1.6979e-04
1500 1.1031e-01 1.6556e+02 5.0000e-01 5.5989e-05
1600 1.1029e-01 1.7659e+02 5.0000e-01 1.5027e-05
1700 1.1027e-01 1.8762e+02 5.0000e-01 5.0119e-05
1800 1.1024e-01 1.9864e+02 5.0000e-01 6.2902e-05
1900 1.1021e-01 2.0967e+02 5.0000e-01 6.4260e-05
2000 1.1020e-01 2.2069e+02 5.0000e-01 5.9679e-05
2100 1.1018e-01 2.3171e+02 5.0000e-01 5.1620e-05
2200 1.1018e-01 2.4272e+02 5.0000e-01 4.1535e-05
2300 1.1016e-01 2.5374e+02 5.0000e-01 3.0416e-05
2400 1.1016e-01 2.6476e+02 5.0000e-01 1.8923e-05
2500 1.1015e-01 2.7577e+02 5.0000e-01 7.5092e-06
2600 1.1015e-01 2.8679e+02 5.0000e-01 3.3410e-06
2700 1.1015e-01 2.9780e+02 5.0000e-01 1.3081e-05
2800 1.1014e-01 3.0882e+02 5.0000e-01 2.1182e-05
2900 1.1014e-01 3.1983e+02 5.0000e-01 2.7239e-05
3000 1.1014e-01 3.3084e+02 5.0000e-01 3.1054e-05
3100 1.1014e-01 3.4186e+02 5.0000e-01 3.2667e-05
3200 1.1013e-01 3.5287e+02 5.0000e-01 3.2340e-05
3300 1.1013e-01 3.6388e+02 5.0000e-01 3.0477e-05
3400 1.1012e-01 3.7490e+02 5.0000e-01 2.7510e-05
3500 1.1013e-01 3.8591e+02 5.0000e-01 2.3841e-05
3600 1.1012e-01 3.9692e+02 5.0000e-01 1.9813e-05
3700 1.1012e-01 4.0793e+02 5.0000e-01 1.5706e-05
3800 1.1011e-01 4.1895e+02 5.0000e-01 1.1733e-05
3900 1.1011e-01 4.2996e+02 5.0000e-01 8.0465e-06
4000 1.1011e-01 4.4097e+02 5.0000e-01 4.7413e-06
4100 1.1012e-01 4.5198e+02 5.0000e-01 1.8668e-06
4200 1.1012e-01 4.6299e+02 5.0000e-01 5.6247e-07
4300 1.1011e-01 4.7400e+02 5.0000e-01 2.5532e-06
4400 1.1011e-01 4.8501e+02 5.0000e-01 4.1226e-06
4500 1.1011e-01 4.9602e+02 5.0000e-01 5.2903e-06
4600 1.1011e-01 5.0704e+02 5.0000e-01 6.0758e-06
4700 1.1011e-01 5.1805e+02 5.0000e-01 6.5002e-06
4800 1.1012e-01 5.2906e+02 5.0000e-01 6.5906e-06
4900 1.1012e-01 5.4007e+02 5.0000e-01 6.3844e-06
5000 1.1012e-01 5.5108e+02 5.0000e-01 5.9296e-06
5100 1.1011e-01 5.6209e+02 5.0000e-01 5.2837e-06
5200 1.1011e-01 5.7310e+02 5.0000e-01 4.5082e-06
5300 1.1011e-01 5.8412e+02 5.0000e-01 3.6639e-06
5400 1.1011e-01 5.9513e+02 5.0000e-01 2.8064e-06
5500 1.1012e-01 6.0614e+02 5.0000e-01 1.9825e-06
5600 1.1012e-01 6.1715e+02 5.0000e-01 1.2283e-06
5700 1.1012e-01 6.2816e+02 5.0000e-01 5.6885e-07
5800 1.1012e-01 6.3917e+02 5.0000e-01 1.8738e-08
5900 1.1012e-01 6.5019e+02 5.0000e-01 4.1624e-07
6000 1.1012e-01 6.6120e+02 5.0000e-01 7.3747e-07
6100 1.1012e-01 6.7221e+02 5.0000e-01 9.5170e-07
6200 1.1012e-01 6.8322e+02 5.0000e-01 1.0694e-06
6300 1.1012e-01 6.9423e+02 5.0000e-01 1.1034e-06
6400 1.1012e-01 7.0525e+02 5.0000e-01 1.0675e-06
6500 1.1011e-01 7.1626e+02 5.0000e-01 9.7590e-07
6600 1.1011e-01 7.2727e+02 5.0000e-01 8.4257e-07
6700 1.1011e-01 7.3828e+02 5.0000e-01 6.8057e-07
6800 1.1011e-01 7.4929e+02 5.0000e-01 5.0189e-07
6900 1.1012e-01 7.6030e+02 5.0000e-01 3.1719e-07
7000 1.1012e-01 7.7131e+02 5.0000e-01 1.3565e-07
7077 1.1012e-01 7.7979e+02 5.0000e-01 2.8284e-09
=================================
Total walltime : 0:04:40
(25.20 steps/s)
- GP train (zz) : 0:00:00
- GP infer (zz) : 0:00:34
- GP train (xz) : 0:00:00
- GP infer (xz) : 0:00:49
=================================
lj_problem.animate()