Available via license: CC BY 4.0

Content may be subject to copyright.

Computer Methods in Applied Mechanics and engineering manuscript No.

(will be inserted by the editor)

Equivariant geometric learning for digital rock physics: estimating

formation factor and effective permeability tensors from Morse graph

Chen Cai ·Nikolaos Vlassis ·Lucas Magee ·Ran Ma ·

Zeyu Xiong ·Bahador Bahmani ·Teng-Fong Wong ·Yusu

Wang ·WaiChing Sun

April 13, 2021

Abstract

We present a SE(3)-equivariant graph neural network (GNN) approach that directly predicting the

formation factor and effective permeability from micro-CT images. FFT solvers are established to compute

both the formation factor and effective permeability, while the topology and geometry of the pore space

are represented by a persistence-based Morse graph. Together, they constitute the database for training,

validating, and testing the neural networks. While the graph and Euclidean convolutional approaches

both employ neural networks to generate low-dimensional latent space to represent the features of the

micro-structures for forward predictions, the SE(3) equivariant neural network is found to generate more

accurate predictions, especially when the training data is limited. Numerical experiments have also shown

that the new SE(3) approach leads to predictions that fulﬁll the material frame indifference whereas the

predictions from classical convolutional neural networks (CNN) may suffer from spurious dependence on

the coordinate system of the training data. Comparisons among predictions inferred from training the CNN

and those from graph convolutional neural networks (GNN) with and without the equivariant constraint

indicate that the equivariant graph neural network seems to perform better than the CNN and GNN without

enforcing equivariant constraints.

Keywords effective permeability, graph neural network, equivariance, objectivity, deep learning

1 Introduction

Formation factor and effective permeability are both important hydraulic and engineering properties of

porous media. While formation factor measures the relative electric resistance of the void space of the porous

media to the reference ﬂuid, the effective permeability measures the ability of the fully ﬂuid-saturated

porous media to transmit ﬂuid under a given pressure gradient. Predicting effective permeability and

conductivity from micro-structures has been a crucial task for numerous science and engineering disciplines.

Ranging from biomedical engineering (cf. Cowin and Doty [2007]), geotechnical engineering (e.g. Terzaghi

et al. [1996]), nuclear waste disposal to petroleum engineering, the knowledge on permeability may affect

the economical values of reservoirs, the success of tissue regeneration, and the likelihood of landslides and

earthquakes [White et al.,2006,Sun et al.,2011a,2014,Kuhn et al.,2015,Paterson and Wong,2005,Suh and

Sun,2021].

Effective permeability can be estimated via the establishment of porosity-formation-factor-permeability

relationship (cf. Jaeger et al. [2009], Bosl et al. [1998]). This relationship can be completely empirical and

established via regression analysis or they can be based on theoretical justiﬁcation such as fractal geometry

Chen Cai, Lucas Magee, Yusu Wang Halicioglu Data Science Institute, University of California, San Diego, California

Nikolaos N. Vlassis, Ran Ma, Zeyu Xiong, Bahador Bahmani, WaiChing Sun

Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, New York

Teng-fong Wong Department of Geosciences, Stony Brook University, Stony Brook, New York

arXiv:2104.05608v1 [cs.LG] 12 Apr 2021

2 Chen Cai et al.

assumption Costa [2006] and percolation threshold [Mavko and Nur,1997]. The recent advancement of

tomographic imaging techniques has made it possible to obtain 3D images of microstructures of various

porous media [Arns et al.,2004,Fredrich et al.,2006,Sun and Wong,2018,Sun et al.,2011b]. These newly

available images provide a new avenue to establish a fuller and precise picture of the relationship between

the pore geometry and macroscopic transport properties. If image segmentation is conducted properly, the

pore space inferred from micro-CT images can be directly used to obtain formation factor and effective

permeability through inverse problems. Finite volume, ﬁnite element as well as recently lattice Boltzmann

and Fast Fourier transform solvers can all be employed to solve Poisson’s and Stokes’ equation to obtain

formation factor and effective permeability. These image-based simulations and inverse problems have led

to an emerging new research area often referred to as Digital Rock Physics (DRP) [Andr

¨

a et al.,2013a,b]. By

using images to predict physical properties, the digital rock physics framework could be viewed as a digital

twin that enables a non-destructive way to infer material properties in a more repeatable and cost-efﬁcient

manner compared to physical tests of which the results are regarded as the ground truth.

Nevertheless, conducting 3D simulations on images can be quite costly, as the computational time and

memory requirement both scale up with the domain size and become the technical barriers for the digital

rock physics approach. Alternatively, the recent advancements on the convolutional neural networks have

led to many attempts to directly inferring relationships between the 3D images and the resultant effective

permeability.

Among the earlier works, Srisutthiyakorn* [2016] shows the application of CNN and multi-layer

perceptron (MLP) frameworks for predicting scalar-valued permeability of rock samples directly from the

2D and 3D images of rock samples, instead of detailed numerical simulations. Wu et al. [2018] proposed

a novel CNN architecture that utilizes other material descriptors such as porosity in the CNN’s fully

connected layer. They show that extra physical or geometrical features may enhance the prediction capacity.

Sudakov et al. [2019] study the effect of feature engineering among various geometrical descriptors on

the accuracy of the permeability regression task. They empirically show that the conventional 3D CNN

outperforms 2D CNN and MLP for permeability prediction. The diffusivity of synthetic 2D porous rock

samples is successfully predicted for a wide range of porosity values via a CNN architecture enhanced by

incorporating ﬁeld knowledge (the same idea as Wu et al. [2018]), pre-processing of the image input, and

customizing the loss function [Wu et al.,2019]. These earlier studies focus on developing surrogate models

for the scalar-valued quantities, while the permeability is usually an anisotropic tensor-valued quantity for

heterogeneous rock samples.

Santos et al. [2020] introduce a convolutional neural network framework called PoreFlow-Net that

directly predicts ﬂow velocity ﬁeld at each voxel of the input image then infer the effective permeability via

inverse problems. They incorporate additional geometrical and physical features, e.g., Euclidean distance

map, in their proposed CNN pipeline to inform the neural network about local and global boundary

conditions. They use the L1-norm loss function that makes prediction more robust to outliers compared

to the mean squared root loss function. They show that the proposed model generalizes well in dealing

with complex geometries not seen in the training stage. While the proposed work shows good matches

between predicted scalar permeability and the benchmark values, the framework has yet to extend to

predict ansiotropic permeability which may require more simulation data and predictions of more complex

ﬂow patterns. Furthermore, as the classical CNN architecture cannot admit rotational symmetry groups to

conform with the frame-indifference property of the ﬂow ﬁeld, an intractable amount of rotated data set

may be required to augmented in 3D applications.

Despite the aforementioned progress, training classical convolutional neural networks that leads to

robust property predictions is not trivial. First, a fundamental issue that may lead to erroneous prediction is

that the classical CNN is not designed to give predictions independent of coordinate frames. For instance, if

the micro-CT image is rotated by a rotation tensor

R

, then the same permeability tensor represented by the

old and new coordination systems should be related by

k0=RkRT(1)

whereas the eigenvalues and the three invariants of the permeability tensors should be independent of

the frame. Furthermore, the effective permeability is expected to remain constant when the images are

in rigid body translation. However, while the earlier designs of convolutional neural network do exhibit

translational equivariant behavior, the predictions may suffer from spurious dependence due to the rotation

Equivariant geometric learning for digital rock physics 3

of the observer [Cohen and Welling,2016]. As such, the resultant predictions on effective permeability and

formation factor tensors may therefore exhibit sensitivity on the frame of reference and hence reduce the

quality of the predictions. Another important issue that deserves attention is the representation of the pore

structures. While it is common to use the binary indicator function the Euclidean voxel space of the binary

image to represent the pore structures, such an approach may not be most efﬁcient and convenient. For

example, the Euclidean representation may incorrectly incorporate ring artifacts into the encoded feature

vector. Furthermore, the micro-CT images are often sufﬁciently large that the 3D convolutional layer may

demand signiﬁcant memory and GPU time and hence make the training expensive [Vlassis et al.,2020,

Frankel et al.,2019].

