# Augmented Computational Design

Pirouz Nourian  Shervin Azadi  Eindhoven University of Technology, 5612 AZ,
Het Kranenveld 8, Eindhoven, NL  Roy Uijtendaal  Nieman Raadgevende
Ingenieurs, 3542 AB, Atoomweg 400, Utrecht, NL  Nan Bai  Delft University of
Technology, 2628 BL, Julianalaan 134, Delft, NL

###### Abstract

This chapter presents methodological reflections on the necessity and utility
of artificial intelligence in generative design. Specifically, the chapter
discusses how generative design processes can be augmented by AI to deliver in
terms of a few outcomes of interest or performance indicators while dealing
with hundreds or thousands of small decisions. The core of the performance-
based generative design paradigm is about making statistical or simulation-
driven associations between these choices and consequences for mapping and
navigating such a complex decision space. This chapter will discuss promising
directions in Artificial Intelligence for augmenting decision-making processes
in architectural design for mapping and navigating complex design spaces.

Keywords: Artificial Intelligence, Design Space Exploration, Generative
Design, Augmented Intelligence, Probabilistic Design 111This is the author
version of the book chapter “Augmented Computational Design: Methodical
Application of Artificial Intelligence in Generative Design.” In Artificial
Intelligence in Performance-Driven Design: Theories, Methods, and Tools
Towards Sustainability, edited by Narjes Abbasabadi and Mehdi Ashayeri. Wiley,
2023

##  1 Introduction

The core of the performance-driven computational design is to trace the
sensitivity of variations of some performance indicators to the differences
between design alternatives. Therefore any argument about the utility of AI
for performance-based design must necessarily discuss the representation of
such differences, as explicitly as possible. The existing data models and data
representations in the field of Architecture, Engineering, and Construction
(AEC), such as CAD and BIM are heavily focused on geometrically representing
building elements and facilitating the process of construction management.
Unfortunately, the field of AEC does not currently have a structured discourse
based on an explicit representation of decision variables and outcomes of
interest. Specifically, the notion of design representation and the idea of
data modelling for representing “what needs to be attained from buildings” is
rather absent in the literature.

This treatise proposes to systematically view the differences between design
alternatives in terms of decision variables, be they spatial or non-spatial.
Based on such an explicit formulation of decision variables, we set forth a
framework for building and utilizing AI in [architectural] generative design
processes for associating decision variables and outcomes of interest as
performance indicators in a reciprocal relationship. This reciprocity is
explained in terms of the duality between two quintessential problems to be
addressed in generative design, i.e. the problem of evaluation of design
alternatives (mapping), and the problem of derivation of design alternatives
(navigation).

Starting from an explicit representation of a design space as an ordered pair
of two vectors respectively denoting decision variables and performance
indicators, we put forth a mathematical framework for structuring data-driven
approaches to generative design in the field of AEC. This framework highlights
two major types of applications for AI in performance-driven design and their
fusion: those capable of augmenting design evaluation procedures and those
capable of augmenting design derivation procedures. Moreover, we introduce the
reciprocity of “flows” and “manifolds” as an intermediary notion for going
beyond the so-called form-function dichotomy. Discussing these notions
necessitates the introduction of a mathematical foundation for the framework
rooted in multi-variate calculus.

The main advantage of this explicit formulation is to enhance the
explainability of AI when utilized in generative design by introducing
meaningful and interpretable latent spaces based on the reciprocal
relationship between manifolds and flows. The balance of
predictive/deterministic power and interpretability/explainability is
discussed in the concrete context of an illustrative example.

A chain of key concepts will be introduced in this chapter, starting from the
notion of decision-making in design, the nature of design variables, the
specifics of spatial decision variables, the notion of design space, and the
two dual actions in the exploration of design spaces: mapping and navigating.

Whilst the introduced framework is quite general, a particular class of
Probabilistic Graphical Models (PGM), Bayesian Belief Networks (BBN), is
introduced to provide a concrete illustrative example of the utility of AI in
AEC. For a deeper insight into this particular approach to data-driven design,
the readers are referred to the two classical books about PGM: [34] [34] &
[24] [24]. The illustrative example is a BBN trained for making a data-driven
replica of the building energy model used by the Dutch government in order to
obtain a rough meta-model to be used in mass-scale policy analysis, e.g. for
advising the government on the relative utility of energy transition subsidies
and planning measures. This example is chosen not because the BBNs are the
most advanced models or the most accurate models for approximating such large
functions. The choice is rather pragmatic in that this model has proven to be
promising from the stance of predictive power while retaining a basic level of
theoretical interpretability and intuitive appeal.

The chapter is structured as follows: we first present a historical context to
establish the necessity of such a data-driven generative design framework;
continue with conceptualizing and mathematically formulating the structure of
the framework, dubbed as Augmented Computational Design (ACD); present an
illustrative example demonstrating the utility of the framework; and conclude
with a discussion on its outlook, open questions, and avenues for further
research.

##  2 Background

Here we revisit the utility of AI for data-driven generative design by
highlighting some key gaps of knowledge in the field of AEC and briefly
mentioning overarching frameworks in computational design and AI that can be
used to address these gaps.

