# Non-Equilibrium Nature of Fracture Determines the Crack Paths

Pengjie Shi  Shizhe Feng  Zhiping Xu<EMAIL_ADDRESS>Applied Mechanics
Laboratory and Department of Engineering Mechanics, Tsinghua University,
Beijing, 100084, China.

###### Abstract

A high-fidelity neural network-based force field, NN-F3, is developed to cover
the strain states up to material failure and the non-equilibrium, intermediate
nature of fracture. Simulations of fracture in 2D crystals using NN-F3 reveal
spatial complexities from lattice-scale kinks to sample-scale patterns. We
find that the fracture resistance cannot be quantified by the energy densities
of relaxed edges as in the literature. Instead, the fracture patterns,
critical stress intensity factors at the kinks, and energy densities of edges
in the intermediate, unrelaxed states offer reasonable measures for the
fracture toughness and its anisotropy.

Fracture is a catastrophic process in nature and engineering, which leaves
facets and kinks along the crack paths. In 2D crystals such as graphene and
h-BN, different types of edges can be cleaved by fracture, including zigzag
(Z), armchair (A), and mixed zigzag-armchair or chiral (C) edges (Fig. 1a).
The relative stabilities of graphene edge structures were explored
experimentally through the abundance of edges created by various techniques
such as fracture [1], mechanical exfoliation [2], and irradiation [3]. Most
observations show almost the same probabilities of zigzag and armchair edges
[4, 5, 6], while a few of them report either zigzag or armchair direction is
preferred over the other [3, 7, 8, 2, 9]. These facts suggest that the
stabilities of zigzag and armchair edges could be quite close, which
contradicts theoretical predictions from first-principles calculations.
Ground-state calculations based on the density functional theory (DFT) show
that electronic and structural relaxation of the armchair edge of graphene
significantly reduces its energy density, $\gamma_{\mathrm{A}}$, which becomes
lower than that of the zigzag edge, $\gamma_{\mathrm{Z}}$ [10, 11, 12, 13, 14,
15, 16]. The disagreement between theory and experiment remains unsolved for
more than a decade.

Figure 1:  (a) A crack-containing graphene specimen under force loading [8].
(b) Fracture toughness $G_{\mathrm{c}}$ of cracks along the zigzag and
armchair motifs, $G_{\mathrm{c}}=K_{\mathrm{I}}^{2}/E$ [17]. The oscillation
in $G_{\mathrm{c}}$ along with the advancing cracks signals the lattice
trapping events. (c) Theoretical predictions of $A_{\gamma}$ [14, 18, 16, 10,
11, 12, 13, 15, 19, 9, 20], and experimental measurements of $A_{G}$ [1, 2].
(d), Uniaxial stress-strain relations measured along the zigzag and armchair
directions.

The selection of crack paths during fracture is closely related to the
relative stability of edges. Under the framework of fracture mechanics, the
crack driving force can be measured by the energy release rate (ERR), $G$,
while the energy cost to activate the fracture is defined as the fracture
toughness, $G_{\mathrm{c}}$ [21]. The value of $G_{\mathrm{c}}$ is difficult
to determine by theory due to the non-equilibrium nature of fracture and thus
commonly measured by experiments [21]. In theoretical studies, the fracture
resistance is usually approximated by the surface or edge energy densities as
$G_{\mathrm{c}}=2\gamma$ (Fig. 1b) [22, 21, 23, 18]. The directional
dependence of $G_{\mathrm{c}}\left(\theta\right)$, which defines the relative
stability of different edges, is expected to align with that of
$2\gamma\left(\theta\right)$ [24, 1].

By analyzing the crack path under specific loading conditions, the relative
stability or the anisotropy of fracture can be deduced. The anisotropy in
$G_{\mathrm{c}}\left(\theta\right)$ and $\gamma\left(\theta\right)$ of
crystals with a honeycomb lattice such as graphene can be quantified by the
ratios between their values at post-fracture zigzag (Z) and armchair (A)
edges, $A_{G}=G_{\mathrm{Z}}/G_{\mathrm{A}}$, and
$A_{\gamma}=\gamma_{\mathrm{Z}}/\gamma_{\mathrm{A}}$ [14], respectively [4, 5,
6]. Recently, direct tensile tests of monolayer graphene and peeling tests of
highly-oriented pyrolytic graphite (HOPG), although unable to resolve the
atomic-scale edge structures, conclude weak anisotropies ($A_{G}=1.06$ [1] and
$0.971$ [2]) from the overall orientation of cracks.

