Using AI based diffusive size factors for extracted networks

This notebook illustrates the use of the deep learning based diffusive conductance algorithm decribed here. PoreSpy’s diffusive_size_factor_AI includes the steps for predicting the diffusive size factors of the conduit images. Note that the diffusive conductance of the conduits can be then calculated by multiplying the size factor by diffusivity of the phase. The function takes in the images of segmented porous medium and returns an array of diffusive size factors for all conduits in the image. Therefore, the framework can be applied to both one conduit image as well as a segmented image of porous medium:


Trained model and supplementary materials

To use the diffusive_size_factor_AI, the trained model, and training data distribution are required. The AI model files and additional files used in this example are available here. The folder contains following files:

  • Trained model weights: This file includes only weights of the deep learning layers. To use this file, the Resnet50 model structure must be built first.

  • Trained data distribution: This file will be used in denormalizing predicted values based on normalized transform applied on training data. The denormalizing step is included in diffusive_size_factor_AI method.

  • Finite difference diffusive conductance: This file is used in this example to compare the prediction results with finite difference method for segmented regions. Note: Finite difference-based size factors can be calculated using PoreSpy’s diffusive_size_factor_DNS method.

Let’s download the tensorflow files required to run this notebook:

    import tensorflow as tf
except ImportError:
    !pip install tensorflow

    import sklearn
except ImportError:
    !pip install scikit-learn

import os

if not os.path.exists("sf-model-lib"):
    !git clone
2024-06-10 03:09:56.172564: I external/local_tsl/tsl/cuda/] Could not find cuda drivers on your machine, GPU will not be used.
2024-06-10 03:09:56.175750: I external/local_tsl/tsl/cuda/] Could not find cuda drivers on your machine, GPU will not be used.
2024-06-10 03:09:56.213321: I tensorflow/core/platform/] This TensorFlow binary is optimized to use available CPU instructions in performance-critical operations.
To enable the following instructions: AVX2 FMA, in other operations, rebuild TensorFlow with the appropriate compiler flags.
2024-06-10 03:09:56.969404: W tensorflow/compiler/tf2tensorrt/utils/] TF-TRT Warning: Could not find TensorRT

Also, since the model weights have been stored in chunks, they need to be recombined first:

import importlib
h5tools = importlib.import_module("sf-model-lib.h5tools")
DIR_WEIGHTS = "sf-model-lib/diffusion"
fname_in = [f"{DIR_WEIGHTS}/model_weights_part{j}.h5" for j in [0, 1]]
h5tools.combine(fname_in, fname_out=f"{DIR_WEIGHTS}/model_weights.h5")

Note that to use diffusive_size_factor_AI, Scikit-learn and Tensorflow must be installed. Import necessary packages and the AI model:

import os
import warnings

import h5py
import numpy as np
import openpnm as op
import porespy as ps
import scipy as sp
from matplotlib import pyplot as plt
from sklearn.metrics import r2_score

path = "./sf-model-lib/diffusion"
path_train = os.path.join(path, 'g_train_original.hdf5')
path_weights = os.path.join(path, 'model_weights.h5')
g_train = h5py.File(path_train, 'r')['g_train'][()]
model = ps.networks.create_model()
[03:09:58] ERROR    PARDISO solver not installed, run `pip install pypardiso`. Otherwise,
                    simulations will be slow. Apple M chips not supported.                                         
ValueError                                Traceback (most recent call last)
Cell In[3], line 18
     16 path_weights = os.path.join(path, 'model_weights.h5')
     17 g_train = h5py.File(path_train, 'r')['g_train'][()]
---> 18 model = ps.networks.create_model()
     19 model.load_weights(path_weights)

File ~/work/porespy/porespy/src/porespy/networks/, in create_model()
    612 from tensorflow.keras.optimizers import Adam
    613 model = _resnet3d()
--> 614 model.compile(loss='mse', optimizer=Adam(lr=1e-4), metrics=['mse'])
    615 return model