###  2.1 Relevance of AI in AEC

The earliest attempts for enhancing accountability and predictive power in
computational design can be traced back to the notions of Scientific
Architecture [11] and the Sciences of the Artificial [40]. Both of these
seminal books explicitly discuss the necessity of forming some kind of a
specific spatial and configurative form of design knowledge, the core of which
boils down to being able to explicitly represent the main subject matter of
spatial design as “spatial configurations”. One of the first phenomenological
and systemic descriptions of design processes explicitly referring to the
notion of performance is the “Function, Behaviour, Structure” framework of
[13] [13], in which the overused notions of form and function are elaborated
in terms of expected and required behaviour/functionality from a system
(dubbed as the function), its design as a form or configuration (dubbed as
structure), and its performance (dubbed as behaviour). The idea of design as a
process of generating the representation of a spatial structure is explicitly
discussed in this framework and the difference between desired behaviour and
the actual behaviour of the structure is discussed as the performance drive
for the process. What can be observed in this phenomenological framework,
predating most recent advancements in computational design, is the fundamental
belief about the innate necessity of creativity in terms of the cognitive
capability of designers for proposing structures capable of working as
desired, based on some kind of tacit knowledge. Congruently, an
anthropological description of design processes refers to the old duality
between form (structure) and function (purpose) of designed artefacts, and the
fact that [in the absence of explicit knowledge and representation schemes],
as [25] [25] has put it, designers are traditionally trained to produce
solutions (draw them) through a “logical leap” often without even
understanding or paying any attention to the design requirements or supposed
levels of quality attainment. Suppose we wanted to evaluate (compare) two
different alternative designs for a hospital, [20], or a home, how do we want
to represent the designs digitally for a computer to evaluate them? Let us
discuss an analogical example: if we wanted to compare two pieces of music in
terms of their beauty, it would be very straightforward to digitize their
notations and feed them to a machine, because the musical notation is already
discretized (digitized), regardless if it is written on paper or etched on the
cylinder of an old-fashioned winding music-box [47]. However, doing the same,
that is comparing two buildings, would be a much more difficult challenge
especially because there is currently no [discrete/textual] notation for
spatial design that can capture the features of spatial configurations.

Instead of the extensive emphasis on the product of architecture as the shapes
of buildings, we turn our attention to the processes of design and put a lens
of “design as [discrete] decision-making” on the debate to avoid commonplace
reduction of design to the production of design drawings. This view forms the
basis of the generative design paradigm as extensively articulated by [31]
[31] & [45] [45]. Similarly, the challenges, opportunities, and promising ways
of utilization of AI (particularly deep-learning and generative models) for
goal-oriented design explorations have been discussed extensively in [37] &
[36].

###  2.2 Historical Context

In this section, we first give a very brief history of the most important and
relevant developments in AI; then lay the foundation of a formulation of
architectural design as a matter of decision-making; discuss the mathematical
implications of this paradigmatic frame for generative design; elaborate on
the notion of decision-making and the duality of derivation and evaluation
problems; and discuss two statistical approaches to design, a possibilistic
approach utilizing Fuzzy Logic or Markovian Design Machines and a
probabilistic approach utilizing Bayesian Belief Networks or Diffusion Models.

We are currently witnessing an era of exponential success in the field of
artificial intelligence that has been evolving for more than 50 years (See
Figure 1). Meanwhile, it is common knowledge that progress is slow in terms of
innovation and scientific knowledge development in the field of AEC.

Figure 1: Highlights in the history of Artificial Intelligence

As has been extensively argued before by [39] [39] and [3] [3], once an
unambiguous language is adopted for discussing the classification of problems,
we can see that many of the problems of the AEC can be dealt with very
adequately (and possibly painstakingly) through conventional mathematics,
physics, and computer science. In other words, the utility or the necessity of
employing AI for dealing with problems that can be dealt with through
conventional mathematical or computational procedures is not only pointless
from a resource-efficiency stance but also questionable from the point of view
of interpretability, transparency, and explainability. To assess the potential
applications of AI in AEC w.r.t. these questions, we highlight the history of
AI (see Figure 1) and refocus on the scope of AI (see Figure 2), at least as
could possibly pertain to AEC.

Figure 2: An Euler diagram of the scope of Artificial Intelligence

Once a problem is formulated adequately, two major determinants can be
considered as to whether it would be sensible to apply AI or not: whether the
data schemata of the problem are structured (vectorized) or unstructured
(textual/visual), and whether the underlying associations between the inputs
and outputs can be modelled through first principles (governing laws of
physics, typically stated in differential equations), stochastic processes, or
Agent-Based Models. If the problem data are unstructured or the conventional
modelling approaches do not have the capability of capturing the complex
associations between the inputs and the outputs of interest, then, especially
when interpretability can be sacrificed over the necessity of predictability,
utilizing AI is quite sensible. The example that we shall discuss in this
chapter may seem somewhat questionable according to these points but on the
other hand, it is too overwhelmingly large and complex that no conventional
approach can deal with it at the aimed level of abstraction. In this case, the
ambition of the project is on such a high level of abstraction for policy
analysis that the inaccuracies and ambiguities of the purely data-driven
approach can be justified because of the insights that can be gained from the
meta-statistical model.

###  2.3 Design as Decision-Making

The commonly overstated notions of difficulty or ill-defined nature of design
problems, see e.g. [39] [39], can be attributed to the fact that most design
tasks are expected to produce a very concrete geometric description of an
object to be built (the form), given only a very abstract description of what
the object is supposed to be used for, how it should work, and what would be
desirable for it to achieve, all of which are often described quite vaguely
(the function), q.v. [26] [26].

