Data–Driven Games in Computational Mechanics
K. Weinberg
a
, L. Stainier
b
, S. Conti
c
and M. Ortiz
c
,
d
a
Chair of Solid Mechanics, Department of Mechanical Engineering, Universität Siegen, Paul-Bonatz-Str.
9-11, Siegen, 57076, Germany
b
Nantes Université, Ecole Centrale Nantes, CNRS, GeM, UMR 6183, Nantes, F-44000, France
c
Institut für Angewandte Mathematik and Hausdor
ff
Center for Mathematics, Universität Bonn, Endenicher Allee
60, Bonn, 53115, Germany
d
California Institute of Technology, Engineering and Applied Science Division, Pasadena, 91125, CA, USA
Abstract
We resort to game theory in order to formulate Data-Driven methods for solid mechanics in
which stress and strain players pursue di
ff
erent objectives. The objective of the stress player
is to minimize the discrepancy to a material data set, whereas the objective of the strain player
is to ensure the admissibility of the mechanical state, in the sense of compatibility and equilib-
rium. We show that, unlike the cooperative Data-Driven games proposed in the past, the new
non-cooperative Data-Driven games identify an e
ff
ective material law from the data and reduce
to conventional displacement boundary-value problems, which facilitates their practical imple-
mentation. However, unlike supervised machine learning methods, the proposed non-cooperative
Data-Driven games are unsupervised,
ansatz
–free and parameter–free. In particular, the e
ff
ective
material law is learned from the data directly, without recourse to regression to a parameterized
class of functions such as neural networks. We present analysis that elucidates su
ffi
cient con-
ditions for convergence of the Data-Driven solutions with respect to the data. We also present
selected examples of implementation and application that demonstrate the range and versatility
of the approach.
Keywords:
data-driven methods, game theory, computational solid mechanics, non-cooperative
data-driven games, unsupervised machine learning, e
ff
ective material law
1. Introduction
Game theory concerns itself with scenarios involving several players seeking strategies that
strive to minimize their respective costs or, equivalently, maximize their respective payo
ff
s. In
general, the cost of one player’s strategy depends on the strategies adopted by the remaining
players and, in consequence, the optimal strategies of the players are coupled to each other. In
one scenario, the players optimize their strategies
cooperatively
by striving to minimize a joint
cost computed as a weighted average of all costs, a condition known as
Pareto optimality
. In
another scenario, the players proceed
non-cooperatively
by each seeking to minimize its own
cost independently. Game theory was pioneered,
inter aliis
, by economists Vilfredo Pareto [1]
and John Nash [2], and mathematician John von Neumann [3], in seminal contributions and
has been of foundational importance in a variety of fields, including economics, social sciences,
evolutionary biology, computer science, and others.
Preprint submitted to Computer Methods in Applied Mechanics and Engineering
June 1, 2023
arXiv:2305.19279v1 [cs.CE] 26 May 2023
Despite its phenomenal success in other fields, game theory has been applied sparingly to
mechanics or not at all. From a mechanics perspective, deterministic game-theoretical problems
may be regarded as instances of
coupled problems
with a particular variational structure (cf. Sec-
tion 2 for a brief review). Among these problems,
inf-sup
problems may be regarded as
zero-sum
games. These correspondences and others open up the opportunity of bringing game-theoretical
concepts, tools and results to bear on a wide range of problems in mechanics, a potential that
remains largely unattained at present.
In this work, we resort to game theory in order to formulate Data-Driven methods for solid
mechanics in which
stress
and
strain players
pursue di
ff
erent objectives: the objective of the
stress player is to minimize the discrepancy to a material data set, whereas the objective of the
strain player is to ensure the admissibility of the mechanical state, in the sense of compatibility
and equilibrium. We show that, unlike the cooperative Data-Driven games proposed in the past
[4, 5, 6, 7, 8], the new non-cooperative Data-Driven games identify an
e
ff
ective material law
from the data and reduce to conventional displacement boundary-value problems, which facil-
itates their practical implementation. In particular, the Data-Driven e
ff
ective material law can
be conveniently implemented as a standard user-supplied material in commercial finite-element
software.
This change of mood notwithstanding, it bears emphasis that, unlike supervised machine
learning methods, the proposed non-cooperative Data-Driven games are unsupervised,
ansatz
–
free and parameter–free. In particular, the e
ff
ective material law is learned from the data
directly
,
without recourse to regression to a parameterized class of functions such as neural networks. In
this sense, the new non-cooperative Data-Driven games follow in the vein of prior cooperative
Data-Driven games [4, 5, 6, 7, 8] by striving to e
ff
ect a direct, unsupervised and model–free
connection between data and prediction.
By identifying stress and strain as players, the proposed non-cooperative Data-Driven games
fall within the set-oriented formulation of mechanics problems, briefly reviewed in Section 3.
The connection between such problems and game theory is introduced in Section 4 in the con-
text of cooperative games, in which stress and strain strive to achieve a common objective of
minimizing distance to a material set while satisfying the field equations of compatibility and
equilibrium. This cooperative strategy reproduces—and provides a game-theoretical interpre-
tation for—set-oriented Data-Driven methods proposed in [4, 5, 6, 7, 8]. The transition from
cooperative to non-cooperative moods is presented in Section 5 by regarding stress and strain as
adversarial players, each pursuing its own objective. Evidently, this strategy is suboptimal with
respect to the cooperative Data-Driven strategy, but it o
ff
ers the significant practical advantage
of reducing to a conventional and well-posed displacement problem, Section 5.2, amenable to
approximation, Section 5.3.
A particularly important case concerns approximations based on empirical point-data sets,
Section 6, e. g., measured empirically or computed from micromechanics. The central ques-
tion then concerns the elucidation of conditions on the data that ensure the convergence of the
Data-Driven solutions to the solution of the underlying—and unknown—material law. We pro-
vide rigorous conditions for convergence with respect to the data for two di
ff
erent scenarios:
i)
Uniformly convergent data
, in which the sampling error decreases as data is added to the
material-data set in a uniform manner controlled by strict upper bounds, Section 6.3.1, and ii)
noisy data with outliers
, in which the data concentrates around the limiting material law in a
weak or average sense that allows for the presence of outliers, Section 6.3.2. In this second sce-
nario, convergence requires regularization in the form of local data averages taken over carefully
chosen local neighborhoods in order to mitigate the e
ff
ect of the outliers, Section 6.2.