The goal of this research is to overcome these two major technical barriers to enable faster and more

accurate and rigorous predictions on the hydraulic properties (effective permeability and formation factor).

In particular, we introduce three new techniques that have yet to be considered for geomechanics predictions,

(1) the representation of pore geometry via Morse graph, (2) the equivariant neural network that generates

predictions equivariant with respect to 3D rotation, and (3) the graph convolutional neural network that

generates low-dimensional features to aid predictions for the formation factor and effective permeability.

Our results show that the graph neural network (GNN) consistently performs better than the classical

convolutional neural network counterpart and the SE(3) equivariant neural network performs the best

among other options.

This paper is organized as follows. We ﬁrst provide a brief account of how the database is generated.

Our goal is to introduce supervised learning where the inputs are the pore space represented by either a

binary indicator function in the Euclidean physical domain or a Morse graph that represents the topology of

the pore space, and the outputs are the formation factor and permeability.

2 Database generation via direct numerical simulations with FFT solvers

To generate the database, we use Fast Fourier Transform Solver to infer both effective permeability and

formation factor from the micro-CT images. For completeness, the procedures to obtain the formation factor

and effective permeability are outlined in the two sub-sections. The results are summarized in Figure 1.

Fig. 1: Eigenvalues distribution for permeability and formation factor inferred from FFT simulations

performed on sub-volumes of a sandstone microCT image. The image voxel resolution is 8-micron and the

sub-volume is of 150X150X150 voxels.

4 Chen Cai et al.

2.1 Effective permeability inferred from micro-CT images

Consider a unit cell

Ω=∏α[−Yα

,

Yα]

deﬁned in the Euclidean space

Rd

with

eα

being the

α

-th basis vector.

A scalar-valued function f:Rd→Ris called a periodic function if

f x+

d

∑

α=1

Yαkαeα!=f(x),∀x∈Rdand k∈Zd. (2)

Furthermore, the Sobolev space of periodic functions Hs

#(Rd)is deﬁned as

Hs

#(Rd) = {f|f∈Hs(Rd),fis periodic}. (3)

The unit cell is further divided into the solid skeleton

Ωs

and the ﬂuid-ﬁlled void

Ωl

, with

Ωf∪Ωs=Ω

and

Ωf∩Ωs=∅

. The solid skeleton is a closed set with a periodic pattern, i.e.

¯

Ωs=Ωs

. The local pressure

p:Ω→Rwithin the unit cell is split into the average part and a ﬂuctuation term

∇xp(x) = G+∇xp∗(x),h∇xp∗(x)iΩ=1

VZΩ∇xp∗(x)dV=0(4)

where

G∈Rd

is the macroscopic pressure gradient and

p∗(x):Ω→R

is the microscopic pressure

perturbation. Suppose that the incompressible viscous ﬂuid ﬂows within the void region with the ﬂow

velocity

v:Ω→Rd

, such that the velocity ﬁeld and the pressure ﬁeld fulﬁll the incompressible Stokes

equation µf∇2v(x) + ∇xp(x) = 0, ∇x·v=0 , ∀x∈Ωf

v(x) = 0 , ∀x∈Ωs,(5)

where µfis the viscosity. Then, the permeability tensor κof the porous media is computed as

hviΩ=−κ

µf

G. (6)

An FFT-based method is utilized to solve the Stokes equation

(5)

within the porous media [Monchiet

et al.,2009]. A fourth order compliance tensor M(x)is ﬁrst introduced

sym (∇xv)=M(x):σ,M(x) = (1

2µfI−1

3I⊗I,∀x∈Ωf

0,∀x∈Ωs,(7)

where

I

is the fourth order identity tensor,

I

is the second order identity tensor, and a small value

ζ

is

introduced to maintain the stress continuity within the solid skeleton. The incompressible Stokes equation

(5) is equivalently written as

Γ(x)∗[M(x):(σ0(x) + δσ(x))] =0⇒Γ(x)∗(M(x):δσ(x))=−Γ(x)∗(M(x):σ0(x))(8)

where

∗

denotes the convolution operator. The fourth order Green’s tensor

Γ

is deﬁned in the frequency

space as [Nguyen et al.,2013]:

ˆ

Γ=2µ0(β⊗β+β⊗β),β=I−1

|ξ2|ξ⊗ξ,(a⊗b)ijkl =1

2aik bjl +ail bjk . (9)

where

ξα=kα/

2

Yα

,

k∈Zd

, and

µ0

is a reference viscosity. For

ξ=0

,

ˆ

Γ=0

because of the zero-mean

property. For an arbitrary second-order tensor ﬁeld

T(x)

,

Γ∗T

is divergence free and its spatial average

vanishes. Also,

Γ∗ε=

0 for any compatible strain ﬁeld

ε(x) = sym(∇xu)

. The macroscopic pressure

gradient

G

is introduced through the initial guess of the stress distribution

σ0

[Monchiet et al.,2009,Nguyen

et al.,2013]:

σ0=−Λ(x)∗f(x), (10)

Equivariant geometric learning for digital rock physics 5

where

f(x)

extends the macroscopic pressure gradient

G

from the void

Ωs

to the unit cell

Ω

, and

Λ(x)

projects

f

to a divergence-free stress ﬁeld. The third order tensor ﬁeld

Λ(x)

and the vector ﬁeld

f(x)

is

derived as

Λijk (ξ) = (i

|ξ|4hδijξk+δik ξj+δjk ξi|ξ|2−2ξiξjξki,ξ6=0

0 , ξ=0, (11)

f(x) = (G,∀x∈Ωf

−1−cs

csG,∀x∈Ωs.(12)

where csdeﬁnes the volume fraction of the solid phase.

Equation

(8)

is linear and is solved by the conjugate gradient method. Figure 2shows the streamlines of

ﬂow velocity obtained four FFT simulations performed on RVE with porosity ranging from 0.208 to 0.257.

To aid the visualization, the color map of the streamline is plotted on the logarithmic scale. Note that the

true ﬂow velocity is equal to Darcy’s velocity divided by the porosity (assuming that the area and volume

fraction of void are equal) [Bear,2013]. Hence, if two microstructures are of the same effective permeability

but with different porosities, the one with the lower porosity is expected to have higher ﬂow velocity. What

we observed in the streamlines plotted in Figure 2nevertheless indicates that the ﬂow speed is lower in the

RVE of lower porosity. This observation is consistent with the porosity-permeability relationship established

in previous work such as Andr¨

a et al. [2013b].

2.2 Formation factor inferred from micro-CT images

The formation factor

F

, which is deﬁned as the ratio between the ﬂuid conductivity and the effective

conductivity of the saturated porous rock, is used to quantify the electric resistivity of a porous media

relative to that of water. The formation factor is considered as a physical property that solely depends on

the geometry and topology of the pore space in the porous media [Archie et al.,1942]. Previous studies have

shown that the formation factor is signiﬁcantly inﬂuenced by the electric double layer between the solid

phase and the liquid phase, and the surface conductivity should be taken into consideration to perform a

micro-CT image-based formation factor prediction at the pore scale [Zhan et al.,2010].

Suppose that all the related physical quantities, including the local conductivity

A

, the current density

J

,

and the electrostatic potential gradient

e

, are deﬁned within the periodic unit cell

Ω

, the charge conservation

equation reads:

∇x×e(x) = 0,J(x) = A(x)·e(x),∇x·J(x) = 0. (13)

The local electrostatic potential gradient ﬁeld

e(x)

is further split into its average

E

and a ﬂuctuation term

e∗(x)∈H1

#(Rd):

e(x) = E+e∗(x),he∗(x)iΩ=0. (14)

The charge conservation equation

(13)

deﬁned within the periodic domain

Ω

is equivalently written as

the periodic Lippmann-Schwinger equation:

G(x)∗[A(x)·(E+δe(x))] =0⇒G(x)∗[A(x)·δe(x)]=−G(x)∗[A(x)·E]. (15)

The fourth order Green’s operator Gis provided in the frequency space as [Vondˇ

rejc et al.,2014]:

ˆ

G(ξ) = ξ⊗ξ

ξ·A0·ξ. (16)

where

A0

is a reference conductivity. For

ξ=0

,

ˆ

G=0

because of the zero-mean property. By applying the

averaging electric potential gradient

E

upon the unit cell, the local current density

J

is obtained by solving

Equation (15) with conjugate gradient method.