Hillier was one of the few shrewd theorists who understood that, at least
after the separation of structural design from architecture in the 19th
century, q.v. [14] [14], what distinguishes building buildings from
architecture is the art and science of configuring spaces, as stated in “Space
is the Machine”, [18]. Once one realizes that the so-called architectural form
is not only a single shape of an iconic object but also a set that includes
the shapes of spaces and eventually the constituent segments of a building
then we can distinguish the superior importance of spatial configurations. As
obvious as it may sound, it seems to be necessary to emphasize that
architectural design is not merely about sculpting a shape but configuring
spaces to accommodate some human activities. This involves some puzzling tasks
such as packing, zoning, and routing spaces of various functions, which we
hereinafter refer to as the task of configuring buildings [3]. For the
problems of shape and configuration to be transformed into decision problems,
they need to be discretized rigorously. In short, we can call a massing
problem a shape problem and a zoning problem a configuration problem (see
Figure 5).

The mainstay of the generative design paradigm is a rigorous reformulation of
a design problem as a discrete topological decision problem rather than a
geometrical problem [31]. Therefore discretization is the process of breaking
down the integrated design problem into multiple smaller yet interdependent
decision problems. An example of such discretization can be a voxel grid that
provides a non-biased and homogeneous representation of spatial units, each of
which poses a decision problem of function allocation [32] &[42].

Moreover, to ensure the correspondence of these discrete decisions we need to
include the topological information about their neighbourhood to represent
their spatial inter-dependencies; similar to topology optimization [33]. At
the limit, such discretization can also be used to model a continuum of
solutions and provide a frequency-based or spectral representation system,
similar to [27], for spatial design much like the musical notation that is
based on notes.

Additionally, it is important to note that design decisions have a strong
spatial dimension, however, they can include the social dimension to represent
the preference of stallholders and enable consensus-building [4]. Given a view
of design as a matter of decision-making, we can readily see two important
types of practical questions that will shed light on the relevance of AI for
decision-making:

  1. 1.

how to map/learn the associations of some hundreds or thousands of constituent
choices of a compound design decision (function approximation and
dimensionality reduction for ex-ante assessment of the impact of decisions)?

  2. 2.

how to navigate a gigantic decision space with thousands of choices and their
astronomically large combinations with a few important consequences in the
picture?

The proposed notion of design as decision-making makes a point of departure
for the rest of the chapter in that it highlights two essential problems of
equal importance and significance that can be tackled by AI and their duality:
Firstly, evaluation problems can be portrayed as mapping problems in Machine
Learning and Deep Learning, where the approximation power of Artificial Neural
Networks (ANN) can be exploited in regression and classification settings.
Secondly, derivation problems can be portrayed as navigation problems in
generative models, concerned with navigating from a low-dimensional
representation of performance indicators towards disaggregated design
decisions.

Encapsulating the complex and often non-linear associations of many design
decisions with a few outcomes of interest or performance indicators is here
dubbed as a mapping problem. Inverting a map (as an approximated function,
e.g. in the form of an ANN), can thus be viewed as an enhanced or augmented
form of design, where the designer is navigated towards many small decisions
just by pointing towards certain data points within a low-dimensional
performance space (see Figure 3). It must be apparent that a navigation
problem in this sense is much harder to solve; almost always impossible in the
absolute sense due to an arbitrarily large increase in information content and
thus a combinatorial explosion of possibilities.

Figure 3: The duality of evaluation and derivation problems in generative
design

###  2.4 AI for Generative Design

Given the formulation of main generative design tasks as _mapping_ and
_navigating_ , we focus on a particular set of AI methods that are
distinguished as to their relevance for these tasks in high-dimensional design
decision spaces. More specifically, within the spectrum of generative design
methods [31], we focus on data-driven _mapping_ and _navigating_ strategies.
As shown in figure 4, for brevity, we will only focus on the data-driven
approaches to design on the right-hand side of the spectrum. Notwithstanding
the other possible applications of [different kinds of] AI in this generative
design spectrum, such as Reinforcement Learning in Policy-Driven design
(playing design games), approximation of evaluation functions in topology or
shape optimization, and Expert Systems in grammatical design, our framework
here is focused on the statistical AI paradigm and so we only discuss the
purely data-driven approaches to generative design.

Two subtle issues must be noted here: firstly, instead of discussing the
utility of the wondrous application of generative models for the entertainment
industry, we shall reflect on how the generative processes based on diffusion
or dimensionality reduction can be controlled for attaining high-performance
designs in an explainable manner. Secondly, the model-driven approaches to
performance-based generative design (topology optimization in particular)
based on first principles, are already utilizing something important from the
realm of nature-inspired computing called Hebbian Learning, which is already
in the scope of [statistical] AI. This point, however important, generally
interesting, and relatively unknown, falls out of the scope of this chapter.

Figure 4: The spectrum of generative design methods and their relation to AI
methods

##  3 Framework

The emphasis on the decision-making approach to design entails that design
tasks can be formulated as a set of [typically unstructured] questions about
the form and materialization/construction of an object (a building) to be
answered. In this chapter, we focus on the questions that pertain to form.

We propose a mathematical framework for generative design that relates
multiple strands of work together. We use _design space_ for referring to an
ordered pair of two vector spaces: a decision space containing vectors or data
points representing design configurations in the form of
$\mathbf{x}\in(0,1]^{n}$ and a performance/quality space containing vectors or
data-point representing combinations of outcomes of interest in the form of
$\mathbf{o}\in[0,1]^{q}$.