2
Finally, we present selected examples of implementation and application that demonstrate
the range and versatility of the approach, Section 7. In particular, the examples illustrate how
the approach can be implemented within a standard displacement finite-element framework in
any dimension, with and without regularization, and using interative solvers such as dynamic
relaxation and Newton-Raphson iteration. The examples also bear out the type of convergence
with respect to the data anticipated by the analysis.
2. Elements of game theory
Game theory
is a well-developed branch of mathematics (cf., e. g., [9] for a general modern
account), but it has not been extensively applied to solid mechanics and may, therefore, stand a
brief review. We specifically collect basic elements of the theory required in subsequent devel-
opments.
For present purposes, it su
ffi
ces to consider two-player, finite dimensional games (cf., e. g.,
[10, 11]). Specifically, we consider two players seeking strategies
u
∈
R
m
and
v
∈
R
n
who
strive to minimize their costs
F
(
u
,
v
) and
G
(
u
,
v
), respectively. They can do so
cooperatively
, by
minimizing a weighted average of their costs
J
(
u
,
v
)
=
λ
F
F
(
u
,
v
)
+
λ
G
G
(
u
,
v
)
, λ
F
≥
0
, λ
G
≥
0
, λ
F
+
λ
G
=
1
,
(1)
i. e., by seeking a joint strategy (
u
∗
,
v
∗
) such that
J
(
u
∗
,
v
∗
)
≤
J
(
u
,
v
)
,
for all
u
∈
R
m
,
v
∈
R
n
,
(2)
a condition known as
Pareto optimality
; or they can do so
non-cooperatively
, by each player
seeking strategies such that
F
(
u
∗
,
v
∗
)
≤
F
(
u
,
v
∗
)
,
for all
u
∈
R
m
,
(3a)
G
(
u
∗
,
v
∗
)
≤
G
(
u
∗
,
v
)
,
for all
v
∈
R
n
,
(3b)
a condition known as
Nash equilibrium
.
We note that, in both (1) and (3), the cost of each player depends on the strategy of the
competitor, which they do not control. The players seek to minimize their own costs either
jointly, as in (1) or without regard for the cost of the competitor, as in (3).
An important class of non-cooperative games is that of two-player zero–sum games. These
are games in which the cost of one player is the negative of the other, i. e., one player loses what
the other player gains. Under these conditions, we have
F
(
u
,
v
)
=
L
(
u
,
v
)
,
G
(
u
,
v
)
=
−
L
(
u
,
v
)
,
(4)
for some
Lagrangian
L
(
u
,
v
), and the Nash equilibrium conditions (3) become
L
(
u
∗
,
v
)
≤
L
(
u
∗
,
v
∗
)
≤
L
(
u
,
v
∗
)
,
(5)
which defines a
saddle-point
or inf
–
sup problem.
Problems (1) and (3) were introduced by economists Vilfredo Pareto [1] and John Nash [2]
in seminal contributions. From a mechanics perspective, problems (1) and (3) are instances of
coupled problems
with a particular variational structure.
3
Example 2.1
(Quadratic cost)
.
Suppose
F
(
u
,
v
)
=
1
2
Au
·
u
+
Cv
·
u
−
f
·
u
,
(6a)
G
(
u
,
v
)
=
1
2
Dv
·
v
+
Bu
·
v
−
g
·
v
,
(6b)
where
A
∈
R
m
×
m
,
C
∈
R
m
×
n
,
B
∈
R
n
×
m
,
D
∈
R
n
×
n
,
A
=
A
T
,
A
>
0,
D
=
D
T
,
D
>
0,
f
∈
R
m
,
g
∈
R
n
, (
·
) denotes the dot product and we write
Cu
·
v
=
(
Cu
)
·
v
,
et cetera
, for short. Then, the
Nash-equilibrium equations are
Au
+
Cv
=
f
,
(7a)
Bu
+
Dv
=
g
,
(7b)
or, in matrix form,
A
C
B
D
! (
u
v
)
=
(
f
g
)
,
(8)
which is a particular type of linear coupled problem characterized by symmetric and positive-
definite diagonal blocks. We also note that
C
,
B
T
in general, with the result that there is no joint
minimum principle for both players to appeal to together. Evidently, a unique Nash equilibrium
(
u
∗
,
v
∗
) exists if and only if the matrix of the system (8) is non-singular.
An alternative form of the problem is
a
(
y
,
z
)
=
b
(
z
)
,
for all
z
∈
Z
,
(9)
where
a
:
Z
×
Z
→
R
and
b
:
Z
→
R
,
Z
=
R
m
×
R
n
, defined as
a
(
y
,
z
)
=
(
α
|
β
)
A
C
B
D
! (
u
v
)
,
b
(
z
)
=
(
f
|
g
)
(
u
v
)
,
(10)
with
y
=
(
α,β
) and
z
=
(
u
,
v
), are non-symmetric bilinear and linear forms, respectively. Then,
by the Lax-Milgram theorem [12] a unique Nash equilibrium exist if and only if
a
(
z
,
z
)
≥
λ
∥
z
∥
2
,
(11)
for some
λ >
0, i. e., if
a
(
·
,
·
) is
coercive
. Suppose that the cost functions of the players are
separately coercive, i. e., there are
λ
A
>
0 and
λ
D
>
0 such that
Au
·
u
≥
λ
A
∥
u
∥
2
,
Dv
·
v
≥
λ
D
∥
v
∥
2
,
(12)
for all
u
∈
R
m
and
v
∈
R
n
, respectively. Suppose, in addition, that there is 0
≤
μ <
1 such that
(
B
+
C
T
)
u
·
v
≤
μ
Au
·
u
+
Dv
·
v
,
(13)
for all
u
∈
R
m
and
v
∈
R
n
. Then,
a
(
z
,
z
)
=
Au
·
u
+
Bu
·
v
+
Cv
·
u
+
Dv
·
v
≥
Au
·
u
+
Dv
·
v
−
(
B
+
C
T
)
u
·
v
≥
(1
−
μ
)
Au
·
u
+
Dv
·
v
≥
(1
−
μ
) min
{
λ
A
,λ
D
}∥
z
∥
2
.