It has been demonstrated that the anisotropic conductivity within the interfacial region has a strong

inﬂuence on the formation factor due to the presence of electric double layers between the solid and liquid

constituents [Weller et al.,2013,Bussian,1983]. However, the thickness of the double layer is usually smaller

than the voxel length. As such, running the FFT electric conductance simulations on the three-phase porous

media consisting of solid, double layer and water is not feasible. In this work, we extend the upscaling

6 Chen Cai et al.

(a) porosity=0.257 (b) porosity=0.228

(c) porosity=0.223 (d) porosity=0.208

Fig. 2: Streamline of the ﬂow velocity for four RVE with porosity = 0.257 (UPPER LEFT), 0.228 (UPPER

RIGHT), 0.223 (LOWER LEFT) and 0.208 (LOWER RIGHT) obtained from FFT direct numerical simulations

in which the effective permeabilities are extracted.

approach in [Zhan et al.,2010] to ﬁrst compute the tensorial effective conductivity of the surface voxel such

that the voxels at the solid-ﬂuid interfacial regions are upscaled to effective values analytically. Then, the

FFT simulation is run to determine the effective conductivity of the entire RVE (see Figure 3).

The interface was ﬁrst identiﬁed from the three-dimensional binary image, which was labeled as yellow

voxels with the unit normal vectors in Figure 3(LEFT). To extract the interface, each voxel was analyzed

along with the 26 closest neighboring voxels in the 3x3x3 window centered around it. The isolated voxels,

whose neighbors are all in the opposite phase, were eliminated. Then the solid voxels neighboring the ﬂuid

were selected to represent the interface if they have at least 9 ﬂuid neighboring voxels. Furthermore, as

shown in Figure 3(RIGHT), the interface voxels were modeled as electric double layers, consisting of a layer

of ﬂuid and a layer with conductivity

σsur f

. The normal component

σn

of the interface conductivity was

calculated as two electrical resistors in series while the tangential component

σt

was computed by resistors

in parallel as shown in (17).

σt=L−χd

Lσf+χd

Lσsur f ,σn= [( L−χd

Lσf)−1+ ( χd

Lσsur f )−1]−1(17)

where

L

is the voxel length,

χd

is the thickness of the surface layer,

σf

is the ﬂuid conductivity. To determine

the normal vectors of the interface, the signed distance function was generated by an open-source software

ImageJ (cf. Abr

`

amoff et al. [2004]), in which the value of each voxel represents its shortest distance to the

Equivariant geometric learning for digital rock physics 7

Fig. 3: Surface and unit normal vectors between solid and ﬂuid phases (LEFT) and structure of electric

double layer (RIGHT).

interface. The normal vectors

n1

were then determined by the gradient of the signed distance function as

shown in Figure 3(LEFT). The upscaled electric conductivity tensors

σ3

of the interface voxel are calculated

via

σ3=σnn1⊗n1+σtn2⊗n2+σtn3⊗n3. (18)

where

n1

is the normal vector of the solid surface and

n2

and

n3

are two orthogonal unit vectors that

spans the tangential plane of the surface. This approach is different from Zhan et al. [2010] in the sense

that the analytical electric conductivity at the solid-ﬂuid boundary is tensorial and anisotropic instead of

volume-averaged and isotropic. Such a treatment enables more precise predictions of the anisotropy of the

surface effect on the electric conductivity that is important for estimating the conductivity tensor.

Once all voxels in the spatial domain are assigned with the corresponding electrical conductivity, we

obtain a heterogeneous domain where electric conductivity may vary spatially. The effective conductivity

tensor can be obtained through computational homogenization via the FFT solver described above. The

effective conductivity

σeff

is deﬁned as a second-order tensor which projects the applied electric potential

gradient Eto the homogenized current density as:

hJ(x)iΩ=σeff ·E. (19)

Since the charge conservation equation

(13)

is linear, the effective conductivity

σeff

is constant for each

micro-CT image. Then, the formation factor

F

is deﬁned as the ratio between the ﬂuid conductivity

σf

and

the effective conductivity σeff as:

F=σfσ−1

eff . (20)

Figure 4shows the simulated streamlines of the electric current along the x-axis for the same RVEs

plotted in Figure 2. As expected, the RVEs with larger porosity tend to be more conductive. However, it

is also obvious that the streamlines in Figures 2and 4also exhibit signiﬁcantly different patterns. This

difference in the streamline pattern could be attributed to the surface effect of the electric conductivity and

could be affected by the periodic boundary condition enforced by the FFT solver.

Regardless of the reasons, the streamline patterns of the inverse problems may dictate the difﬁculty of

the neural network predictions of the formation factor and permeability and will be explored further in

Section 5.

Remark 1

The homogenized permeability

κ

is non-symmetric in general since the macroscopic pressure

gradient

G

and the homogenized velocity

hvi

are not power conjugate to each other. In fact, the homogenized

stress

hσi

and the homogenized velocity gradient

h∇xvi

are power conjugate and their inner product

equals to the input power due to the Hill-Mandel condition. Similar results are also reported previously

[White et al.,2006,Sun and Wong,2018]. On the other hand, the formation factor

F

is symmetric, since the

homogenized current density

hJi

and the applied electric potential gradient

E

are power conjugate to each

other.

8 Chen Cai et al.

(a) porosity=0.257 (b) porosity=0.228

(c) porosity=0.223 (d) porosity=0.208

Fig. 4: Streamline of the gradient of electric current for four RVE with porosity = 0.257 (UPPER LEFT), 0.228

(UPPER RIGHT), 0.223 (LOWER LEFT), and 0.208 (LOWER RIGHT) obtained from FFT direct numerical

simulations in which the effective conductivities are extracted.

3 Discrete Morse Graph Generation

To predict permeability, one could directly apply a CNN-type architecture on the input 3D images. However,

recognizing that the effective permeability is in fact determined by the pores among grains, it is thus intuitive

that a representation capturing such a pore network is more informative and provides better inductive bias

for the learning algorithm.

Our framework will ﬁrst construct a graph-skeleton representation of the “pore” networks from the 3D

images, using the discrete Morse-based algorithm of [Dey et al.,2018], which we refer to as DM-algorithm

herein. Below we ﬁrst brieﬂy provide the intuition behind this algorithm. We then describe the enrichment

of such a DM-graph representation, which we will be used as input for a GNN architecture. Note that our

experimental results (presented in Tables 1and 2) show that such a graph skeleton + GNN framework

indeed leads to much better prediction accuracy compared to an image + CNN-based framework.

3.1 Discrete Morse based graph skeletonization

To provide the intuition of the DM-algorithm of Wang et al. [2015], Dey et al. [2018], ﬁrst consider the

continuous case where we have a density function

ρ:Rd→R

. In our setting, an image can be thought of

Equivariant geometric learning for digital rock physics 9

Fig. 5: An example of discrete Morse graph reconstruction on 2D data. (A) A single image from our 3D stack.

(B) The signed distance from the boundary of (A). (C) The image (B) is converted into a triangulation and

density function. The discrete Morse graph reconstruction then captures the mountain ridges of the density

function. (D) These ridges capture maximal distance from the boundary of (A).

as a discretization of a function on

R3

. If we view the graph of this density function as a terrain in

Rd+1

(see Figure 5A), then its graph skeleton can be captured by the “mountain ridges” of this density terrain,

as intuitively, the density on such ridges is higher than the density off the ridges. Mathematically, such

mountain ridges can be described by the so-called 1-stable manifolds from Morse theory, which intuitively

correspond to curves connecting maxima (mountain peaks) to saddles (of the index (

d−

1)) and to other

maxima, forming boundaries separating different basins/valleys in the terrain. To simplify the terrain

and only capture “important” mountain ridges, we use the persistent homology theory [Zomorodian and

Carlsson,2005], one of the most important developments in the ﬁeld of topological data analysis in the past

two decades, and simplify those less important mountain peaks/valleys (deemed as noise). As the graph is

computed via the global mountain ridge structures, this approach is very robust to gaps in signals, as well

as non-uniform distribution of signals.