The mathematical lens that we shall put on the issue is to redefine both of
these notions to provide a much more specific and workable idea for discussing
the utility or futility of applying AI to design problem-solving. It is
hopefully easy for the reader to accept that a regular discretization of a so-
called _design space_ (which is an unfortunately common misnomer, but here
somewhat pragmatically useful) provides a straightforward and simple
discretization of design decisions as vectors in the form of
$\mathbf{x}:=[x_{i}]_{n\times 1}\in(0,1]^{n}$ or $\mathbf{x}\in\\{0,1\\}^{n}$,
where $n$ is the number of discrete cells in the design space in which
virtually any conceivable shape can be constructed at a certain level of
resolution. Without loss of generality, however, the decision variables are
not necessarily spatial and can be assumed to be relativized float variables
in the range of minimum and maximum admissible parameter values of the
functions that together result in the shape and configuration of a building.
Even if a multi-colour (multi-label, multi-functional) space is the subject of
the design problem, then multiple categories/colours of such vectors can be
seen together as a matrix of decision variables, whose rows have to add up to
1 (see Figure 5).

Once this terminology is established, it is easy to observe that, in Machine
Learning (ML) terms, the problem of performance-based design can be seen as
two problems that are dual to one another, a multi-variate regression problem
for figuring out an approximation function that can map a few outputs to many
inputs (here we call this mapping or the evaluation problem), and a pseudo-
inversion problem for finding the combination of inputs that could result in
desired output data points (here we call this navigation or the derivation
problem), see Figure 3.

When approached as a data-driven problem-solving task, both problems are
somewhat hard and impossible to solve in the absolute sense of the word,
unless we think about them as loss minimization or approximation problems. The
navigation/design problem is much harder than the mapping/evaluation problem.
The main idea here is to advocate for training (fitting) meta-models (neural
networks) to sets of sampled pairs of inputs and outputs to first approximate
a complex design space as a map between decision data points and performance
data points, and then find the pseudo-inverse of this map or navigate it in
the reverse direction to be able to find designs (decision data points) that
perform in a desired way. In other words, mathematically, we look at the
performance-based design process as a pairing between a decision space and a
performance space, where a map is conceptualized as a function
$f:(0,1]^{n}\mapsto[0,1]^{q})$ such that $\mathbf{o}=f(\mathbf{x})$. The
pseudo-inverse map is thus dubbed as $f^{-1}:[0,1]^{q}\mapsto(0,1]^{n}$, such
that $\mathbf{x}=f^{-1}(\mathbf{o})$.

For the sake of brevity and also generalizability to non-spatial design
problems, we shall focus on massing problems and leave coloured configuration
problems out of the picture momentarily (see Figure 5 for the distinction).

Furthermore, by considering two abstract and high-level descriptions of a
design task in our proposed regular discretization frameworks, we can
formulate two mathematical tasks:

  * •

Mapping Design Spaces: approximating the function that can model the
associations between the many input design variables and a few outcomes of
interest;

  * •

Navigating Design Spaces: approximating the inverse function that can guide
the generation of valid configurations in the decision space given desired
data points in the performance space.

###  3.1 Design Space Exploration

Here we explain the mathematical meaning of the two dual problems that
together can be called design space exploration tasks: mapping and navigation.

##### Mapping

The problem of mapping associations between a large set of independent input
decision variables and dependent output performance indicators is key to
performance-driven design. Any explainable and accountable design methodology
should have the capacity to guarantee the attainment of some quality or
performance indicators. From a mathematical and statistical point of view, we
might prefer to have an explainable and interpretable model of such relations
that can be fitted into our data or ideally a simulation model to predict
outputs from input data. However, in some cases, especially where a multitude
of very different quality/performance indicators are involved, and when one
does not have an established basis for simulation modelling, statistical
(data-driven) modelling seems to be the only option. And so, when the
complexity of the model passes a certain threshold of non-linearity and a
multitude of input outputs, we might prefer to trade interpretability for
predictive power. That is exactly where ANNs as families of adjustable non-
linear functions stand out as viable function approximators. Training a
network is practically a matter of minimizing a loss/error function by
adjusting the parameters of a family of functions (that is set out by the so-
called architecture/structure of the ANN).

Even though this approximation is inherently non-linear and global, it is
illuminating to think of an alternative [locally] linear approximation based
on the Jacobian Matrix. Suppose that
$\mathbf{o}=f(\mathbf{x}):=[f_{k}(\mathbf{x})]_{q\times
1}=[f_{k}([x_{i}]_{n\times 1})]_{q\times 1}$ is a vector of multiple scalar
functions of vector input variables. Then a basic idea of approximation is to
approximate this function locally around an input data point by its Jacobian.
This matrix operator gives the basis for a hyper-plane equation that provides
the $n$-dimensional Euclidean tangent space of the underlying function, very
much like a multi-variate regression hyperplane, albeit the latter would be
fitted to the entire dataset.

Note that the Machine Learning task here would be a multi-variate regression
task in this case, i.e. predicting the dependent given the independent
variables. To understand the difficulty of the mapping then consider that the
Jacobian matrix $\mathbf{J}:=[J_{k,i}]_{q\times n}=[\frac{\partial
f_{k}}{\partial x_{i}}]_{q\times n}=[\nabla^{T}f_{q}]_{q\times 1}$ would just
provide the best local linear approximation of an otherwise globally non-
linear map from $\mathbb{R}^{n}$ to $\mathbb{R}^{q}$, i.e. $n$ decision
variables to $q$ quality criteria or performance indicators.