(14)
Thus, the
coercivity condition
(13) ensures that
a
(
·
,
·
) be coercive with
λ
=
(1
−
μ
) min
{
λ
A
,λ
D
}
and, by Lax-Milgram, it ensures the existence of a unique Nash equilibrium.
□
4
3. Set–oriented formulation of problems in mechanics
We consider finite-dimensional mechanical systems comprising
m
components, e. g., struc-
tural members, material points,
et similia
, whose state is characterized by two work-conjugate
fields
ε
≡ {
ε
e
∈
R
d
,
e
=
1
,...,
m
}
and
σ
≡ {
σ
e
∈
R
d
,
e
=
1
,...,
m
}
. We refer to the
space of pairs
Z
e
=
{
z
e
≡
(
ε
e
,σ
e
)
∈
R
d
×
R
d
}
as the
local phase space
of component
e
, and
Z
=
Z
1
×···×
Z
m
=
R
N
×
R
N
,
N
=
md
, as the
global phase space
of the system. We suppose that
a suitable norm is defined in
Z
, e. g.,
∥
z
∥
=
m
X
e
=
1
w
e
∥
z
e
∥
2
e
1
/
2
=
m
X
e
=
1
w
e
C
e
ε
e
·
ε
e
+
C
−
1
e
σ
e
·
σ
e
1
/
2
,
(15)
where
w
e
>
0 are weights and
C
e
∈
R
d
×
d
sym
,
+
are positive-definite symmetric matrices,
e
=
1
,...,
m
.
3.1. Classical solutions
We begin by assuming linearized kinematics and compatibility and equilibrium constraints
of the general form
m
X
e
=
1
w
e
B
T
e
σ
e
=
f
,
(16a)
ε
e
=
B
e
u
+
g
e
,
e
=
1
,...
m
,
(16b)
where
u
∈
R
n
is the array of degrees of freedom of the system,
w
e
are positive weights,
B
e
∈
R
d
×
n
is a discrete gradient operator,
B
T
e
is a discrete divergence operator,
f
∈
R
n
is a force array
resulting from distributed sources and Neumann boundary conditions and the arrays
g
e
∈
R
d
follow from Dirichlet boundary conditions. In terms of global arrays,
wB
T
σ
=
f
,
(17a)
ε
=
Bu
+
g
,
(17b)
where we write
w
=
diag(
w
1
,...,
w
m
),
B
=
(
B
1
,...,
B
m
),
ε
=
(
ε
1
,...,ε
m
),
σ
=
(
σ
1
,...,σ
m
) and
g
=
(
g
1
,...,
g
m
).
Classically, the problem is closed by assuming a material law of the form
σ
e
=
ˆ
σ
e
(
ε
e
)
,
(18)
or, in terms of global arrays,
σ
=
ˆ
σ
(
ε
)
=
{
ˆ
σ
1
(
ε
1
)
,...,
ˆ
σ
m
(
ε
m
)
}
,
(19)
where ˆ
σ
e
(
·
) are material-specific functions. Existence and uniqueness of displacement solutions
then follows under suitable restrictions on
B
e
and ˆ
σ
e
(
·
), cf. Prop. 5.4.
3.2. Set–oriented reformulation
An alternative set–oriented representation of the material law ˆ
σ
(
ε
) is to view it as a graph
D
in phase space
Z
, or
material set
. In this representation, the material law is regarded as a material-
specific
N
-dimensional manifold, or graph, in the 2
N
-dimensional phase space
Z
. Similarly, the
5
constraints (16) are material independent and define an a
ffi
ne subspace
E
of
Z
, or
constraint set
.
The constraint set
E
encodes all the data of the problem, including geometry, loading and bound-
ary conditions. From elementary linear algebra considerations, it follows that the constraint set
E
is an a
ffi
ne subspace of
Z
of dimension
N
and co-dimension
N
[8]. The actual states
z
=
(
ε,σ
)
of the system in phase space, if they exist, lie in the intersection
D
∩
E
, i. e., are the admissi-
ble states that are consistent with the material law, or, equivalently, the material states that are
compatible and in equilibrium. Evidently, classical solutions exist if the sets
D
and
E
have a
non-empty intersection, i. e., if they are
transversal
[6].
4. Cooperative Data–Driven games in mechanics
Suppose that, as is often the case in practice, the graph
D
of the material law is not known
in its entirety, but only through an approximating sequence of data sets. For instance, the sets
D
may consist of increasing collections of points (
ε,σ
) in phase space
Z
obtained, e. g., by
experimental measurement. In general, the intersection between the admissible set
E
and the
material data sets
D
may be empty, in which case no classical approximating solution exists in
the sense of Section 3. One way to circumvent this excessive rigidity of the classical paradigm
is to relax the notion of ’solution’ and replace intersection by a regularized optimality criterion
[4, 6]. We regard the resulting paradigm as
well-posed
if the corresponding approximate Data–
Driven solutions converge to the exact solution as the data sets
D
sample the exact material-law
graph
D
with increasing fidelity. In this section, we appeal to game-theoretical concepts in order
to formulate well-posed Data-Driven approaches.
4.1. Cooperative game-theoretical reformulation
Data–Driven problems of the type proposed in [4, 5, 6, 7, 8] can be interpreted as cooperative
game problems in the sense of
Pareto optimality
. Thus, suppose that the material behavior is
characterized by a material law with graph
D
in phase space
Z
and that the equilibrium, com-
patibility, Dirichlet and Neumann constraints are represented by an a
ffi
ne subspace
E
of
Z
of
dimension
N
and co-dimension
N
. Let
I
D
(
y
)
=
(
0
,
if
y
∈
D
,
+
∞
,
otherwise
,
(20)
and
I
E
(
z
)
=
(
0
,
if
z
∈
E
,
+
∞
,
otherwise
,
(21)
be the corresponding
indicator functions
. Suppose, in addition, that we are given a
discrepancy
function
Φ
:
Z
×
Z
→
R
with the properties: i)
Φ
is convex; ii)
Φ
is non-negative; and iii)
Φ
(
y
,
z
)
=
0 i
ff
y
=
z
. We may then introduce the cost functions
F
(
y
,
z
)
=
I
D
(
y
)
+ Φ
(
y
,
z
)
,
(22a)
G
(
y
,
z
)
=
I
E
(
z
)
+ Φ
(
y
,
z
)
.