In the discrete setting, our input is a cell complex as a discretization of the continuous domain. For

example, our input is a 3D image, which is a cubic complex decomposition of a subset of

R3

, consisting of

vertices, edges, square faces, and cube cells. The 1-stable manifolds in Morse theory are differential objects,

and sensitive to discretization. Instead, one can use the

discrete Morse theory

developed by Robin Forman

[Forman,1995], which is not a discretization of the classical Morse theory, but rather a combinatorial analog

of it. Due to the combinatorial nature, the resulting algorithm is very stable and simpliﬁcation via persistent

homology can easily be incorporated. The ﬁnal algorithm is conceptually clean and easy to implement.

Some theoretical guarantees of this algorithm are presented in [Dey et al.,2018] and it has already been

applied to applications, e.g, in geographic information system and neuron-image analysis [Wang et al.,2015,

Dey et al.,2017,2019,Banerjee et al.,2020].

3.2 Morse Graph representation of pore space

For each simulation, we have a corresponding 150 x 150 x 150 binary image stack representing the rock

boundary. We ﬁrst compute the signed distance to the boundary on this domain, shift all non-zero values

by 255 minus the maxima signed distance in the domain (this assures the new maxima signed distance is

255), and apply a Gaussian ﬁlter with

σ=

1. We then run the discrete Morse algorithm with a persistence

threshold δ=48.

On the output Morse graph for each simulation, we assign features used later by the GNN. The features

assigned to each node are its coordinates

(x

,

y

,

z)

, its (gaussian smoothed) signed distance value, its original

signed distance value, whether or not the vert is a maximum, and three node ﬂow values. Node ﬂow value

is calculated from the underlying vector ﬁeld computed during the Morse graph calculation. Every node

in the domain will be connected to a single node in the Morse output in the vector ﬁeld. The ﬁrst node

10 Chen Cai et al.

Fig. 6: The Morse graph pipeline we use on our data. (A) A single image from our 3D stack. (B) The 3D

volume of the entire image stack. (C) The signed distance function from the boundary in (B). (D) The signed

distance function is shifted by 255 - the max value of the signed distance function at all locations is nonzero.

Then a Gaussian ﬁlter with sigma equal to 1 is applied. This is the input function for discrete Morse graph

reconstruction. (E) The discrete Morse graph reconstruction output (persistence threshold = 48) overlayed

on the original signed distance function. (F) The discrete Morse graph reconstruction output on the ﬁrst

octant of the original signed distance function. By zooming in we see that the graph is accurately capturing

the signed distance function.

ﬂow feature simply counts the number of nodes that ﬂow into that node the second counts the number of

non-boundary nodes that ﬂow into the node, and the thirds sums the function values of nodes that ﬂow

into the node. The features assigned to each edge are whether or not it is a saddle, and its persistence value

if it is a saddle (all non-saddle edges have a persistence value

≤δ

, and we simply assign -1), length, and

minima, average, and total function value of vertices along the edge for both the input Morse function and

the original signed distance function. Prior to removing degree two nodes discussed in the next paragraph,

each edge has length 1, and zero is assigned to the minima, average, and total function features. These

values are properly updated at every removal of a degree two node.

Finally, for the sake of computational efﬁciency in training the GNN, we reduce the number of nodes

and edges in the graph by removing nodes of degree two that are not maxima or adjacent to a saddle edge.

The only node features from the original graph that need to be updated are the node ﬂow feature. For each

node removed in the simpliﬁed graph, it lies along a unique edge connecting to vertices in the simpliﬁed

graph. We add the removed node’s ﬂow features to the node ﬂow features of the node in the direction of the

maxima on this edge. The edge features mentioned previously are also updated to capture the information

lost by removing nodes.

4 Equivariant GNN

We now introduce the notation of equivariance and strategies to build equivariant neural networks. The

equivariant neural networks we designed here take the Morse graph described in Section 3.2 as input and

Equivariant geometric learning for digital rock physics 11

outputs the formation factor and effective permeability tensors. One of the key challenges of predicting

property tensor lies in the frame indifference. Given a frame

B1

, we can represent the position of each graph

node with respect to

B1

and obtain the prediction from the model with respect to

B1

. If we change the frame

from

B1

to a different frame

B2

, the model prediction with respect to

B2

will certainly be different in terms

of numbers but in principle, two predictions under different frames represent the same geometric object.

The requirement that model prediction is independent of arbitrary-chosen frames is natural and can be

mathematically formulated as equivariance.

However, the traditional neural network has no such inductive bias built-in, and therefore not ideal for

permeability prediction. To achieve this goal of frame indifference, we will adapt the recent progress on the

equivariant neural network, a set of neural networks that is equivariant to symmetry transformations. We

ﬁrst outline the notion of the group, its representation, and feature types, and then introduce equivariance

and strategies of building an equivariant neural network.

Group, its representation, and feature types:

Let

G

be a set. We say that

G

is a group with law of

composition

?

if the following axioms hold: 1

)

closure of

?

: the assignment

(g

,

h)→g?h∈G

, deﬁnes a

function

G×G→G

. We call

g?h

the product of

g

and

h

. 2

)

existence of identity: there exists a

eG∈G

such that for every

g∈G

, we have

g?eG=g=eG?g

. We call

eG

an identity of

G

. 3

)

existence of inverse:

for every

g∈G

, there exists an

h∈G

such that

h?g=eG=g?h

. We call such

h

an inverse of

g

. 4

)

associativity: for any g,h,k∈Gwe have (g?h)?k=g?(h?k).

A group representation

ρ:G→GL(N)

is a map from a group

G

to the set of

N×N

invertible matrices

GL(N)

. Critically

ρ

is a group homomorphism; that is, it satisﬁes the following property

ρ(g1?g2) =

ρ(g1)ρ(g2)

for any

g1

,

g2∈G

where the multiplication on the right side of equality denotes the matrix

multiplication. Speciﬁcally for 3D rotations group

SO(

3

)

, we have a few interesting properties: 1) its

representations are orthogonal matrices, 2) all representations can be decomposed as

ρ(g) = Q>"M

`

D`(g)#Q, (21)

where

Q

is an orthogonal,

N×N

change-of-basis matrix called Clebsch-Gordan coefﬁcients.

Dl

for

l=

0, 1, 2, ... is a

(

2

l+

1

)×(

2

l+

1

)

matrix known as a Wigner-D matrix.

⊕

is the direct sum of matrices

along the diagonal. Features transforming according to

Dl

are called type-

l

features. Type-0 features (scalars)

are invariant under rotations and type-1 features (vectors) rotate according to 3D rotation matrices. A rank-2

tensor decompose into representations of dimension 1 (trace), 3 (anti-symmetric part), and 5 (traceless

symmetric part). In the case of symmetric tensors such as permeability and formation factor, they are 0

⊕

2

features.

Equivariance:

Given a set of transformations

Tg:V → V

for

g∈G

where

G

is a group. a function

φ:V → Y is equivariant if and only if for every g∈Gthere exists a transformationSg:Y → Y such that

Sg[φ(v)] = φTg[v]for all g ∈G,v∈ V

In this paper, we are interested in building neural networks that are equivariant with respect to the 3D

rotation group SO(3).

Efforts have been made to build equivariant neural networks for different types of data and symmetry

groups. [Thomas et al.,2018,Weiler et al.,2018,Fuchs et al.,2020,Cohen et al.,2019,Finzi et al.,2020,Weiler

and Cesa,2019,Cohen et al.,2019,Worrall and Welling,2019]. In the case of 3D roto-translations tensor ﬁeld

network (TFN) [Thomas et al.,2018] is a good choice of parametrizing map

φ

. It by construction satisﬁes the

equivariance constraint and recently has been shown universal [Dym and Maron,2020], i.e., any continuous

equivariant function on point clouds can be approximated uniformly on compact sets by a composition of

TFN layers.