Energy densities of edges cleaved by fracture cannot be directly measured in
experiments, and the use of theoretically calculated $2\gamma(\theta)$ as the
fracture toughness remains questionable. In fact, experimentally measured
fracture toughness is usually much higher than the value of $2\gamma$ even for
brittle crystals where plastic dissipation is absent [23, 18, 25, 26] (Fig.
1b). Large-scale molecular dynamics (MD) simulations may help address the
issue if provided with force fields of high accuracy and low cost. Empirical
force fields (FFs) reported in the literature cannot capture the non-
equilibrium nature and high, non-uniform lattice distortion at the crack front
[27, 14, 19, 9, 18, 16, 20, 28]. The values of $A_{\gamma}$ calculated using
Stillinger-Weber or Tersoff potentials are the same as the bond-cutting
estimation, $\sqrt{3}/2$, and do not correctly capture the bonding
characteristics of materials [29, 28]. By including the chemistry of
interatomic bonding, the adaptive intermolecular reactive empirical bond order
(AIREBO) predicts $A_{\gamma}<1$ [14, 18, 16], while the reactive force field
(ReaxFF) yields opposite results, $A_{\gamma}>1$ [19, 9, 20] (Fig. 1c).
Compared with the experimental measurements [1, 2], DFT calculations predict a
relatively strong anisotropy with $A_{\gamma}>1.1$, where electronic and
structural relaxation of the edges are taken into account [10, 11, 12, 13, 14,
15, 16] [17]. The DFT predictions are also quantitatively different from the
AIREBO and ReaxFF results. Recently, the implementation of neural network-
based force fields [30] shows the capability to resolve the accuracy-cost
dilemma and led to significant progress in several fields [31, 32, 33, 34].
However, the lack of a reasonable description of the non-equilibrium nature
and exploration of the full space of strain states leaves the atomistic
approach to fracture still immature.

Here we develop a neural network-based force field for fracture (NN-F3) for 2D
crystals including graphene and h-BN based on first-principles calculations
and an active-learning framework [30]. The tensorial nature of strain states
and the undercoordination nature of cleavaged edges [14] demand a large
training set of DFT data and are addressed by an active-learning workflow
[35]. Our training sets for NN-F3 include structures with strained lattices
(the uniaxial strain in the range of $0-0.25$ along different lattice
directions), cleaved edges (zigzag and armchair segments as well as the kinks
between them), and cracks ($203,554$ datasets in total). The Deep Potential
Smooth Edition (DeepPot-SE) model [36, 37] is used to train NN-F3, the
performance of which is validated by reporting mean absolute errors (MAEs) of
the energies per atom, the edge energy densities, and the interatomic forces
below $2\,\mathrm{meV}$, $2.2\,\mathrm{meV/\AA}$ and $43\,\mathrm{meV/\AA}$,
respectively. The relative error (RE) in the stress-strain relations is under
$2\%$ (Fig. 1d). The workflow thus assures an accurate description of the
crack-tip processes [17].

Figure 2:  (a) The simulation setup of fracture tests. (b) The relation
between $\theta_{\mathrm{Z}}$ and $\theta_{\mathrm{Edge}}$ obtained from the
MD simulations for graphene using NN-F3 and theoretical predictions using Eq.
2. (c) The atomic-level structure of edges with
$\theta_{\mathrm{Edge}}=19.11^{\circ}$. (d) The asymmetry between A- and
B-site atoms along armchair edges at the crack tip.

Fracture Anisotropy of Graphene\- The fracture of graphene is explored by
quasi-static uniaxial tension using the Large-scale Atomic/Molecular Massively
Parallel Simulator (LAMMPS) [38, 17]. In order to host relatively long cracks,
wide samples ($W\approx 50\,\mathrm{nm}$) are constructed (Fig. 2). One atom
at the left edge is removed to initialize the crack. Periodic boundary
conditions (PBCs) are enforced along the tensile direction. The span $L$ is in
the range of $4-8~{}\mathrm{nm}$ (and $15-20~{}\mathrm{nm}$ to see the size
dependence) to accommodate different lattice orientations
($\theta_{\mathrm{Z}}\in\left[0^{\circ},30^{\circ}\right]$, measured from the
zigzag motif, Fig. 2a, b).