File /opt/hostedtoolcache/Python/3.10.14/x64/lib/python3.10/site-packages/keras/src/optimizers/, in Adam.__init__(self, learning_rate, beta_1, beta_2, epsilon, amsgrad, weight_decay, clipnorm, clipvalue, global_clipnorm, use_ema, ema_momentum, ema_overwrite_frequency, loss_scale_factor, gradient_accumulation_steps, name, **kwargs)
     43 def __init__(
     44     self,
     45     learning_rate=0.001,
     60     **kwargs,
     61 ):
---> 62     super().__init__(
     63         learning_rate=learning_rate,
     64         name=name,
     65         weight_decay=weight_decay,
     66         clipnorm=clipnorm,
     67         clipvalue=clipvalue,
     68         global_clipnorm=global_clipnorm,
     69         use_ema=use_ema,
     70         ema_momentum=ema_momentum,
     71         ema_overwrite_frequency=ema_overwrite_frequency,
     72         loss_scale_factor=loss_scale_factor,
     73         gradient_accumulation_steps=gradient_accumulation_steps,
     74         **kwargs,
     75     )
     76     self.beta_1 = beta_1
     77     self.beta_2 = beta_2

File /opt/hostedtoolcache/Python/3.10.14/x64/lib/python3.10/site-packages/keras/src/backend/tensorflow/, in TFOptimizer.__init__(self, *args, **kwargs)
     21 def __init__(self, *args, **kwargs):
---> 22     super().__init__(*args, **kwargs)
     23     self._distribution_strategy = tf.distribute.get_strategy()

File /opt/hostedtoolcache/Python/3.10.14/x64/lib/python3.10/site-packages/keras/src/optimizers/, in BaseOptimizer.__init__(self, learning_rate, weight_decay, clipnorm, clipvalue, global_clipnorm, use_ema, ema_momentum, ema_overwrite_frequency, loss_scale_factor, gradient_accumulation_steps, name, **kwargs)
     33     warnings.warn(
     34         "Argument `decay` is no longer supported and will be ignored."
     35     )
     36 if kwargs:
---> 37     raise ValueError(f"Argument(s) not recognized: {kwargs}")
     39 if name is None:
     40     name = auto_name(self.__class__.__name__)

ValueError: Argument(s) not recognized: {'lr': 0.0001}

Create test image

We can create a 3D image using PoreSpy’s poly_disperese_spheres generator:

shape = [120, 120, 120]
dist = sp.stats.norm(loc=7, scale=5)
im = ps.generators.polydisperse_spheres(shape=shape,
fig, ax = plt.subplots(1, 1, figsize=[4, 4])
ax.imshow(im[:, :, 20], origin='lower', interpolation='none')

Extract the network

We then extract the pore network of the porous medium image using PoreSpy’s snow2 algorithm. snow2 returns the segmented image of the porous medium as well as extracted network data.

snow = ps.networks.snow2(im, boundary_width=0, parallelization=None)
regions = snow.regions
net =

Apply diffusive_size_factor_AI

AI_based diffusive size factors of conduits in the extracted network can then be calculated applying diffusive_size_factor_AI on the segmented regions. We can then define throat.diffusive_size_factor_AI property and assign the predicted size_factor to this property.

conns = net['throat.conns']
size_factors = ps.networks.diffusive_size_factor_AI(regions,
net['throat.diffusive_size_factor_AI'] = size_factors
47/47 [==============================] - 42s 841ms/step

The resulting network can then be imported to OpenPNM for later use such as diffusive mass transport simulations problems. Let’s visualize the network:

pn =
fig, ax = plt.subplots(1, 1, figsize=[5, 5])
ax = op.visualization.plot_connections(network=pn, alpha=0.8, color='grey', ax=ax)
ax = op.visualization.plot_coordinates(network=pn, ax=ax, color='b', markersize=50)

Compare with finite difference method

Now that the extracted network includes AI_based diffusive size factor data, we can use the network to compare the accuracy of diffusive_size_factor_AI, shape factor method,and geometry method (no shape factor) in contrast to finite difference method. Assuming a generic phase with diffusivity of 1, the diffusive conductance of the conduits will be equal to their diffusive size factors. The diffusive conductance of the conduits can be calculated using OpenPNM’s generic_diffusive method. The diffusive conductance of the conduits using shape factor based method assuming cones and cylinders shapes for pores and throats can be calculated as follows:

net : < at 0x7f0256f989a0>
  #  Properties                                                   Valid Values
  2  throat.conns                                                    744 / 744
  3  pore.coords                                                     199 / 199
  4  pore.region_label                                               199 / 199
  5  pore.phase                                                      199 / 199
  6  throat.phases                                                   744 / 744
  7  pore.region_volume                                              199 / 199
  8  pore.equivalent_diameter                                        199 / 199
  9  pore.local_peak                                                 199 / 199
 10  pore.global_peak                                                199 / 199
 11  pore.geometric_centroid                                         199 / 199
 12  throat.global_peak                                              744 / 744
 13  pore.inscribed_diameter                                         199 / 199
 14  pore.extended_diameter                                          199 / 199
 15  throat.inscribed_diameter                                       744 / 744
 16  throat.total_length                                             744 / 744
 17  throat.direct_length                                            744 / 744
 18  throat.perimeter                                                744 / 744
 19  pore.volume                                                     199 / 199
 20  pore.surface_area                                               199 / 199
 21  throat.cross_sectional_area                                     744 / 744
 22  throat.equivalent_diameter                                      744 / 744
 23  throat.diffusive_size_factor_AI                                 744 / 744
  #  Labels                                                 Assigned Locations
  2  pore.all                                                              199
  3  throat.all                                                            744
pn['pore.diameter'] = pn['pore.inscribed_diameter']
pn['throat.diameter'] = pn['throat.inscribed_diameter']
pn['throat.coords'] = pn['throat.global_peak']
phase = op.phase.Phase(network=pn)
phase['pore.diffusivity'] = 1
phase['throat.diffusivity'] = 1
g_SF = np.copy(phase['throat.diffusive_conductance'])

To find the diffusive conductance of the conduit using geometry method (no shape factor) we assume cylindrical pores and throats:

cn = pn.conns
L1, Lt, L2 = pn['pore.diameter'][
    cn[:, 0]] / 2, pn['throat.length'], pn['pore.diameter'][cn[:, 1]] / 2
D1, Dt, D2 = pn['pore.diameter'][
    cn[:, 0]], pn['throat.diameter'], pn['pore.diameter'][cn[:, 1]]
A1, At, A2 = np.pi * D1**2 / 4, np.pi * Dt**2 / 4, np.pi * D2**2 / 4
g_Geo = 1 / (L1 / A1 + L2 / A2 + Lt / At)

The diffusive conductance of the conduit using AI-based method:

g_AI = np.copy(phase['throat.diffusive_conductance'])

The finite difference-based diffusive size factors were calculated using PoreSpy’s size factor method diffusive_size_factor_DNS. However, due to the long runtime of the DNS function the results were saved in the example data folder and used in this example. The Following code was used to estimate the finite difference-based values using PoreSpy:

g_FD = ps.networks.diffusive_size_factor_DNS(regions, conns)

Now let’s compare the diffusive conductance calculated from geometry-based method, shape factor based-method, and AI-based method with reference finite difference method:

fname = os.path.join(path, 'g_finite_difference120-phi7.hdf5')
g_FD = h5py.File(fname, 'r')['g_finite_difference'][()]
max_val = np.max([g_FD, g_AI, g_Geo, g_SF])
fig, ax = plt.subplots(1, 3, figsize=[10, 4])

ax[0].plot(g_FD, g_Geo, '*', [0, max_val], [0, max_val], 'r')
ax[0].set_xlim([0, max_val])
ax[0].set_ylim([0, max_val])
ax[0].set_xlabel('finite difference $g_d$')
ax[0].set_ylabel('geometry based $g_d$')
ax[0].set_title('$R^2$ = ' + str(np.round(r2_score(g_FD, g_Geo), 2)))

ax[1].plot(g_FD, g_SF, '*', [0, max_val], [0, max_val], 'r')
ax[1].set_xlim([0, max_val])
ax[1].set_ylim([0, max_val])
ax[1].set_xlabel('finite difference $g_d$')
ax[1].set_ylabel('shape factor based $g_d$')
ax[1].set_title('$R^2$ = ' + str(np.round(r2_score(g_FD, g_SF), 2)))

ax[2].plot(g_FD, g_AI, '*', [0, max_val], [0, max_val], 'r')
ax[2].set_xlim([0, max_val])
ax[2].set_ylim([0, max_val])
ax[2].set_xlabel('finite difference $g_d$')
ax[2].set_ylabel('AI based $g_d$')
ax[2].set_title(r'$R^2$ = ' + str(np.round(r2_score(g_FD, g_AI), 2)));

As shown in the scatter plots, the AI-based diffusive conductance method predicts the conductance values with a higher accuracy than geometry-based and shape factor-based methods. A comprehensive comparison between these methods for a large dataset can be found here.