Equivariant Neural Network

: Just as convolution neural network (CNN) is composed of linear, pooling

and nonlinear layers, one also has to follow the same principle to build an equivariant neural network. The

major challenge is to characterize the equivariant linear map and design the architecture to parametrize such

maps. We sketch how to build linear equivariant layers below. Note that the input of equivariant neural

network is a Morse graph constructed from 3D voxel images. For simplicity of presentation, we follow the

original paper Tensor Field Network (see below) and use the point cloud as our input. Adding connection

between point clouds can be easily done by modifying Equations 22 and 23.

12 Chen Cai et al.

Tensor Field Network

: Tensor Field Network (TFN) is one of the equivariant neural networks targeted

for point clouds. Tensor Field Network maps feature ﬁelds on point clouds to feature ﬁelds on point clouds

under the constraint of SE(3) equivariance, the group of 3D rotations and translations. For the point clouds

{xi}N

i=1of size N, the input is a ﬁeld f:R3→Rdof the form

f(x) =

N

∑

j=1

fjδx−xj(22)

where

{fj}

are node features such as atom types or degrees of Morse graphs,

δ

is the Dirac delta function,

and

{xj}

are the 3D point coordinates. In order to satisfy the equivarance constraint, each

fj∈Rd

has to

be a concatenation of vectors of different types, where a subvector of type-

l

is denoted as

fl

j

. A TFN layer

takes type-

k

feature to type-

l

feature via the learnable kernel

W`k:R3→R(2l+1)(2k+1)

. The type

l

output at

position xiis

Fig. 7: We visualize the real-valued spherical harmonics

Y1

,

Y2

,

Y3

in each row, where the color and density

indicates the sign and absolute value of the functions. Spherical harmonics are solutions of Laplace equations

for functions on the sphere. They form a complete set of orthogonal functions and are used to parametrize

the weights of an equivariant neural network.

f`

out,i=∑

k≥0ZW`kx0−xifk

in x0dx0

| {z }

k→`convolution

=∑

k≥0

n

∑

j=1

W`kxj−xifk

in,j

| {z }

node j→node imessage

(23)

It has been shown that kernel

W`k

[Weiler et al.,2018,Kondor,2018,Thomas et al.,2018] has to lie in the

span of an equivariant basis nW`k

Jok+`

J=|k−`|. Mathematically,

W`k(x) =

k+`

∑

J=|k−`|

ϕ`k

J(kxk)W`k

J(x), where W`k

J(x) =

J

∑

m=−J

YJm (x/kxk)Q`k

Jm . (24)

Each basis kernel

W`k

J:R3→R(2`+1)×(2k+1)

is formed by taking the linear combination of Clebsch-

Gordan matrices

Q`k

Jm

of shape

(

2

`+

1

)×(

2

k+

1

)

, where the

J

,

mth

linear combination coefﬁcient is the

Equivariant geometric learning for digital rock physics 13

mth

dimension of the

Jth

spherical harmonic. Note that the only learnable part in

W`k

is the radial function

ϕ`k

J(kxk)

.

Q`k

Jm

and

YJm

(and therefore

W`k

J(x)

) are precomputed and ﬁxed. See [Thomas et al.,2018,Fuchs

et al.,2020] for more details.

CNN

: The non-Euclidean graph neural network architectures’ performance is tested against a classically

used Euclidean 3D convolutional neural network. The 3D convolutional network has been previously

employed in the identiﬁcation of material parameters [Santos et al.,2020]. Convolutional layers have

successfully been implemented for feature extraction from both 2D and 3D images. However, they can be

prone to noise, grid resolution issues and do not guarantee frame indifference in general.

The architecture employed in this control experiment predicts the formation factor and permeability

tensors directly from the 3D microstructure image. The input of this architecture is a 3D binary voxel image

(150

×

150

×

150 pixels) and the output is either the formation factor tensor or the permeability tensor. The

architecture consists of ﬁve 3D convolution layers with ReLU activation functions. Each convolutional layer

is followed by a 3D max pooling layer. The output of the last pooling layer is ﬂattened and then fed into

two consecutive dense layers (50 neurons each) with ReLU activations for ﬁnal prediction.

GNN

: We also build a baseline GNN to compare against equivariant GNN. There are various choices

of Graph Convolution layers, and we experiment popular choices such at GCN [Kipf and Welling,2016],

GraphSage [Hamilton et al.,2017], GAT [Veli

ˇ

ckovi

´

c et al.,2017], CGCNN [Xie and Grossman,2018], and

GIN [Xu et al.,2018]. Empirically we ﬁnd GIN works the best. The building block of our graph neural

networks is based on the modiﬁcation of Graph Isomorphism Network (GIN) that can handle both node

and edge features. In particular, we ﬁrst linear transform both node feature and edge feature to be vectors of

the same dimension. At the k-th layer, GNNs update node representations by

h(k)

v=ReLU

MLP(k)

∑

u∈N (v)∪{v}

h(k−1)

u+∑

e=(v,u):u∈N (v)∪{v}

h(k−1)

e

(25)

where

N(v)

is a set of nodes adjacent to

v

, and

e= (v

;

v)

represents the self-loop edge. MLP stands for

multilayer perceptron. Edge features h(k−1)

eis the same across the layers.

We use average graph pooling to obtained the graph representation from node embeddings, i.e.,

hG=

MEAN nh(K)

v|v∈Go. We set the number of layers to be 20 and the embedding dimension to be 50.

Other experiment details

: The metric used for training and test to measure the agreement of prediction

and true permeability is loss(

ˆ

y

,

y

) =

||y−ˆ

y||F

||y||F

, where

y

,

ˆ

y

stands for the true permeability and predicted

permeability tensor and

||y||F

stands for Frobenius norm, i.e.,

||y||F=q∑3

i=1∑3

j=1yij 2=ptrace(y∗y)

.

Note that

||y||F=||Uy||F=||yU ||F

for any unitary matrix

U

. We use 67% data for training and the rest for

test. We try 5 different random split to obtain more accurate result. To tune the hyperparameters (learning

rate, number of layers...) of different models, We take 1/3 of training data for validation and pick the best

hyperparameter combinations on validation data. We use software e3nn [Geiger et al.,2020] for Tensor Field

Network and pytorch geometric [Fey and Lenssen,2019] for baseline graph neural network.

Equivaraince Error

: Following Fuchs et al. [2020], we use

∆EQ =kLsΦ(f)−ΦLs(f)k2/kLsΦ(f)k2

to

measure equivariance error where

Ls

,

Φ

and

f

denotes the group action on output feature ﬁeld, neural

network, and input feature ﬁeld respectively. we measure the exactness of equivariance by applying

uniformly sampled SO(3)-transformations to input and output. The distance between the two, averaged

over samples, yields the equivariance error. The

∆EQ

for TFN is 1.1

∗

10

−7

, indicating TFN is equivariant

with respect to SO(3) up to the small error.

Visualization

: Given a positive semi-deﬁnite matrix

M

, we can visualize it as an ellipsoid located at

the origin, where the distance away from boundary in direction

v

(unit length) is

vTMv

. We visualize the

prediction of equivariant neural network for Morse graphs of different orientations, shown in ﬁgure 8and 9.

It can be seen that as we rotate the Morse graph, the output of equivariant neural network , visualized as an

ellipsoid, also rotates accordingly.

Results on the augmented dataset

: As the dataset is small, and may not play an advantage to the

equivariant neural network, we augmented each graph with property

P

by randomly rotating the input by

R

, and take the

RPRT

as the ground truth property of the rotated input. Results are shown in Figure 10. We

14 Chen Cai et al.

Fig. 8: Rotating Morse graph along y-axis (see top ﬁgures) will make the prediction of equivariant neural

networks rotate accordingly (see bottom ﬁgures). The second-order tensor predicted by the equivariant

neural network is visualized as an ellipsoid.