Figure 3:  The ratio of SIFs ($K_{\mathrm{II}}/K_{\mathrm{I}}$) along with the
advancing crack at $\theta_{\mathrm{Z}}=10.89^{\circ}$ (a) and
$\theta_{\mathrm{Z}}=25.87^{\circ}$ (b). (c and d) Cleaved edges with zigzag
and armchair segments. The domain defined to calculate the SIFs is annotated
[17]. (e and f) Theoretical predictions of the relations between $\Delta$ and
$K_{\mathrm{II}}/K_{\mathrm{I}}$ from Eq. 4 using $A_{G}=0.96$. (g)
Theoretical predictions of the relation between $A_{G}$ and
$\left(K_{\mathrm{II}}/K_{\mathrm{I}}\right)_{c}$ using Eq. 4.

MD simulation results show that the cracks are straight at the sample scale
(Figs. 3) [17]. Their overall orientations are denoted by
$\theta_{\mathrm{Edge}}$ (measured from the zigzag motif, see Fig. 2a).
However, the cracks may deflect at the lattice scale, leaving kinks between
the zigzag and armchair segments behind (Fig. 2c). The relations between
$\theta_{\mathrm{Edge}}$ and $\theta_{\mathrm{Z}}$ are summarized in Fig. 2b,
which can be classified into three regimes. For
$\theta_{\mathrm{Z}}\in[0^{\circ},10^{\circ}]$ or $[27^{\circ},30^{\circ}]$,
the crack advances along the zigzag ($\theta_{\mathrm{Edge}}=0^{\circ}$) or
armchair ($\theta_{\mathrm{Edge}}=30^{\circ}$) direction, respectively. For
$\theta_{\mathrm{Z}}\in[10^{\circ},27^{\circ}]$, the crack advances between
them ($\theta_{\mathrm{Edge}}\in\left(0^{\circ},30^{\circ}\right)$). Cleavage
of zigzag edges dominates if the loading direction is uniformly sampled, which
is attributed to the fact of $G_{\mathrm{Z}}<G_{\mathrm{A}}$ [17]. This
finding conforms with the observations in the peeling experiments of HOPG
where the polycrystalline texture is randomly oriented [2]. However, the
energy densities of relaxed edge $\gamma(\theta)$ obtained from DFT
calculations display an opposite trend of
$\gamma_{\mathrm{Z}}>\gamma_{\mathrm{A}}$ [10, 11, 12, 13, 14, 15, 16, 17].
This inconsistency indicates that $\gamma(\theta)$ fails to correctly
characterize the anisotropy in fracture resistance.

The crack driving force under uniaxial tensile stress $\sigma_{y}$ is
$G\left(\theta_{\mathrm{Edge}}\right)\sim\cos^{2}\left(\theta_{\mathrm{Z}}-\theta_{\mathrm{Edge}}\right)\sigma_{y}^{2}$
[17]. Following the criterion of maximum ERR (MERR), the crack will advance in
the direction with $G\left(\theta_{\mathrm{Edge}}\right)\geq
G_{\mathrm{c}}\left(\theta_{\mathrm{Edge}}\right)$. In the honeycomb lattice
of graphene, the cleaved edges consist of zigzag and armchair segments, and
the value of $G_{\mathrm{c}}(\theta_{\mathrm{Edge}})$ can be estimated as the
average value of $G_{\mathrm{A}}$ and $G_{\mathrm{Z}}$ weighted by their
lengths [2, 14], that is

$G_{\mathrm{c}}\left(\theta_{\mathrm{Edge}}\right)=2G_{\mathrm{A}}\left[\sin\left(\theta_{\mathrm{Edge}}\right)+A_{G}\sin\left(30^{\circ}-\theta_{\mathrm{Edge}}\right)\right].$
(1)