(22b)
Evidently, for given
z
the function
F
(
·
,
z
) requires
y
to be in
D
, whereupon it penalizes its distance
to
E
; and for given
y
the function
G
(
y
,
·
) requires
z
to be in
E
, whereupon it penalizes its distance
to
D
.
6
The Data–Driven solution (
y
∗
,
z
∗
) is now the minimizer of the function
J
(
y
,
z
)
=
w
F
F
(
y
,
z
)
+
w
G
G
(
y
,
z
)
,
(23)
with
w
F
>
0,
w
G
>
0 and
w
F
+
w
G
=
1, i. e., (
y
∗
,
z
∗
) is the solution of the cooperative game
J
(
y
∗
,
z
∗
)
≤
J
(
y
,
z
)
,
for all
y
,
z
∈
Z
,
(24)
in the sense of Pareto optimality. In this strategy,
y
∗
is the material state that is closest to being
admissible and
z
∗
is the admissible state that is closest to being material, as desired. We note that,
owing to the constraints (1) on the Pareto weights and the invariance of the indicator functions
under multiplication by positive constants, the combined cost functional (23) evaluates to
J
(
y
,
z
)
= Φ
(
y
,
z
)
+
I
D
(
y
)
+
I
E
(
z
)
,
(25)
independent of the Pareto weights, and the solution (
y
∗
,
z
∗
) is also in Nash equilibrium.
An appeal to direct methods of the calculus of variations shows that the solutions exist if, in
addition to the assumed convexity properties,
Φ
satisfies a growth condition of the form [4, 6]:
Φ
(
y
,
z
)
→
+
∞
if
∥
y
∥
+
∥
z
∥ →
+
∞
with
y
∈
D
and
z
∈
E
. This condition generalizes the
intersection notion of transversality between the sets
D
and
E
to sets that may be discontinuous,
e. g., point sets, and may lack a proper intersection. Evidently, every classical solution in the
intersection
D
∩
E
is also a Data–Driven solution.
4.2. The long and short games
We can reduce the problem to the determination of admissible states
z
by eliminating out the
material state variable
y
in the cooperative game (24). By analogy to the game of golf, we may
think of point data sets
D
as a collection of holes in a links. The player
z
plays the
long game
,
or game of approach to the holes. For a given outcome
z
of an approach shot, the player
y
then
plays the
short game
of putting the ball as close as possible to the nearest hole.
The
short game
is, therefore, defined by the functional
H
(
z
)
=
inf
y
∈
Z
J
(
y
,
z
)
=
I
E
(
z
)
+
φ
D
(
z
)
,
(26)
where
φ
D
(
z
)
=
inf
y
∈
D
Φ
(
y
,
z
)
(27)
measures the deviation of
z
from the data set
D
. The remaining
long game
is
H
(
z
∗
)
≤
H
(
z
)
,
for all
z
∈
Z
,
(28)
which endeavors to minimize the deviation
φ
D
(
z
) from
D
among all admissible state
z
∈
E
.
We note that the sequence of long and short games is equivalent to (24), since the game is
cooperative.
4.3. Minimum–distance Data–Driven game
The Data–Driven paradigm formulated in [4, 6] may be rephrased as a cooperative game by
setting
Φ
(
y
,
z
)
=
∥
y
−
z
∥
2
.
(29)
7
Evidently,
Φ
is convex, non-negative and
Φ
(
y
,
z
)
=
0 if
y
=
z
, as required. In addition,
Φ
satisfies
the required growth condition i
ff
D
and
E
are transverse. In addition,
φ
D
(
z
)
=
inf
y
∈
D
∥
y
−
z
∥
2
=
dist
2
(
z
,
D
)
,
(30)
where dist(
z
,
D
) denotes the distance between
z
and the set
D
. The resulting long game is,
therefore
z
∗
∈
argmin
{
dist
2
(
z
,
D
) :
z
∈
E
}
,
(31)
i. e., finding the admissible state that is closest to the material set. If, for instance, the distance is
computed from the norm (15) and the material is elastic with
D
=
{
y
=
(
ε,σ
) :
σ
=
C
ε
}
,
(32)
then a straightforward calculation gives [4]
φ
D
(
z
)
=
1
2
m
X
e
=
1
w
e
C
−
1
e
(
σ
e
−
C
e
ε
e
)
·
(
σ
e
−
C
e
ε
e
)
≡
1
2
∥
σ
−
C
ε
∥
2
.
(33)
Evidently,
H
vanishes on
D
and grows away from it, as required. Thus,
H
(
z
) measures the
distance from
z
to the material data set
D
and the long-game endeavors to minimize that distance
for all admissible states
z
∈
E
.
4.4. Convex Data–Driven games
The preceding example is a particular case of a more general class of convex games. Suppose
that the material behavior is characterized by a convex strain energy function
W
(
ε
) with well
defined dual stress energy function
W
∗
(
σ
). Then, the material set is the graph
D
=
{
y
=
(
ε,σ
)
∈
Z
:
σ
=
DW
(
ε
)
}
=
{
y
=
(
ε,σ
)
∈
Z
:
ε
=
DW
∗
(
σ
)
}
.
(34)
By convexity and the Fenchel-Taylor theorem [13], we have
f
(
ε,σ
)
≡
W
(
ε
)
+
W
∗
(
σ
)
−
σ
·
ε
≥
0
,
(35)
and
f
(
ε,σ
)
=
0 i
ff
(
ε,σ
)
∈
D
.