Fig. 9: Rotating Morse graph along z-axis (see top ﬁgures) will make the prediction of equivariant neural

networks rotate accordingly (see bottom ﬁgures). The second-order tensor predicted by the equivariant

neural network is visualized as an ellipsoid

tested two ways of augmentation 1) augment all datasets and 2) only augment training dataset. The results

are shown in Figures 10 and 11.

We ﬁnd for the equivariant neural network, the test error is signiﬁcantly lower than the non-equivariant

ones. As the argumentation multiplier increase from 2 to 5, the error reduces signiﬁcantly to nearly perfect

performance. For non-equivariant neural networks, although the data augmentation improves the perfor-

mance measured by the metric, the result is still far away from (one order of magnitude worse) the results

obtained from the equivariant neural network.

In the case of augmenting only training data, we ﬁnd that the performance slowly improves as the

number of copies increases for non-equivariant GNN, but the equivariant neural network without data

argumentation is still better than the GNN. This demonstrates the data efﬁciency of equivariant neural

network.

Equivariant geometric learning for digital rock physics 15

Fig. 10: Augment all dataset. The loss after augmenting all data by

k

copies (shown in x-axis as augmentation

multiplier). The blue/orange curve indicates the performance of GNN under different number of copies

(k=2, 5, 10, 20, 50). Equivariant neural network outperforms GNN by a large margin.

Fig. 11: Augment only training data. The loss after augmenting all data by

k

copies (shown in x-axis as

augmentation multiplier). We can see as

k

increases, the loss of GNN on test data slightly decreases, but

still not better than the equivariant neural network without data augmentation, whose result is shown in

dashed line.

5 Prediction results compared with CNN

The permeability and formation factor predictions made by the trained CNN, GNN, and Equivariant GNN

are shown in Tables 1and 2respectively. Generally speaking, one may observe that GNN outperforms CNN,

and Equivariant GNN outperforms GNN. We believe the Morse graph representation, although only being

a rough representation of 3D images, captures the essence of the geometrical and topological information

that aids downstream tasks. This trend is more obvious in the formation factor calculation than that of the

effective permeability.

Another advantage of GNN over CNN is the computational efﬁciency: performing convolution on 3D

images takes about 100s per epoch while GNN takes only 4s. Even though using Morse graph comes with

the overhead of computing Morse graphs (takes roughly 3 minutes per graph), the overall computational

time of working with graph representation is still desirable.

Previously published works that employ convolutional neural networks for effective permeability

predictions are often trained with data set way smaller than those used in other scientiﬁc disciplines. For

instance, the convolutional neural network predictions on velocity ﬁeld employ only 1080 3D images with

80

3

voxels in Santos et al. [2020] whereas similar research work on predicting effective permeability by

Srisutthiyakorn* [2016] and Sudakov et al. [2019] employs database that consists of 1000 images with 100

3

16 Chen Cai et al.

Table 1: Permeability. Comparison between three models. The error for baseline CNN, baseline GNN,

equivariant GNN are shown for ﬁve different splits of data. The improvement indicates the improvement of

equivariant GNN over GNN.

seed CNN GNN Equivariant GNN Improvement

1 0.312 0.246 0.218 11.3%

2 0.354 0.247 0.221 10.5%

3 0.339 0.251 0.226 10.0%

4 0.379 0.247 0.228 7.7%

5 0.382 0.298 0.252 15.4%

mean 0.353 0.258 0.229 11.2%

Table 2: Formation factor. Comparison between three models. The error for baseline CNN, baseline GNN,

equivariant GNN are shown for ﬁve different splits of data. The improvement indicates the improvement of

equivariant GNN over GNN.

seed CNN GNN Equivariant GNN Improvement

1 0.081 0.048 0.039 17.6%

2 0.091 0.049 0.044 9.9%

3 0.129 0.050 0.043 14.6%

4 0.127 0.051 0.042 18.3%

5 0.151 0.047 0.039 17.4%

mean 0.116 0.049 0.041 15.6%

voxels and 9261 3D images with 100

3

voxels respectively. In our cases, we employ 300 3D images with 150

3

voxel data for the training and test where only 2/3 of the data are used for training the neural network and

1/3 of the data are used in the test cases that measure the performance reported in Tables 1and 2. As a

comparison, the common datasets showcasing the advantage of equivariant neural network is QM9 are

much larger (133,885 small molecule graphs) and more diverse than the dataset employed in this study.

However, since both the pore-scale numerical simulations and experiments are expensive [Arns et al.,

2004,Sun et al.,2011b,Andr

¨

a et al.,2013b,Sun et al.,2011a,Vlassis et al.,2020,Wang et al.,2021,Fuchs

et al.,2021,Heider et al.,2021], one may argue that the ability of the neural network to function properly

with a relatively smaller dataset is crucial for practical purposes. Although the data set may not play to

our advantage, equivariant neural network still outperforms GNN. The advantage of equivariant neural

network over GNN is much more prominent in the case of the augmented dataset, shown in Figure 10. In

addition, equivariance with respect to SE(3) group guarantees the material frame indifference.

Figures 12 and 13 showcase results of 100

blind predictions

on formation factor and effective perme-

ability on binary micro-CT images that are excluded from the training database. In both cases, we compare

the ratio of the major, immediate and minor principle values of the effective permeability predicted by

CNN, GNN, and equivariant GNN over the benchmark calculation done by the FFT solver. If the results are

perfect, the ratio is 1. In both formation factor and effective permeability predictions, there are few trends

that worth noticing. First, the formation factor predictions are generally more accurate than that of the

effective permeability. Second, the beneﬁts of using the GNN and equivariant GNN are more obvious in the

formation factor predictions than that of the effective permeability. This might be attributed to the fact that

the surface conductivity leads to the topology of the ﬂux in the inverse problem correlates more strongly

with the Morse graph.

Another common trend we noticed is that the predictions on the two graphs neural network tend to

underestimate the largest eigenvalues of the formation factor and more likely to overestimate the middle

and smallest eigenvalues, whereas the CNN generally leads to underestimation of all eigenvalues with the

smallest eigenvalues being the most underestimated. This trend, however, is not observed in the effective

permeability predictions in Figure 13.

Equivariant geometric learning for digital rock physics 17

Fig. 12: For each test data, we plot the ratio of predicted eigenvalues over true eigenvalues of the formation

factor tensor. From left to right: prediction of CNN, GNN, and equivariant GNN.

On the other hand, the CNN predictions on effective permeability do not exhibit a similar trend. While

the errors for the CNN permeability predictions do not exhibit a clear pattern, the equivariant GNN and,

to a lesser extent, the GNN predictions both show that the largest eigenvalues tend to be more likely to

be underestimated whereas the smallest one is likely to be overestimated. These results suggest that the

prediction seems to be less accurate when the anisotropic effect is strong.

To examine this issue further, we compare the spectrum of the eigenvalues of both formation factor

and effective permeability in Figure 14. It can be seen that formation factor has a much smaller eigenvalue

ratio compared to permeability. While the ratio of the eigenvalues of the effective permeability may vary by

about an order of magnitude, that of formation factor is generally within a factor of 2 and hence suggesting

the nearly isotropic nature of the formation factor. Since the formation factor predictions are about one order

more accurate than the effective permeability predictions, these results suggest that the degree of anisotropy

may play an important role in the accuracy of the GNN and equivariant predictions. This limitation might be

circumvented by including more anisotropic data in the training dataset or different graph representations

that better captures the anisotropy of the pore space. Researches on both topics are ongoing but are out of

the scope of this study.

In addition to the eigenvalues, we also compare the predictions of the principal directions of both

formation factor and effective permeability. Notice that the permeability tensors obtained from the inverse

problems are often non-symmetric (see White et al. [2006] and Sun and Wong [2018]) whereas the formation

factors we obtained from the FFT are very close to symmetric despite machine precision error. Given the

true and predicted permeability/formation factor tensor, we compute its eigenvectors (unit length) and

show distribution (in boxplot) of angle error. Angle error measures how close the predicted eigenvector

ˆ

v

is to true eigenvector

v

, measured in terms of 1

− |v·ˆ

v| ∈ [

0, 1

]

. Equivariant GNN is better at predicting

the eigenvector corresponding to the largest eigenvalue (but not necessarily best for other eigenvectors),

which might be the reason that it performs better than the other two models. Notice that the spectrum of the

18 Chen Cai et al.

Fig. 13: For each test data, we plot the ratio of predicted eigenvalues over true eigenvalues of the effective

permeability tensor. From left to right: prediction of CNN, GNN, and equivariant GNN.