This result presumes that the formation and interaction energies of lattice
kinks are negligible in comparison with the edge energies [14, 39]. The
direction of crack propagation, $\theta_{\mathrm{Edge}}$, can thus be obtained
from the lattice orientation, $\theta_{\mathrm{Z}}$, by finding the minimum of
$\sigma_{y}^{2}$ that satisfies $G=G_{\mathrm{c}}$, that is

$\theta_{\mathrm{Edge}}=\arg\min\sigma_{y}^{2}=\arg\min\frac{G_{\mathrm{c}}\left(\theta_{\mathrm{Edge}}\right)}{\cos^{2}\left(\theta_{\mathrm{Z}}-\theta_{\mathrm{Edge}}\right)}.$
(2)

The predictions using $A_{G}=0.96$ and $0.93$ fit the simulation results for
$\theta_{\mathrm{Edge}}$ smaller and larger than the critical value of
$\theta_{\rm c}=19.11^{\circ}$ (Fig. 2b), respectively. At
$\theta_{\mathrm{Edge}}=19.11^{\circ}$, the numbers of zigzag and armchair
segments are the same along the edge (Fig. 2c). The smaller values of $A_{G}$
at $\theta_{\mathrm{Edge}}>19.11^{\circ}$ may be attributed to the asymmetry
between the A and B sites at the armchair edges (Fig. 2d), which can elevate
the fracture toughness and promote deflection. Similar effects of the edge
asymmetry on crack deflection and toughening were also observed in h-BN [23,
17] and $\mathrm{WS_{2}}$ [27].

Origin of Edge Kinks\- The high-fidelity NN-F3 allows us to explore the edge
structures at the atomic level. Zigzag and armchair segments as well as
lattice kinks connecting them can be resolved at length scales where fracture
mechanics can be applied for analysis. Large-scale MD simulations excluding
the size effects show periodic crack patterns [17] , and highlight the
advantages of NN-F3 in offering high accuracy at the first-principles level
and low computation cost that allows direct simulations up to the experimental
scale [17, 1] The criterion of MERR [40, 24, 1] suggests that the direction of
a propagating crack defined in the local coordinate system (Fig. 3c and 3d) is

$\alpha=\arg\max\frac{G\left(\alpha\right)}{G_{\mathrm{c}}\left(\alpha\right)},$
(3)

where $G(\alpha)$ is evaluated by the SIFs in the tensile and shear modes
($K_{\mathrm{I}}$ and $K_{\mathrm{II}}$, respectively) as the out-of-plane
displacement is ignored [20]. The values of $K_{\mathrm{I}}$ and
$K_{\mathrm{II}}$ can be determined by fitting the crack-tip displacement
field with the Williams power expansion [41, 17].

For loading conditions with crack directions not aligning with the zigzag or
armchair motifs, we find that the presence of a mode-II feature could deflect
the mode-I crack [21, 1]. The effect can be measured by the ratio
$K_{\mathrm{II}}/K_{\mathrm{I}}$ extracted from MD simulations. Two
representative examples are shown in Figs. 3c and 3d, where the cleaved edges
are dominated by the zigzag and armchair segments, respectively. The value of
$K_{\mathrm{II}}/K_{\mathrm{I}}$ oscillates as the crack propagates (Figs. 3a
and 3b), indicating that the deflection is activated as the ratio approaches
the threshold values
$\left(K_{\mathrm{II}}/K_{\mathrm{I}}\right)_{\mathrm{c}}$. The threshold
depends on the loading conditions and lattice orientations and is higher for
cracks advancing in the zigzag direction than that along the armchair ones.
The asymmetry between the A and B sites of the armchair edge further breaks
the symmetry (Fig. 2d) and yields two thresholds (Fig. 3b), which confirms the
effect of edge asymmetry on $A_{G}$ (Fig. 2b).