(36)
Hence,
f
(
ε,σ
) quantifies the deviation of (
ε,σ
) from
D
. Consider the discrepancy function
Φ
(
y
,
z
)
=
f
(
ε
−
α,σ
−
β
)
=
W
(
ε
−
α
)
+
W
∗
(
σ
−
β
)
−
(
σ
−
β
)
·
(
ε
−
α
)
,
(37)
with
y
=
(
α,β
) and
z
=
(
ε,σ
). An appeal to duality and the Fenchel-Taylor theorem gives, with
w
=
(
ξ,η
),
φ
D
(
ε,σ
)
=
inf
y
∈
D
Φ
(
y
,
z
)
=
inf
y
∈
D
f
(
ε
−
α,σ
−
β
)
=
lim
λ
→
+
∞
inf
y
∈
Z
f
(
ε
−
α,σ
−
β
)
+
λφ
D
(
α,β
)
=
lim
λ
→
+
∞
inf
y
∈
Z
sup
w
∈
Z
η
·
(
ε
−
α
)
−
W
∗
(
η
)
+
ξ
·
(
σ
−
β
)
−
W
(
ξ
)
−
(
σ
−
β
)
·
(
ε
−
α
)
+
λφ
D
(
α,β
)
=
8
lim
λ
→
+
∞
sup
w
∈
Z
inf
y
∈
Z
n
η
·
ε
−
W
∗
(
η
)
+
σ
·
ξ
−
W
(
ξ
)
−
σ
·
ε
+
λ
λ
−
1
(
σ
−
η
)
·
α
+
λ
−
1
(
ε
−
ξ
)
·
β
+
W
(
α
)
+
W
∗
(
β
)
−
β
·
α
o
=
(38)
lim
λ
→
+
∞
sup
w
∈
Z
n
η
·
ε
−
W
∗
(
η
)
+
σ
·
ξ
−
W
(
ξ
)
−
σ
·
ε
+
λφ
∗
D
λ
−
1
(
η
−
σ
)
,λ
−
1
(
ξ
−
ε
)
o
=
sup
w
∈
Z
n
η
·
ε
−
W
∗
(
η
)
+
σ
·
ξ
−
W
(
ξ
)
−
σ
·
ε
+
D
η
φ
∗
D
0
,
0)(
η
−
σ
)
+
D
ξ
φ
∗
D
(0
,
0)(
ξ
−
ε
)
o
=
sup
w
∈
Z
n
η
·
ε
+
σ
·
ξ
−
W
∗
(
η
)
−
W
(
ξ
)
−
σ
·
ε
o
=
W
(
ε
)
+
W
∗
(
σ
)
−
σ
·
ε
=
φ
D
(
ε,σ
)
.
Thus, it follows that the corresponding long game (28) endeavors to minimize the deviation from
D
, as measured by
φ
D
, among all admissible states
z
∈
E
.
5. Non-cooperative Data–Driven games
We now change moods and propose a new class of non-cooperative Data–Driven games as
an alternative to the cooperative Data–Driven games described in the foregoing. We specifically
focus on the long game (28). We recall that this long game results from explicitly or implicitly
minimizing out the material state
y
over
D
, so that the resulting cost function
H
(
z
) represents the
deviation of an admissible state
z
∈
E
from the material set
D
. The long game that remains is
then to minimize such deviation among all admissible states.
5.1. Data–Driven game with adversarial stresses and strains
In the cooperative mood adopted in Section 4, the minimization of the cost
H
(
z
), with
z
=
(
ε,σ
), is pursued jointly in
ε
and
σ
. The compatibility constraint can be enforced constructively
by setting
ε
=
Bu
, with
u
a displacement field. In addition, the equilibrium constraint on
σ
can
be enforced by means of Lagrange multipliers. This implementation of the game results in two
standard linear problems for the displacements
u
and the Lagrange multipliers
v
, regardless of
the nature of the material data [4].
As an alternative, here we explore a reformulation of Data-Driven game in which the players
(
ε,σ
) are adversarial: The objective of the stress player
σ
is to minimize the discrepancy to the
data set
D
for fixed
ε
; the objective of the strain player
ε
is to ensure the admissibility of the state
z
for fixed
σ
. The corresponding non-cooperative game is
φ
D
(
ε
∗
,σ
∗
)
≤
φ
D
(
ε
∗
,σ
)
,
for all
σ
∈
R
N
,
(39a)
φ
E
(
ε
∗
,σ
∗
)
≤
φ
E
(
ε,σ
∗
)
,
for all
ε
∈
R
N
,
(39b)
where the function
φ
E
penalizes deviations from
E
. For instance, proceeding as in the cooperative
game (24), we have
φ
E
(
ε,σ
)
=
I
E
(
z
) with
z
=
(
ε,σ
), or explicitly in stress-strain variables,
φ
E
(
ε,σ
)
=
(
0
,
if
ε
=
Bu
,
wB
T
σ
=
f
,
+
∞
,
otherwise
.
(40)
9
The corresponding Nash-equilibrium conditions are
∂φ
D
∂σ
(
ε,σ
)
=
0
,
(41a)
ε
=
Bu
,
wB
T
σ
=
f
,
(41b)
and the optimal strategies (
ε
∗
,σ
∗
) are the solutions of these equations, if any. Indeed, for
φ
E
(
ε,σ
)
as in (40) to be finite we must have
z
∈
E
, whence (41b) follows.
An alternative means of deriving (41b) is by regularization of the indicator function and a
subsequent passage to the limit. Thus, we may define the sequence of regularized functionals
φ
E
,δ
(
ε,σ
)
=
(
δ
2
(
B
T
w
C
B
)
u
·
u
+
w
σ
·
Bu
−
f
·
u
,
if
ε
=
Bu
,
+
∞
,
otherwise
,
(42)
with
δ
↓
0 and the sti
ff
ness matrix (
B
T
w
C
B
) introduced for dimensional consistency. We note
that compatibility can be enforced constructively, leading to the reduced functional
φ
E
,δ
(
u
,σ
)
=
δ
2
(
B
T
w
C
B
)
u
·
u
+
w
σ
·
Bu
−
f
·
u
.
(43)
The corresponding Nash equilibrium condition is
∂φ
E
,δ
∂
u
(
u
,σ
)
=
δ
(
B
T
w
C
B
)
u
+
wB
T
σ
−
f
=
0
,
(44)
and (41b) is recovered by passing to the limit
δ
→
0.
Example 5.1
(Distance deviation function)
.
With the deviation function (33), the Nash equilib-
rium condition (41a) specializes to
2
w
e
C
−
1
e
(
σ
e
−
C
e
ε
e
)
=
0
,
e
=
1
,...,
m
,
(45)
and the e
ff
ective constitutive relation is given locally by Hooke’s law
ˆ
σ
e
(
ε
e
)
=
C
e
ε
e
,
e
=
1
,...,
m
,
(46)
as expected. In addition, we note that the Nash equilibrium conditions (41) can be expressed
jointly as
0
wB
T
−
wB
w
C
−
1
! (
u
σ
)
=
(
f
0
)
,
(47)
which is of the form (8). These equations follow jointly as Euler-Lagrange equations of the
Lagrangian
L
(
u
,σ
)
=
w
σ
·
Bu
−
f
·
u
−
1
2
w
C
−
1
σ
·
σ,
(48)
corresponding to the inf–sup problem
inf
u
sup
σ
L
(
u
,σ
)
=
inf
u
1
2
w
C
Bu
·
Bu
−
f
·
u
,
(49)
which identifies the e
ff
ective problem as a zero–sum game.