Fig. 14: Let

v0

,

v1

,

v2

denote the eigenvalues of true permeability/formation factor from small to large. We

plot the distribution of eigenvalue ratios over all data.

eigenvalues for the formation factor is very narrow and hence predicting the eigenvectors of the formation

factor could be more difﬁcult but of less importance. On the other hand, since the permeability exhibits

a higher degree of anisotropy, the eigenvector predictions are more crucial for practical purposes. Closer

examination of the eigenvector predictions performed by CNN, GNN, and the equivariant GNN indicates

that the equivariant neural network may lead to modest improvements in the accuracy of the orientation of

the principal directions (see Figure 15). Presumably, one may expect that this improvement can be more

signiﬁcant with more data. A systematic study on the relationship between the amount of training of data

and performance may provide a fuller picture on this issue but is out of the scope of this research.

Equivariant geometric learning for digital rock physics 19

Fig. 15: The plot of angle error for different property and models. First/second/third eigenvector denotes

the eigenvectors corresponding to the smallest, intermediate, and largest eigenvalues.

6 Conclusions

This paper introduces the equivariant GNN and Morse graph representation to enable a direct image-

to-prediction workﬂow that provides formation factor and effective permeability images of a series of

sandstone images with improved accuracy. There are several departures from the standard convolutional

neural network that may attribute to the improved performance. First, the introduction of Morse graphs to

represent the pore topology and geometrical information provide a more efﬁcient way to represent and store

the microstructures. Second, incorporating SE(3) equivariance constraint into the neural network is the key

to enforce the material frame indifference and ensure that the machine learning predictions are not affected

by the observer’s frame. This work demonstrates how graph and group representations can be leveraged to

enforce physics principles for mechanistic predictions relevant to image-based engineering analysis.

7 Acknowledgements

Chen Cai would like to thank the helpful discussion on theory and implementation of equivariant neural

network with Maurice Weiler, Pim de Haan, Fabian Fuchs, Tess Smidt and Mario Geiger. This collaborative

work is primarily supported by grant contracts OAC-1940203 and OAC-1940125. The additional efforts and

labor hours of the Columbia Research team members are supported by the NSF CAREER grant from the

Mechanics of Materials and Structures program at National Science Foundation under grant contract CMMI-

1846875, the Dynamic Materials and Interactions Program from the Air Force Ofﬁce of Scientiﬁc Research

under grant contracts FA9550-19-1-0318 and Army Research Ofﬁce under grant contract W911NF-18-2-0306.

These supports are gratefully acknowledged.

References

Michael D Abr

`

amoff, Paulo J Magalh

˜

aes, and Sunanda J Ram. Image processing with imagej.

Biophotonics

international, 11(7):36–42, 2004.

Heiko Andr

¨

a, Nicolas Combaret, Jack Dvorkin, Erik Glatt, Junehee Han, Matthias Kabel, Youngseuk Keehm,

Fabian Krzikalla, Minhui Lee, Claudio Madonna, et al. Digital rock physics benchmarks—part i: Imaging

and segmentation. Computers & Geosciences, 50:25–32, 2013a.

Heiko Andr

¨

a, Nicolas Combaret, Jack Dvorkin, Erik Glatt, Junehee Han, Matthias Kabel, Youngseuk

Keehm, Fabian Krzikalla, Minhui Lee, Claudio Madonna, et al. Digital rock physics benchmarks—part ii:

Computing effective properties. Computers & Geosciences, 50:33–43, 2013b.

Gustave E Archie et al. The electrical resistivity log as an aid in determining some reservoir characteristics.

Transactions of the AIME, 146(01):54–62, 1942.

20 Chen Cai et al.

Christoph H Arns, Mark A Knackstedt, W Val Pinczewski, and Nicos S Martys. Virtual permeametry on

microtomographic images. Journal of Petroleum Science and Engineering, 45(1-2):41–46, 2004.

S. Banerjee, L. Magee, D. Wang, X. Li, B. Huo, J. Jayakumar, K. Matho, M. Lin, K. Ram, M. Sivaprakasam,

J. Huang, Y. Wang, and P. Mitra. Semantic segmentation of microscopic neuroanatomical data by

combining topological priors with encoder-decoder deep networks.

Nature Machine Intelligence

, 2:

585–594, 2020.

Jacob Bear. Dynamics of ﬂuids in porous media. Courier Corporation, 2013.

William J Bosl, Jack Dvorkin, and Amos Nur. A study of porosity and permeability using a lattice boltzmann

simulation. Geophysical Research Letters, 25(9):1475–1478, 1998.

AE Bussian. Electrical conductance in a porous medium. Geophysics, 48(9):1258–1268, 1983.

Taco Cohen and Max Welling. Group equivariant convolutional networks. In

International conference on

machine learning, pages 2990–2999. PMLR, 2016.

Taco S Cohen, Maurice Weiler, Berkay Kicanaoglu, and Max Welling. Gauge equivariant convolutional

networks and the icosahedral cnn. arXiv preprint arXiv:1902.04615, 2019.

Antonio Costa. Permeability-porosity relationship: A reexamination of the kozeny-carman equation based

on a fractal pore-space geometry assumption. Geophysical research letters, 33(2), 2006.

Stephen C Cowin and Stephen B Doty. Tissue mechanics. Springer Science & Business Media, 2007.

T. K. Dey, J. Wang, and Y. Wang. Graph reconstruction by discrete morse theory. In

Proc. Internat. Sympos.

Comput. Geom., pages 31:1–31:15, 2018.

Tamal Dey, Jiayuan Wang, and Yusu Wang. Road network reconstruction from satellite images with machine

learning supported by topological methods. In

Proc. 27th ACM SIGSPATIAL Intl. Conf. Adv. Geographic

Information Systems (GIS), pages 520–523, 2019.

Tamal K. Dey, Jiayuan Wang, and Yusu Wang. Improved road network reconstruction using discrete morse

theory. In

Proc. 25th ACM SIGSPATIAL Intl. Conf. Adv. Geographic Information Systems (GIS)

, pages

58:1–58:4, 2017.

Nadav Dym and Haggai Maron. On the universality of rotation equivariant point cloud networks.

arXiv

preprint arXiv:2010.02449, 2020.

Matthias Fey and Jan Eric Lenssen. Fast graph representation learning with pytorch geometric.

arXiv

preprint arXiv:1903.02428, 2019.

Marc Finzi, Samuel Stanton, Pavel Izmailov, and Andrew Gordon Wilson. Generalizing convolutional neural

networks for equivariance to lie groups on arbitrary continuous data.

arXiv preprint arXiv:2002.12880

,

2020.

R. Forman. A discrete Morse theory for cell complexes.

Geometry, Topology and Physics for Raoul Bott.

,

1995.

Ari L Frankel, Reese E Jones, Coleman Alleman, and Jeremy A Templeton. Predicting the mechanical

response of oligocrystals with deep learning. Computational Materials Science, 169:109099, 2019.

JT Fredrich, AA DiGiovanni, and DR Noble. Predicting macroscopic transport properties using microscopic

image data. Journal of Geophysical Research: Solid Earth, 111(B3), 2006.

Alexander Fuchs, Yousef Heider, Kun Wang, WaiChing Sun, and Michael Kaliske. Dnn2: A hyper-parameter

reinforcement learning game for self-design of neural network based elasto-plastic constitutive descrip-

tions. Computers & Structures, 249:106505, 2021.

Fabian B Fuchs, Daniel E Worrall, Volker Fischer, and Max Welling. Se (3)-transformers: 3d roto-translation

equivariant attention networks. arXiv preprint arXiv:2006.10503, 2020.