To estimate the values of
$\left(K_{\mathrm{II}}/K_{\mathrm{I}}\right)_{\mathrm{c}}$ and their relations
with $A_{G}$, a dimensionless quantity $\Delta$ is introduced based on Eq. 3
as [17]

$\alpha=\arg\max\left[\frac{G\left(\alpha\right)}{G_{\mathrm{c}}\left(\alpha\right)}\frac{2G_{\mathrm{A}}E}{K_{\mathrm{I}}^{2}}\right]=\arg\max\left[\Delta\left(\alpha,A_{G},\frac{K_{\mathrm{II}}}{K_{\mathrm{I}}}\right)\right],$
(4)

The direction of cracks determined from $A_{G}$ and
$K_{\mathrm{II}}/K_{\mathrm{I}}$ follows the armchair or zigzag motifs
($\alpha=0^{\circ}$ or $30^{\circ}$) due to the discrete nature of lattices.
The relations between $K_{\mathrm{II}}/K_{\mathrm{I}}$ and $\Delta$ in the
armchair- and zigzag-dominated regimes with $A_{G}=0.96$ are summarized in
Figs. 3e and 3f, where the thresholds
$\left(K_{\mathrm{II}}/K_{\mathrm{I}}\right)_{\mathrm{c}}$ are identified as
$0.092$ and $0.18$, respectively. Alternatively, the values of $A_{G}$ can be
obtained from $\left(K_{\mathrm{II}}/K_{\mathrm{I}}\right)_{\mathrm{c}}$ that
is directly determined by experiments or simulations (Fig. 3g and Table 1).
The results show that the value of $A_{G}$ does not match the anisotropy
measured from the energies of relaxed edges in direct NN-F3 or DFT
calculations, $A_{\gamma}=\gamma_{\mathrm{Z}}/\gamma_{\mathrm{A}}=1.113$
(Figs. 1c) [17].

Figure 4:  (a) Energy densities of relaxed ($\gamma$) and unrelaxed ($\Gamma$) graphene edges. TEM images of edges in 2D graphene (b), h-BN (c), and MoS2 (d) crystals adapted from [8, 23, 42]. (e) Values of $A_{\gamma}$ and $A_{\Gamma}$ of graphene, h-BN and MoS2 obtained from DFT calculations. Table 1: The number of kinks in the sample with width $W\approx 50$ nm, and the relation between $\left(K_{\mathrm{II}}/K_{\mathrm{I}}\right)_{\rm c}$ and $A_{G}$ under loading conditions defined by $\theta_{\mathrm{Z}}$. $\theta_{\mathrm{Z}}$ | # kinks | $\left(K_{\mathrm{II}}/K_{\mathrm{I}}\right)_{\rm c}$ | $A_{G}$
---|---|---|---
$25.87^{\circ}$ | 8 | $\left[0.080,0.100\right]$ | $\left[0.940,0.966\right]$
$10.89^{\circ}$ | 13 | $\left[0.205,0.210\right]$ | $\left[0.935,0.940\right]$
$9.82^{\circ}$ | 2 | $\left[0.195,0.200\right]$ | $\left[0.945,0.950\right]$

Energy Densities of Unrelaxed Edges\- The mismatch between $A_{G}$ and
$A_{\gamma}$ implies that the anisotropy in the edge energies density fails to
capture the atomistic kinetics of fracture, which selects the crack path.
Since the work of fracture should not depend on posterior edge relaxation
processes after the event of cleavage, the energy densities of unrelaxed
edges, $2\Gamma(\theta)$, are calculated using NN-F3 or DFT calculations and
compared to $2\gamma(\theta)$ for relaxed edges. The results,
$A_{\Gamma}=0.959<1$, suggest a weak anisotropy in the fracture toughness,
agreeing well with the experimental evidence [2, 1] and the simulation results
(Fig. 4a). The values of $2\Gamma_{\rm Z}$ and $2\Gamma_{\rm A}$ also conform
qualitatively well with $G_{\rm Z}$ and $G_{\rm A}$, respectively, by ignoring
the lattice-trapping effects (Fig. 1b). We investigate several measures of
fracture energies [17], and conclude that the energy density of unrelaxed
edges, $A_{\Gamma}$, can be a good indicator of fracture resistance. The
consistency between $A_{G}$ and $A_{\Gamma}$ indicates that $2\Gamma$
characterizes the non-equilibrium nature of the fracture. Specifically, for
$\theta_{\mathrm{Edge}}<19.11^{\circ}$, the value $A_{G}=0.959$ fitted from
$\theta_{\mathrm{Edge}}-\theta_{\mathrm{Z}}$ relation matches well with
$A_{\Gamma}$ (Fig. 2b). For $\theta_{\mathrm{Edge}}>19.11$, the fitting result
of $A_{G}=0.93$ is slightly smaller than $A_{\Gamma}=0.959$, which is
attributed to the nature of asymmetric fracture where cracks advancing along
the armchair motif prefer to deflect into the zigzag directions (Fig. 3b). The
relations between $\theta_{\mathrm{Edge}}$ and $\theta_{\mathrm{Z}}$
summarized in Fig. 5 show that predictions by assuming $A_{G}=A_{\gamma}$ do
not prefer zigzag edges, while the results using $A_{G}=A_{\Gamma}$ confirm
the experimental results [2]. We also find that the energy densities of
unrelaxed edges are very close to the measured fracture toughness
($G_{\mathrm{c}}\left(\theta\right)\approx 2\Gamma\left(\theta\right)$) (Figs.
1b, 2b and 3), although the strain states at the crack tip are different from
that in lattice decohesion [43].