□
10
Example 5.2
(Convex deviation function)
.
Suppose now that the deviation function is given by
(35) as in Example 4.4, the Nash equilibrium condition (41a) specializes to
2
w
e
(
DW
∗
e
(
σ
e
)
−
ε
e
)
=
0
,
e
=
1
,...,
m
,
(50)
and the e
ff
ective constitutive relation is given locally by
ˆ
σ
e
(
ε
e
)
=
DW
e
(
ε
e
)
,
e
=
1
,...,
m
,
(51)
as expected. As in the preceding problem, the Nash equilibrium conditions (41) follow jointly
from the Lagrangian
L
(
u
,σ
)
=
m
X
e
=
1
w
e
σ
e
·
B
e
u
−
W
∗
e
(
σ
e
)
−
f
·
u
(52)
corresponding to the inf–sup problem
inf
u
sup
σ
L
(
u
,σ
)
=
inf
u
m
X
e
=
1
w
e
W
e
(
B
e
u
)
−
f
·
u
,
(53)
which again identifies the e
ff
ective problem as a zero–sum game.
□
5.2. Existence
Solving (41a) for the stresses, we obtain a relation of the form
σ
=
ˆ
σ
(
ε
)
,
(54)
which defines an
e
ff
ective
, or
learned
, constitutive law. Conditions on
φ
D
(
ε,σ
) for the function
ˆ
σ
(
ε
) to exist and be continuous are given by the implicit function theorem [14]. Eq. (41b) then
requires
wB
T
ˆ
σ
(
Bu
)
=
f
,
(55)
which defines an e
ff
ective displacement problem. A first fundamental question is whether the
problem (55) is well-posed in the sense of existence and uniqueness of solutions.
We note that problem (55) is not required to have a variational structure, e. g., to derive from
a minimum energy principle. Nevertheless, existence of solutions follows if the local material
laws ˆ
σ
e
(
ε
e
) are
coercive
, in the sense of material stability, and the structure is likewise stable,
in the sense of not allowing zero-strain mechanisms. Uniqueness of the solution follows if, in
addition, the material law is
strictly monotone
.
A general framework for existence and convergence of approximations is set forth by the
following propositions, which are adapted from [12], §9.1, to the present setting.
Lemma 5.3
(Zeros of a vector field)
.
Suppose that a continuous function v
:
R
n
→
R
n
satisfies
v
(
u
)
·
u
≥
0
,
if
∥
u
∥
=
r
,
(56)
for some r
>
0
. Then, there exists a point u
∗
∈
R
n
,
∥
u
∗
∥≤
r, such that v
(
u
∗
)
=
0
.
The proof of the lemma is based on Brouwer’s fixed point theorem and can be found in [12],
§9.1.
11
Proposition 5.4
(Existence of solutions)
.
Let w
e
>
0
and
ˆ
σ
e
:
R
d
→
R
d
continuous functions,
e
=
1
,...,
m. Let f
∈
R
n
and B
:
R
n
→
R
md
. Suppose that:
i) Material stability. There are a
>
0
, b
≥
0
such that, for all
ε
e
∈
R
d
,
ˆ
σ
e
(
ε
e
)
·
ε
e
≥
a
∥
ε
e
∥
2
e
−
b
.
(57)
ii) Structural stability. There is c
>
0
such that, for all u
∈
R
n
,
∥
u
∥
2
≤
c
∥
Bu
∥
2
.
(58)
Then, problem (55) has a solution u
∗
∈
R
n
that satisfies the bound
∥
u
∗
∥
2
≤
a
c
−
ε
−
1
1
4
ε
∥
f
∥
2
+
bV
,
(59)
for all
ε <
a
/
c and V
=
P
m
e
=
1
w
e
. Suppose, in addition:
iii) Strict monotonicity. There is
θ >
0
such that, for all
ε
′
e
,
ε
′′
e
∈
R
d
,
ˆ
σ
e
(
ε
′
e
)
−
ˆ
σ
e
(
ε
′′
e
)
·
(
ε
′
e
−
ε
′′
e
)
≥
θ
∥
ε
′
e
−
ε
′′
e
∥
2
e
.
(60)
Then, the solution is unique.
We note that condition (i) stipulates coercivity of the material laws, whereas condition (ii)
requires the absence of mechanisms, i. e., displacements that occur at zero strain. The proof
of the proposition is illustrative of the roles played by material and structural stability and is
therefore included next in full.
Proof.
Let (
e
1
,...,
e
n
) be the standard orthonormal basis of
R
n
. Define the continuous function
v
:
R
n
→
R
n
by setting
v
i
(
u
)
=
w
ˆ
σ
(
Bu
)
·
Be
i
−
f
·
e
i
,
(61)
for every
u
∈
R
n
. From (i), we find
v
(
u
)
·
u
=
w
ˆ
σ
(
Bu
)
·
Bu
−
f
·
u
≥
a
∥
Bu
∥
2
−
bV
−
f
·
u
,
(62)
and by (ii),
v
(
u
)
·
u
+
bV
+
f
·
u
≥
a
c
∥
u
∥
2
.
(63)
For every
ε >
0, we have the estimate
|
f
·
u
|≤
ε
∥
u
∥
2
+
1
4
ε
∥
f
∥
2
,
(64)
which, inserted into (63), gives
v
(
u
)
·
u
+
bV
+
ε
∥
u
∥
2
+
1
4
ε
∥
f
∥
2
≥
a
c
∥
u
∥
2
.
(65)
Rearranging terms,
v
(
u
)
·
u
≥
a
c
−
ε
∥
u
∥
2
−
bV
−
1
4
ε
∥
f
∥
2
,
(66)
12
Hence,
v
(
u
)
·
u
≥
0 if
∥
u
∥
=
r
, provided that we select
ε
small enough and
r
>
0 su
ffi
ciently large.