The Jacobian approximation is numerically computable provided the underlying
function is smooth and differentiable. For brevity, as commonly done, we have
omitted the fact that the Jacobian can be evaluated at a certain input data
point and that it is expected to be the best linear approximation of the
function in question in the vicinity of that point. If we abbreviate the
notation for the Jacobian as such a functional, then we can denote the
approximate linear function at any given data point as follows (as a first-
order Taylor Series expansion): $\mathbf{o}(\mathbf{x})|_{x\sim
x_{o}}\simeq\mathbf{J}(\mathbf{x_{0}})\left(\mathbf{x}-\mathbf{x_{0}}\right)$,
or simply put, as $\mathbf{o}\simeq\mathbf{J}\mathbf{x}$, if we assume
$\mathbf{x}$ to represent the vector of difference between the input data
point with the centre of the neighbourhood.

The Jacobian approximation is also illuminating for another important reason:
it allows us to approximate the Jacobian in a different sense, i.e. in the
sense of dimensionality/rank reduction using the Singular Value Decomposition
(SVD) to see a clearer picture of the main factors playing the most
significant roles in attaining the outcomes of interest, in other words
identifying the inputs variables to which the outcomes of interest are most
sensitive. Even though we do not explicitly perform this operation in our
demonstrative example using the SVD, it is still illuminating to see what SVD
can do for this insightful approximation and dimensionality reduction for two
reasons:

  * •

The SVD approximation of the Jacobian allows us to make a cognitive and
interpretable map of the most important causes of the effects of interest

  * •

The SVD approximation of the Jacobian allows us to conceptualize a pseudo-
inverse function to navigate the design space from the side of performance
data points.

The SVD (low-rank) approximation of the Jacobian Matrix can be denoted as
below:

$\mathbf{J}:=\mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^{T},$ (1)

where $\mathbf{U}_{q\times q}:=[\mathbf{u}_{k}]_{1\times q}$ and
$\mathbf{V}_{n\times n}:=[\mathbf{v}_{i}]_{1\times n}$ are orthogonal matrices
(i.e. $\mathbf{U}\mathbf{U}^{T}=\mathbf{I}_{q\times q}$ and
$\mathbf{V}\mathbf{V}^{T}=\mathbf{I}_{n\times n}$), and $\boldsymbol{\Sigma}$
is a matrix of size $q\times n$ with only $p=min\\{q,n\\}$ non-zero diagonal
entries denoted as $\sigma_{c}$ and called singular values, which are the
square roots of the eigenvalues of both $\mathbf{J}^{T}\mathbf{J}$ and
$\mathbf{J}\mathbf{J}^{T}$, sorted in descending order, q.v. [28].

$\mathbf{J}\simeq\sum_{c\in[0,r)}\sigma_{c}\mathbf{u}_{c}\mathbf{v}_{c}^{T},$
(2)

where, $r\leq p$. It must be noted that the sum is not meant to be exhaustive
in that the sum of the first significant terms would achieve the purpose of
dimensionality reduction of the decision space by showing a low-dimensional
picture of the correlations between decision variables and their performance
consequences. So, instead of decomposing the Jacobian up to $p$, we can choose
to have a lower dimensional approximation up to some arbitrary smaller number
$r$.

##### Navigation

Navigating a high-dimensional design space from the side of the performance
space towards the decision space for deriving design decisions (see Figure 3)
is clearly a very challenging task, almost always impossible in the absolute
sense of solving the equation $\mathbf{J}\mathbf{x}=\mathbf{o}$, if the
decision variables $\mathbf{x}$ are the unknowns.

It is easy to see that the Moore-Penrose pseudo-inverse of the approximated
Jacobian matrix can be computed as a matrix of size $n\times q$ by easily
using the SVD factorized matrix:

$\mathbf{J}^{\dagger}:=\mathbf{V}\boldsymbol{\Sigma}^{\dagger}\mathbf{U}^{T},$
(3)

where $\boldsymbol{\Sigma}^{\dagger}$ is simply formed as a diagonal matrix of
size $n\times q$ with the reciprocals of the singular values. Similarly, the
approximate pseudo-inverse of the Jacobian can be computed as:

$\mathbf{J}^{\dagger}\simeq\sum_{c\in[0,r)}\sigma_{c}^{-1}\mathbf{v}_{c}\mathbf{u}_{c}^{T}.$
(4)

However, in the same way, a minimal loss approximate solution exists for such
equations when the matrix is rectilinear, $\mathbf{J}^{\dagger}\mathbf{o}$ is
expected to be the least-square solution to the linearized Jacobian
approximate of a navigation problem. Even though the system might in theory
have a solution, the odds of finding a unique solution are practically very
skewed towards having an indeterminate system with many more inputs than
outputs, and thus the system will have many approximate solutions rather than
a unique exact solution.

This is of course in line with the intuition of most human beings about the
inherent difficulty of design problems for which there is no unique solution.
Note that in all these theoretical treatments we implicitly assumed that all
data points within the decision space correspond to valid designs whereas in
reality, it might be more difficult to ensure finding valid solutions
(feasible in the sense of complying with constraints) rather than good
solutions. In other words, constraint solving tends to be more difficult than
optimization in a feasible region of decision space.