Figure 5:  $\theta_{\mathrm{Edge}}-\theta_{\mathrm{Z}}$ relations of graphene
obtained by assuming $A_{G}=A_{\gamma}$ (dash line) and $A_{G}=A_{\Gamma}$
(solid line) [17]. The data points are measurements from the peeling
experiments of HOPG [2].

Conclusion\- Using a high-fidelity neural network-based force field developed
in this work, we find that the kinetics of fracture is much determined by the
intermediate, unrelaxed states of the crack tip. The energy density of relaxed
edges widely used in the literature fails to offer a reasonable measure of
fracture toughness. Instead, the $\theta_{\mathrm{Edge}}-\theta_{\mathrm{Z}}$
relation, $\left(K_{\mathrm{II}}/K_{\mathrm{I}}\right)_{\rm c}$, and
$\Gamma\left(\theta\right)$ offer reasonable measures of the fracture
anisotropy, that are, $A_{G}=0.96$, $A_{G}=0.935-0.966$, and
$A_{\Gamma}=0.959$ (for graphene), respectively. The first two measures can be
obtained from experiments or simulations with the atomic-level resolution,
while the third one can be considered as material parameters and determined by
first-principles calculations. This work highlights the multiscale and non-
equilibrium nature of the fracture and the theory and methodology developed
for graphene are extended to other 2D crystals such as h-BN and MoS2 (Figs.
4b-4e) [17].

This study was supported by the National Natural Science Foundation of China
through grants 11825203, 11832010, 11921002, and 52090032. The computation was
performed on the Explorer 100 cluster system of the Tsinghua National
Laboratory for Information Science and Technology.

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##  Appendix A Participants at Machine Learning for Science: Bridging Data-
driven and Mechanistic Modelling (Dagstuhl Seminar 22382, 18-23 September
2022)

_Organisers: Philipp Berens, Kyle Cranmer, Neil D. Lawrence, Jessica
Montgomery, Ulrike von Luxburg._

Thank you to all those who contributed to Dagstuhl Seminar 22382, discussions
at which are the foundation for this paper:

  * •

Mauricio A Álvarez (University of Manchester, GB)

  * •

Bubacarr Bah (AIMS South Africa – Cape Town, ZA)

  * •

Jessica Beasley (Collective Next – Boston, US)

  * •

Philipp Berens (Universität Tübingen, DE)

  * •

Maren Büttner (Helmholtz Zentrum München & Universität Bonn)

  * •

Kyle Cranmer (University of Wisconsin – Madison, US)

  * •

Thomas G. Dietterich (Oregon State University – Corvallis, US)

  * •

Carl Henrik Ek (University of Cambridge, GB)

  * •

Stuart Feldman (Schmidt Futures – New York, US)

  * •

Asja Fischer (Ruhr-Universität Bochum, DE)

  * •

Philipp Hennig (Universität Tübingen, DE)

  * •

David W. Hogg (New York University, US)

  * •

Christian Igel (University of Copenhagen, DK)

  * •

Samuel Kaski (Aalto University, FI)

  * •

Ieva Kazlauskaite (University of Cambridge, GB)

  * •

Hans Kersting (INRIA – Paris, FR)

  * •

Niki Kilbertus (TU München, DE & Helmholtz AI München, DE)

  * •

Neil D. Lawrence (University of Cambridge, GB)

  * •

Gilles Louppe (University of Liège, BE)