By Lemma 5.3, there is
u
∗
∈
R
n
such that
v
(
u
∗
)
=
0, i. e.,
u
∗
is a solution of (55). In addition,
from (66) the solution satisfies the bound
a
c
−
ε
∥
u
∗
∥
2
≤
1
4
ε
∥
f
∥
2
+
bV
,
(67)
which implies (59), as advertised. Assume, in addition, that (iii) holds. Suppose that there are
two solutions
u
′
,
u
′′
∈
R
n
. Then, for all
v
∈
R
n
,
w
ˆ
σ
(
Bu
′
)
·
Bv
=
w
ˆ
σ
(
Bu
′′
)
·
Bv
=
f
·
v
,
(68)
whence
w
ˆ
σ
(
Bu
′
)
−
ˆ
σ
(
Bu
′′
)
·
Bv
=
0
.
(69)
Setting
v
=
u
′
−
u
′′
and using (ii) and (iii)
0
=
w
ˆ
σ
(
Bu
′
)
−
ˆ
σ
(
Bu
′′
)
·
B
(
u
′
−
u
′′
)
≥
θ
∥
Bu
′
−
Bu
′′
∥
2
≥
θ
c
2
∥
u
′
−
u
′′
∥
2
,
(70)
which requires
∥
u
′
−
u
′′
∥
=
0 and
u
′
=
u
′′
, as advertised.
5.3. Approximation
Suppose now that the local material laws ˆ
σ
e
(
ε
e
) in problem (55) are not known exactly but
only approximately through a convergent sequence of approximate local material laws ˆ
σ
e
,
h
(
ε
e
).
The approximate local material laws set forth a sequence of approximating problems
wB
T
ˆ
σ
h
(
Bu
)
=
f
,
(71)
where we write ˆ
σ
h
(
ε
)
=
( ˆ
σ
e
,
h
(
ε
e
))
m
e
=
1
. We wish to ascertain conditions under which the solutions
u
∗
h
of problems (71) converge to the solution
u
of the limiting problem (55).
The following proposition sets forth conditions under which approximation of the local ma-
terial laws results in convergent approximate solutions.
Proposition 5.5
(Approximation)
.
Let w
e
>
0
and
ˆ
σ
e
,
h
:
R
d
→
R
d
continuous functions, e
=
1
,...,
m, h
∈
N
. Let f
∈
R
n
and B
:
R
n
→
R
md
. Suppose that:
i) The sequence
( ˆ
σ
e
,
h
)
is uniformly stable in the sense of Prop 5.4(i), i. e., there are a
>
0
,
b
≥
0
independent of h such that, for all
ε
e
∈
R
d
,
ˆ
σ
e
,
h
(
ε
e
)
·
ε
e
≥
a
∥
ε
e
∥
2
e
−
b
.
(72)
ii) B is stable in the sense of Prop 5.4(ii).
iii) The local material laws
ˆ
σ
e
,
h
are continuous and converge to local material laws
ˆ
σ
e
uni-
formly on compact sets.
Then, the solutions u
∗
h
of problem (71) converge, up to subsequences, to a solution of the limiting
problem (55).
13
We recall that, by the Arzelà-Ascoli theorem, (iii) is ensured if the sequences ( ˆ
σ
e
,
h
) are
uniformly bounded and equi-continuous, in which case the limiting laws ˆ
σ
e
are also continuous.
If the approximate local material laws ( ˆ
σ
e
,
h
) are di
ff
erentiable, then equi-continuity is ensured if
the derivatives of ( ˆ
σ
e
,
h
) are uniformly bounded.
Proof.
By Prop. 5.4, the approximating problems (71) have solutions
u
∗
h
, not necessarily unique,
satisfying
v
h
(
u
∗
h
)
=
0, with
v
h
defined from ˆ
σ
e
,
h
as in (61), and uniformly bounded as in (59). By
this latter property, there is a subsequence (not renamed) that converges to some
u
∗
∈
R
n
. By
uniform convergence,
v
(
u
∗
)
=
lim
h
→∞
v
h
(
u
∗
h
)
=
0
,
(73)
with
v
defined from ˆ
σ
e
as in (61), which shows that
u
∗
is in fact a solution of the limiting problem
(55).
The main conclusion a
ff
orded by the preceding propositions is that
existence and convergence
can be elucidated, for any stable structure and applied loading, locally at the material point level
,
simply by examining the properties and convergence of the local material laws. We also remark
that, by applying Cauchy’s test, convergence of the approximating sequences of local material
laws can be established without knowing the limiting local material laws explicitly. Such inferred
convergence then guarantees that the approximating solutions in turn converge to the solutions
of the (unknown) limiting problem.
Examples of approximation and convergence are presented in Section 6.3.3 for the important
case of approximation by empirical point data.
6. Approximation by empirical point data
An important property of the set–oriented Data–Driven games defined in the foregoing is
that they remain applicable when the exact material data set, e. g., the graph of the underly-
ing constitutive relation, is replaced by an approximation
D
thereof in the form of a point set,
e. g., measured empirically or computed from micromechanics. A central question is then to
ascertain conditions ensuring the convergence of the Data-Driven solutions to the solution of the
(unknown) limiting material law. In this section, we specifically consider two di
ff
erent scenar-
ios: i)
Uniformly convergent data
, in which the sampling error decreases as data is added to the
material-data set in a uniform manner controlled by strict upper bounds, and ii)
noisy data with
outliers
, in which the data concentrates around the limiting material law in a weak or average
sense that allows for the presence of outliers.
6.1. Distance-based Data-Driven game
Suppose that the material behavior is local, i. e.,
D
=
D
1
×···×
D
m
, with local material data
sets
D
e
=
{
y
e
,
i
∈
Z
e
:
i
=
1
,...,
N
e
}
. Then,
φ
D
(
z
)
=
m
X
e
=
1
φ
D
,
e
(
z
e
)
(74)
with
φ
D
,
e
(
z
e
)
=
inf
{
Φ
e
(
y
e
,
z
e
) :
y
e
∈
D
e
}
,
(75)
14
and the Nash equilibrium condition for the stress player reduces to
ˆ
σ
e
(
ε
e
)
=
η
e
,
with (
ξ
e
,η
e
)
∈
argmin
{
Φ
e
(
y
e
,
z
e
) :
y
e
∈
D
e
}
.
(76)
In the special case of a distance-based discrepancy function, (76) reduces to
ˆ
σ
e
(
ε
e
)
=
η
e
,
with (
ξ
e
,η
e
)
∈
D
e
and
ξ
e
closest to
ε
e
.