###  3.2 Spatial Design Variables

If the question of the design problem directly pertains to the shape of the
configuration of an object, we can still construct decision variables to be
handled within the proposed framework for mapping and navigating design
spaces.

The idea of bringing spatial decision variables in a generative design process
is to consider first the nature of the objects being designed as manifolds,
i.e. locally similar spaces (homeomorphic) to Euclidean spaces of low
dimensions (2D planes or 3D hyper-planes) but globally more complex, possibly
having holes, handles, and cavities (shells). Three types of these manifolds
are of special interest for generative design, such as those that conduct walk
flows (explicit or implicit pedestrian corridors in buildings and cities),
light flows (rays of sunlight, sky-view, or other visibility targets), and
force flows in structures. Our conceptual framework proposes that these flows
are conducted within spatial manifolds as below:

  1. 1.

Walkable Space Manifolds (2D) conduct walks (accessibility questions)

  2. 2.

Air Space Manifolds (3D) conduct light rays (visibility questions)

  3. 3.

Material Space Manifolds (3D) conduct forces (stability questions)

This consideration allows us to see the way this object is supposed to
function would largely be determined by how this manifold is configured in
that the way the manifold in question conducts the flows of walks in a
walkable floor space, flows of light rays in a visible air space, or flows of
forces in a reliable material space. Thus we can highlight the specific
concept of flow in a network representation dual to the discrete
representation of a manifold as an unambiguous alternative intermediary
instead of any vague notion of function to study and measure.

Apart from mathematical elegance, this approach also provides multiple
computational advantages that are very much in line with the recent
advancements in the field of generative models in AI. In a nutshell, the
discrete representation of the so-called design space not only provides a
workable representation of all possible forms but also a workable
representation of some inherent functional properties of the represented
manifold that should logically determine how it could function as a building
(or a structure). The manifold representation can be mathematically denoted as
a polygon mesh of vertices, edges, and faces $\mathcal{M}=(V,E,F)$ (for a
2-manifold) or a polyhedral mesh of vertices, edges, faces, and cells
$\mathcal{M}=(V,E,F,C)$ that can have a dual graph representation in the form
of $\Gamma=(N,\Lambda)$.

Figure 5: An illustrative discrete design domain and its associated decision
space distinguished for shaping/massing and zoning/configuring problems

This description should principally sound natural if we articulate the purpose
of a design task as follows: finding the ideal form (configuration and shape)
of a manifold to conduct some flows in a desirable pattern. In this way, we
are diverting our attention from the containers of space (the building) into
what it contains (i.e. the space and its spatial configuration). This change
of focus allows us to see the direct correspondence between the so-called form
and function of a design or, better put, the form (i.e. configuration & shape)
and expected quality/performance of a spatial configuration.

In what follows, we shall go much beyond the vector data inputs consisting of
only numerical variables, especially in the context of our illustrative
example. In fact, without loss of generality, the ideas of mapping and
navigating design spaces in an approximate sense go beyond decision variables
pertaining to continuous decision variables and those pertaining to the
spatial configuration and geometric shape of spatial manifolds. The same ideas
can be applied to design problems that are about decision-making in a much
more general sense as discussed above. Note that the illustrative example that
we have demonstrated at the end of the chapter has a heterogeneous mix of
spatial and mostly non-spatial decision variables as well as a mix of
categorical and numerical decision variables.

###  3.3 Statistical Approaches to Design

Amongst the statistical approaches to design, we can distinguish the
possibilistic approaches from the probabilistic ones.

  * •

probabilistic approaches: BBN, Variational Auto-Encoders (VAE), and Diffusion
Models

  * •

possibilistic approaches: Markovian Design Machines, Fuzzy Design (see MAGMA
below)

##### Possibilistic Approach

The essence of the possibilistic approach to design is using a multi-valued or
non-binary logic framework for making design decisions, typically in the sense
of making discrete choices about discrete segments of space, for example, the
Markovian Design Machines of [6], the Spatial Agents Academy of [44], and
MAGMA (Multi-Attribute Gradient-Driven Mass Aggregation) through Fuzzy Logic
as introduced briefly in [30] [30] and [42] [42]. Both of these methodologies
apply non-binary logic from a possibilistic point of view, in the sense that
they take design inputs that are valued in the range of $[0,1]$ but treat them
as possibility measures rather than probability measures. The two big ideas
behind these two methods are the utilization of Markov Chains, Markov
Decision-Processes, and Fuzzy T-Norms for coping with uncertainty and human-
like reasoning in simulated negotiations between spatial agents.

##### Probabilistic Approach

The probabilistic models briefly mentioned here are all related to the concept
of conditional probability, the Bayes theorem, and [generalized] stochastic
processes that resemble Markov Chains [46, 30]. In a nutshell, the core of
these models is about updating some posterior probabilities indicating beliefs
about the truth of some statements by prior probabilities multiplied by the
likelihood of compelling evidence, scaled by the probability of the existence
of the evidence. When probabilistic neuron-like nodes in Probabilistic
Graphical Models are combined, these new posterior probabilities or
probability distributions can be fed into other layers of a network to create
ANN architectures. A basic idea here is to gradually reduce the dimensionality
of input data into an abstract low-dimensional representation (encoding, or
mapping, albeit into a typically unintuitive and interpretable latent space)
and then gradually use the inverse of the forward diffusion-like processes to
denoise a vector in the low dimensional hidden space. The latter process is
called denoising or decoding and it matches our description of navigation
processes, albeit without the direct control of the meaning of the latent
space vectors. A breakthrough in this domain can come from enhancing the
explainability of the latent space low-dimensional representations. This idea,
however interesting, falls way outside the scope of this short treatise.
Therefore, here we only provide a theoretical minimum for understanding the
demonstrative example (i.e. a shallow Bayesian Belief Network).