  * •

Jakob Macke (Universität Tübingen, DE)

  * •

Dina Machuve (DevData Analytics – A, TZ)

  * •

Eric Meissner (University of Cambridge, GB)

  * •

Siddharth Mishra-Sharma (MIT – Cambridge, US)

  * •

Jessica Montgomery (University of Cambridge, GB)

  * •

Jonas Peters (University of Copenhagen, DK)

  * •

Aditya Ravuri (University of Cambridge, GB)

  * •

Markus Reichstein (MPI für Biogeochemistry – Jena, DE)

  * •

Bernhard Schölkopf (MPI für Intelligente Systeme – Tübingen, DE)

  * •

Francisco Vargas (University of Cambridge, GB)

  * •

Soledad Villar (Johns Hopkins University – Baltimore, US)

  * •

Ulrike von Luxburg (Universität Tübingen, DE)

  * •

Verena Wolf (Universität des Saarlandes – Saarbrücken, DE)

##  Appendix B Research questions arising from the ‘AI for science research
agenda’ discussion during the Dagstuhl workshop

### Building AI systems for science

  * •

How can AI systems accurately generalise from finite observations? How can
they detect causality or structure from finite observations?

  * •

What is the computational cost of complexity, and what methods can help manage
this?

  * •

What forms of system calibration and uncertainty quantification are useful in
the context of scientific discovery? Are theoretical guarantees necessary?

  * •

What new forms of explainability or interpretability could facilitate the
deployment of AI in science?

  * •

How could AI support generalisation from a small number of observations? What
methods could enable few- or one-shot learning?

  * •

How can AI researchers build meaningful models from data to accurately
represent causal mechanisms in the system of study? How can researchers
identify the most effective model for their system of study?

  * •

What does it mean to understand a model? How can researchers combine
explainability with complexity?

  * •

How can AI methods be made robust and easy to use in deployment by domain
scientists?

  * •

How can advances in simulation methods be applied in domains where the system
at hand is less easily described by equations?

  * •

What advances are needed to expand the use of simulations in science? How can
AI help simulate laboratory experiments or environments, helping make more
efficient different elements of the scientific process? How might this be
expanding in the long-term, for example to planning experimental design or
helping identify where data is missing?

  * •

How can ‘digital siblings’ be used to explore the impact of different
interventions on complex systems?

### Combining human and machine intelligence

  * •

How can AI researchers best extract, formalise and assimilate the knowledge
that domain researchers have acquired? What forms of knowledge representation
can formalise scientific understandings of the world, translating these to
objective functions for AI systems? What forms of human-AI engagement can make
use of the ‘qualitative’ knowledge – or intuitions about a system – that
domain researchers have accumulated?

  * •

How can AI capture the qualitative understanding that researchers have of
their domain to more accurately or effectively characterise a system?

  * •

How can AI be effectively deployed to mine the existing research knowledge
base – for example, papers, databases, and so on – to extract new insights?

  * •

Where can automation support research progress? Which elements of the
scientific process could be automated, and where is human input vital?

  * •

What forms of collaboration are needed to effectively specify helpful outputs
from an AI system?

  * •

How can insights from AI analysis be returned to researchers in an actionable
way? What mix of AI design, engineering, social interaction, and education can
make effective interfaces between domain researchers and AI systems?

  * •

How can the outputs of AI systems be made interpretable for scientific users?

  * •

How can AI researchers better understand and design for the forms of
interpretability that resonate with domain researchers?

  * •

What processes of collaboration or co-design can help describe what scientists
‘need to know’ from an AI system?

  * •

What best practices or methods can be deployed to effectively communicate
uncertainty from AI systems to human users?

###  B.1 Influencing practice and adoption

  * •

What are the craft skills in AI for science? What ‘know-how’ is necessary to
make AI work effectively in practice?

  * •

What skills-building or forms of outreach can help take AI tools out of the AI
community and into ‘the lab’?

  * •

How has machine learning been used most effectively for research and
innovation? What best practices, or lessons, do existing efforts in AI for
science offer?

  * •

Which AI tools are suitable for which purposes, disciplines, or experimental
designs? Is it possible to create a taxonomy for science?

  * •

Are there generalisable methods or conclusions that can be taken from domain-
specific efforts to deploy AI for science?