(77)
Thus, the e
ff
ective constitutive law ˆ
σ
e
(
ε
e
)
learned
by the Data–Driven game consists of looking
in the data set
D
e
for the point (
ξ
e
,η
e
) such that
ξ
e
is closest to
ε
e
and taking the corresponding
stress
η
e
as the value of ˆ
σ
e
(
ε
e
).
6.2. Max-ent regularization, clustering and smoothing
Remarkably, the game just defined for point-data results in a ’learned’ stress-strain relation
(76), or (77), that is discontinuous and multiply-valued, e. g., if the query strain is equidistant
from more than one strain in the data set. This lack of continuity places the e
ff
ective stress-strain
relation (76) out of scope of Prop. 5.5, which requires continuity, and necessitates the use of
specialized solvers such as dynamic relaxation, cf. Sections 7.2.1 and 7.2.2, or smoothing over a
carefully chosen strain range, cf. Section A.1. An additional di
ffi
culty arises from outliers in the
data, i. e., points that deviate markedly from the general trend in the data. Indeed, uncontrolled
outliers can bias the Data-Driven solution and forestall convergence.
These di
ffi
culties can be overcome by means of a
max-ent
regularization of the problem [5].
As before, we suppose that the data set is a collection of local point–data sets,
D
=
D
1
×···×
D
m
,
with
D
e
=
{
y
e
,
i
∈
Z
e
:
i
=
1
,...,
N
e
}
. We begin by noting that the local deviation function (75)
can equivalently be expressed as
φ
D
,
e
(
z
e
)
=
inf
n
N
e
X
i
=
1
p
e
,
i
Φ
e
(
y
e
,
i
,
z
e
) :
N
e
X
i
=
1
p
e
,
i
=
1
,
p
e
,
i
≥
0
,
i
=
1
,...,
N
e
o
.
(78)
The weights
{
p
e
,
i
}
N
e
i
=
1
quantify how well a point
z
e
of local phase space
Z
e
is represented by a point
y
e
,
i
in the corresponding local material data set
D
e
, or, conversely, the relevance of a point
y
e
,
i
in the local material data set to a given point
z
e
in local phase space. Evidently, the minimizing
weights concentrate on the material points that minimize their deviation from
z
e
, whereupon (75)
is recovered.
The measure-theoretical representation (78) can now be conveniently regularized through the
addition of a small entropy term, with the result
φ
D
,
e
(
z
e
;
β
e
)
=
inf
n
N
e
X
i
=
1
p
e
,
i
Φ
e
(
y
e
,
i
,
z
e
)
+
β
−
1
e
log
p
e
,
i
:
N
e
X
i
=
1
p
e
,
i
=
1
o
,
(79)
with
β
e
→
+
∞
. The regularization term can indeed be interpreted as the negative of Shannon’s
information-theoretical entropy [15], hence the term
maximum-entropy
, or ’max-ent’, regular-
ization. The entropy term ensures that the distribution of weights is as unbiased as possible. The
regularized functional thus has the structure of a free energy, with the first term representing the
15
internal energy of the system. In this interpretation, the problem (79) expresses the principle of
minimum free energy,
β
e
plays the role of a local
reciprocal temperature
and the limit
β
e
→
+
∞
is the corresponding zero-temperature, or
athermal
, limit.
We additionally recall that the entropy term measures the
Kullback-Leibler discrepancy
[16,
17] between the empirical and weighted measures
μ
D
,
e
=
N
e
X
i
=
1
δ
y
e
,
i
and
ν
D
,
e
=
N
e
X
i
=
1
p
e
,
i
δ
y
e
,
i
.
(80)
By this interpretation, the entropy term in (79) aims to minimize the discrepancy between the
two measures (80), i. e., to render the weights
p
e
,
i
as uniform as possible. The local deviation
function
Φ
e
(
y
e
,
i
,
z
e
) in turn ensures that points
y
e
,
i
in the local data set
D
e
that are distant from
the query point
z
e
are accorded less weight than nearby points. Evidently, the optimal weights
now follow as the result of a competition between the deviation function and entropy, with
β
−
1
e
playing the role of a local Pareto weight.
Conveniently, the minimizing weights and the minimum of the functional (79) follow explic-
itly as
p
∗
e
,
i
(
z
e
;
β
e
)
=
1
Z
e
(
z
e
;
β
e
)
e
−
β
e
Φ
e
(
y
e
,
i
,
z
e
)
,
(81a)
Z
e
(
z
e
;
β
e
)
=
N
e
X
i
=
1
e
−
β
e
Φ
e
(
y
e
,
i
,
z
e
)
,
(81b)
φ
∗
D
,
e
(
z
e
;
β
e
)
=
−
β
−
1
e
log
Z
e
(
z
e
;
β
e
)
,
(81c)
which are a discrete Gibbs distribution, partition function and equilibrium free energy, respec-
tively. In addition, the Nash equilibrium condition (41a) corresponds to minimizing the local
free energy
to with respect to the stress
σ
e
at fixed strain
ε
e
. Appealing to the optimality of the
weights, we compute
∂φ
∗
D
,
e
∂σ
e
(
z
e
;
β
e
)
=
N
e
X
i
=
1
p
∗
e
,
i
(
z
e
;
β
e
)
∂
Φ
D
,
e
∂σ
e
(
y
e
,
i
,
z
e
)
=
0
,
(82)
where we write
z
e
=
(
ε
e
,σ
e
).
It follows from (81a) that points in the data set
D
e
that deviate from
z
e
much more than
β
−
1
/
2
e
have negligible weight, whereas, conversely, the local behavior at
z
e
is dominated by the local
cluster of data points in the
β
−
1
/
2
e
–neighborhood of
z
e
, hence the term
clustering
. In particular,
outliers, or points outside that neighborhood, have negligible weight.
Example 6.1
(Minimum–distance deviation)
.
For a local discrepancy function of the form
Φ
e
(
y
e
,
z
e
)
=
∥
y
e
−
z
e
∥
2
e
,
(83)
corresponding to a global discrepancy function of the form (29), the Nash equilibrium condition
(82) reduces to
σ
e
=
N
e
X
i
=
1
p
∗
e
,
i
(
z
e
;
β
e
)
σ
e
,
i
,
(84)
16