##  4 Demonstration

In this section, we will present a demonstration of the utility of the
proposed framework to indicate how a discrete decision-making approach can
facilitate generative design processes. As a disclaimer, it must be noted that
this example is not chosen for technical reasons related to AI but rather due
to its real and societal and environmental important purpose for policy
analysis concerning energy transition planning actions at the level of a
country, and sustainability strategies at the building level.

###  4.1 Case Study

Understanding the energy performance of architectural designs is crucial in
ensuring a sustainable future. Building Energy Modelling (BEM) is a multi-
purpose approach used by designers and policymakers for checking building code
compliance, certifying energy performance, subsidy policy making, and building
management. The Dutch government has recently introduced the NTA 8800
calculation model for quantitatively determining the energy performance and
code compliance of buildings [12]. The NTA 8800 aims to provide a transparent,
verifiable, and enforceable building energy performance model, based on the
European Energy Performance of Buildings Directive (EPBD), the European
Committee of Standardization (CEN), and the Dutch Normalization Institute
(NEN) published standards [29]. These regulations describe methods to
calculate the energy performance of buildings, set energy requirements for new
buildings, and make agreements about energy label obligations in existing
buildings. The NTA 8800 only concerns building-related measures, as expressed
in the EPBD, Annex A [43].

The NTA 8800 document has been implemented as an MS Excel tool by the Dutch
government (commissioned by Nieman B.V. consultants). This calculation model
translates the public European standard document into a calculation tool. The
calculation tool is not publicly available and it is not documented. Since we
were given temporary and bounded access to this model we chose to approximate
it and construct a meta-model out of it. The model consists of 269 unique
input parameters about the spatial and technical building design
configurations, based on which the model returns three scalar response values
about the energy performance of the building design; BENG 1 (maximum
permissible energy demand in $kWh/m^{2}y$), BENG 2 (maximum permissible
primary energy consumption in $kWh/m^{2}y$), and BENG 3 (minimum permissible
share of renewable energy use as a percentage). The acronym BENG refers to
some national performance indicators for Nearly Zero-Energy Buildings (Bijna
Energie Neutrale Gebouwen in Dutch).

NTA 8800 model has three main limitations: (1) it can only process and compute
information about a single specific scenario at a time; (2) it returns scalar
values about the energy performance that is untraceable to input parameters;
and (3) missing input values could result in errors or non-realistic response
values. These three limitations make the model impractical for designers and
policy analysts, particularly in the early stages of design. This
impracticality is because, in conceptual design and policy analysis, designers
need to (1) explore and iterate various options simultaneously; (2) need
feedback on the degree of influence of each design decision; and (3) cannot
provide detailed information yet about later stage design choices, such as the
technical systems.

The framework of Augmented Computational Design (ACD) is particularly useful
here as it allows us to relate the aggregated performance changes of the few
NTA 8800 outputs of interest to the changes in the many design decision
parameters of its input. In this particular case, we adopt a probabilistic
meta-modelling (function approximation) approach based on the methodology
suggested by [9].

###  4.2 Methodology

##### Meta-Modelling

Meta models are models that describe the structure, behaviours, or other
characteristics of related models, providing a higher-level abstraction for
constructing and interpreting complex numerical models that approximate more
sophisticated models often based on simulations. A meta-model is to serve as a
simplified, computationally efficient _model of the model_ [9], also referred
to as a surrogate model [22]. The process of creating a meta-model is referred
to as meta-modelling [15]. Some alternative meta-modelling techniques include
interpolation methods such as spline models [5], polynomial regression [23],
or Krigging [2].

Within the ACD framework, such meta-models provide structured ways to perform
the two most important tasks of the generative design: _mapping_ and
_navigation_.

In general, a standard meta-model can be described as:
$\mathbf{o}=\mathbf{f}(\mathbf{x})\simeq\mathbf{g}(\mathbf{x})$, where
$\mathbf{o}$ is the aggregated simulation response, $\mathbf{f}$ denotes a
computational simulation-based model conceptualized as a vector function and
$\mathbf{g}$ is the approximated model function (cf. Figure 3.) With this
notation, the objective of meta-modelling is to build the $\mathbf{g}$ in such
a way that it produces reasonably close values of $\mathbf{o}$. In the case of
augmented computational design, meta-modelling can be adopted as a methodology
of design _mapping_ that provides a differentiable and ideally reversible
$\mathbf{g}$ that can be used in the _navigating_ process. In other words, the
meta-modelling should structurally relate the choices and consequences in such
a way that the choices can be derived from the desired consequences; hence
providing a data-driven basis for generative design. The next part
demonstrates a probabilistic meta-modelling approach to navigation tasks in
high-dimensional design decision spaces, based on the methodology introduced
by [9] [9].