Mario Geiger, Tess Smidt, Benjamin K. Miller, Wouter Boomsma, Kostiantyn Lapchevskyi, Maurice Weiler,

Micha? Tyszkiewicz, and Jes Frellsen. github.com/e3nn/e3nn, May 2020. URL

https://doi.org/10.

5281/zenodo.3723557.

William L Hamilton, Rex Ying, and Jure Leskovec. Inductive representation learning on large graphs.

arXiv

preprint arXiv:1706.02216, 2017.

Yousef Heider, Hyoung Suk Suh, and WaiChing Sun. An ofﬂine multi-scale unsaturated poromechanics

model enabled by self-designed/self-improved neural networks.

International Journal for Numerical

and Analytical Methods in Geomechanics, 2021.

John Conrad Jaeger, Neville GW Cook, and Robert Zimmerman.

Fundamentals of rock mechanics

. John

Wiley & Sons, 2009.

Equivariant geometric learning for digital rock physics 21

Thomas N Kipf and Max Welling. Semi-supervised classiﬁcation with graph convolutional networks.

arXiv

preprint arXiv:1609.02907, 2016.

Risi Kondor. N-body networks: a covariant hierarchical neural network architecture for learning atomic

potentials. arXiv preprint arXiv:1803.01588, 2018.

Matthew R Kuhn, WaiChing Sun, and Qi Wang. Stress-induced anisotropy in granular materials: fabric,

stiffness, and permeability. Acta Geotechnica, 10(4):399–419, 2015.

Gary Mavko and Amos Nur. The effect of a percolation threshold in the kozeny-carman relation.

Geophysics

,

62(5):1480–1482, 1997.

Vincent Monchiet, Guy Bonnet, and Guy Lauriat. A fft-based method to compute the permeability induced

by a stokes slip ﬂow through a porous medium. Comptes Rendus M ´

ecanique, 337(4):192–197, 2009.

Trung-Kien Nguyen, Vincent Monchiet, and Guy Bonnet. A fourier based numerical method for computing

the dynamic permeability of periodic porous media.

European Journal of Mechanics-B/Fluids

, 37:90–98,

2013.

Mervyn S Paterson and Teng-fong Wong.

Experimental rock deformation-the brittle ﬁeld

. Springer Science

& Business Media, 2005.

Javier E Santos, Duo Xu, Honggeun Jo, Christopher J Landry, Ma

ˇ

sa Prodanovi

´

c, and Michael J Pyrcz.

Poreﬂow-net: A 3d convolutional neural network to predict ﬂuid ﬂow through porous media.

Advances

in Water Resources, 138:103539, 2020.

Nattavadee Srisutthiyakorn*. Deep-learning methods for predicting permeability from 2d/3d binary-

segmented images. In

SEG technical program expanded abstracts 2016

, pages 3042–3046. Society of

Exploration Geophysicists, 2016.

Oleg Sudakov, Evgeny Burnaev, and Dmitry Koroteev. Driving digital rock towards machine learning:

Predicting permeability with gradient boosting and deep neural networks.

Computers & geosciences

,

127:91–98, 2019.

Hyoung Suk Suh and WaiChing Sun. An immersed phase ﬁeld fracture model for microporomechanics

with darcy–stokes ﬂow. Physics of Fluids, 33(1):016603, 2021.

Wai Ching Sun, Jose E Andrade, and John W Rudnicki. Multiscale method for characterization of porous mi-

crostructures and their impact on macroscopic effective permeability.

International Journal for Numerical

Methods in Engineering, 88(12):1260–1279, 2011a.

WaiChing Sun and Teng-fong Wong. Prediction of permeability and formation factor of sandstone with

hybrid lattice boltzmann/ﬁnite element simulation on microtomographic images.

International Journal

of Rock Mechanics and Mining Sciences, 106:269–277, 2018.

WaiChing Sun, Jos

´

e E Andrade, John W Rudnicki, and Peter Eichhubl. Connecting microstructural at-

tributes and permeability from 3d tomographic images of in situ shear-enhanced compaction bands using

multiscale computations. Geophysical Research Letters, 38(10), 2011b.

WaiChing Sun, Qiushi Chen, and Jakob T Ostien. Modeling the hydro-mechanical responses of strip and

circular punch loadings on water-saturated collapsible geomaterials.

Acta Geotechnica

, 9(5):903–934,

2014.

Karl Terzaghi, Ralph B Peck, and Gholamreza Mesri.

Soil mechanics in engineering practice

. John Wiley &

Sons, 1996.

Nathaniel Thomas, Tess Smidt, Steven Kearnes, Lusann Yang, Li Li, Kai Kohlhoff, and Patrick Riley. Tensor

ﬁeld networks: Rotation-and translation-equivariant neural networks for 3d point clouds.

arXiv preprint

arXiv:1802.08219, 2018.

Petar Veli

ˇ

ckovi

´

c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua Bengio.

Graph attention networks. arXiv preprint arXiv:1710.10903, 2017.

Nikolaos N Vlassis, Ran Ma, and WaiChing Sun. Geometric deep learning for computational mechanics part

i: Anisotropic hyperelasticity.

Computer Methods in Applied Mechanics and Engineering

, 371:113299,

2020.

Jaroslav Vond

ˇ

rejc, Jan Zeman, and Ivo Marek. An fft-based galerkin method for homogenization of periodic

media. Computers & Mathematics with Applications, 68(3):156–173, 2014.

Kun Wang, WaiChing Sun, and Qiang Du. A non-cooperative meta-modeling game for automated

third-party calibrating, validating and falsifying constitutive laws with parallelized adversarial attacks.

Computer Methods in Applied Mechanics and Engineering, 373:113514, 2021.

22 Chen Cai et al.

S. Wang, Y. Wang, and Y. Li. Efﬁcient map reconstruction and augmentation via topological methods. In

Proc. 23rd ACM SIGSPATIAL, page 25. ACM, 2015.

Maurice Weiler and Gabriele Cesa. General e (2)-equivariant steerable cnns. In

Advances in Neural

Information Processing Systems, pages 14334–14345, 2019.

Maurice Weiler, Mario Geiger, Max Welling, Wouter Boomsma, and Taco S Cohen. 3d steerable cnns: Learning

rotationally equivariant features in volumetric data. In

Advances in Neural Information Processing

Systems, pages 10381–10392, 2018.

Andreas Weller, Lee Slater, and Sven Nordsiek. On the relationship between induced polarization and

surface conductivity: Implications for petrophysical interpretation of electrical measurements.

Geophysics

,

78(5):D315–D325, 2013.

Joshua A White, Ronaldo I Borja, and Joanne T Fredrich. Calculating the effective permeability of sandstone

with multiscale lattice boltzmann/ﬁnite element simulations. Acta Geotechnica, 1(4):195–209, 2006.

Daniel Worrall and Max Welling. Deep scale-spaces: Equivariance over scale. In

Advances in Neural

Information Processing Systems, pages 7366–7378, 2019.

Haiyi Wu, Wen-Zhen Fang, Qinjun Kang, Wen-Quan Tao, and Rui Qiao. Predicting effective diffusivity of

porous media from images by deep learning. Scientiﬁc reports, 9(1):1–12, 2019.

Jinlong Wu, Xiaolong Yin, and Heng Xiao. Seeing permeability from images: fast prediction with convolu-

tional neural networks. Science bulletin, 63(18):1215–1222, 2018.

Tian Xie and Jeffrey C Grossman. Crystal graph convolutional neural networks for an accurate and

interpretable prediction of material properties. Physical review letters, 120(14):145301, 2018.

Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks?

arXiv preprint arXiv:1810.00826, 2018.

Xin Zhan, Lawrence M Schwartz, M Naﬁ Toks

¨

oz, Wave C Smith, and F Dale Morgan. Pore-scale modeling

of electrical and ﬂuid transport in berea sandstone. Geophysics, 75(5):F135–F142, 2010.

Afra Zomorodian and Gunnar Carlsson. Computing persistent homology.

Discrete & Computational

Geometry, 33(2):249–274, 2005.