##### Bayesian Belief Networks

A Bayesian Belief Network (BBN) is a kind of Probabilistic Graphical Model
(PGM) that is effectively an ANN in the form of a Directed Acyclic Graph (DAG)
with neuron-like nodes that can compute Joint Probability Distributions (JPDs)
from input probability distributions or discrete Probability Density Functions
(PDF), which is then attributed to an output probability distribution through
a Conditional Probability Distribution (CPD) computing posterior
probabilities/beliefs through the Bayes theorem, hence the name Bayesian. The
set of edges of a BBN forms the model architecture or structure that
represents the particular probabilistic dependencies between the discrete
probability distributions attributed to the starts and ends of the nodes. This
structure is typically set by the modeller based on their knowledge of the
process; while the conditional probability distributions (transition
probability matrices) are learnt from the experimental data. BBNs can help us
semi-automatically reason about uncertain knowledge or data [35]. This makes
it possible to perform probabilistic inference, such as computing the JPDs of
some outputs (effects) given some inputs (causes). The name of these ANN comes
from the idea of updating beliefs or hypotheses posterior to observing
evidence; more precisely, utilizing the Bayes theorem for updating conditional
probabilities in network structures, in a fashion similar to modelling and
evaluating Markov Chains, albeit with the difference that Markov Chains
operate as uni-partite networks but each neuron of a BBN is a bipartite
network coupled with an outer product calculator.

The neurons of a BBN consist of two layers, the first of which can be dubbed a
presynaptic layer that combines input discrete probability distributions
(through an outer product) and forms a JPD and then flattens the JPD to form a
vector-shaped probability distribution. The second, i.e. the synaptic layer
then is a CPD, i.e. practically a rectangular probability transition matrix
that maps this flattened JPD to the output probability distribution. A BBN
then consists of such neurons connected in a DAG. Training a BBN means finding
the entries of the CPD in such a way as to minimize the loss of recovery of
the output probability distribution from the input distributions. The appeal
of BBNs is twofold: on the one hand, they allow the inclusion of expert
knowledge and intuition in the architecture of the network and on the other
hand the training of the network makes the network adapted to the objective
data. In this case, we limit the architecture of the network to a single layer
of neurons to keep the network invertible.

Figure 6: An illustrative example of a Bayesian Belief Network, eliciting the
nature of nodes and the network architecture, an example adapted from here

##### Workflow

Research from [8] [8] illustrates the four process steps involved in
developing a BBN meta-model. In this use case, we alter this methodology as
creating a BBN with all 269 input parameters is infeasible. We add an
intermediary step of sensitivity analysis to identify the most influential
input parameters before constructing the BBN. Thus we follow these steps in
order (see Figure 7): (1) sample the input parameter space, (2) run
simulations to generate the output values, (3) sensitivity analysis and
selection of influential input parameters, (4) train the BBN, and finally (5)
evaluate the model’s robustness. As [9] [9] highlights, it is important to
model a shallow BBN as a complete bipartite graph connecting all input nodes
to all output nodes, effectively limiting the topology to two layers. This
would allow us to make a reversible approximation that can be used to derive
the necessary input configuration for any desired performance output.
Additionally, the fixed values can also include some of the input variables
turning them into design constraints.

Figure 7: Overview of the workflow adopted from the framework introduced by
[8]

##### Step 1: Sampling the Parameter Space

We need to set up a Design of Experiment (DoE), to generate simulation data to
study the relationships between various input variables and output variables
[17]. The experiment involves running several simulations at randomised input
configurations [38]. Before running the simulation, it is important to
carefully select a sampling method, to determine these input configurations,
since the chosen strategy influences the quality of the meta-model [10]. Since
it is assumed that the decision space is unknown, the intention is to be
inclusive of all regions of the decision space as possible. The sampling
algorithm should generate a well-varied response data set that captures all
the information about the relationships between the input parameters and the
responses. For this study, 20.000 quasi-random input samples were generated
based on Sobol’s sequences [41] to ensure the homogeneity of the samples.

##### Step 2: Run NTA 8800 Simulation Model

Vectorization is an important part of the ACD, we represent the decision
variables and outputs of interests as vectors (See Section 3.) Each sample
point can be interpreted as a vector of scalar input values $\mathbf{x}$. Each
batch of such vectors is fed to the NTA 8800 model to generate the vector of
corresponding building performance outputs $\mathbf{o}$. After running the
primary simulation model for the sampled input data points, the response data
is collated and linked to the input samples to form an input-output dataset
for regression modelling (as in Machine Learning).

##### Step 3: Sensitivity Analysis

The creation of a meta-model from 269 parameters, each with scalar input
values, requires a simulation of all possible combinations (the number of
options to the power of 269). Even limiting the number of options for each
parameter to two, results in an immense number of possible combinations,
calculated at $5.39\times 10^{80}$. To contextualize the magnitude of this
number, it is more than the estimated number of atoms in the observable
universe.

The sheer magnitude of this number makes storage and training of Bayesian
Belief Networks (BBNs) infeasible. Therefore, in this study, we use global
sensitivity analysis to apportion the uncertainty in outputs to the
uncertainty in each input factor over their entire range. This allows us to
remove the parameters with the lowest influence on energy performance. The
sensitivity analysis method is implemented in the workflow based on the SALib
library [16].

This results in a meta-model with fifteen parameters instead of 269, making it
feasible to store and train the BBN, but on the other hand, reducing the
accuracy and scope of the model. However, the most influential fifteen
parameters are responsible for 90,45% to 92,30% of the final energy
performance score. Hence, we decided on the inclusion of the fifteen specific
parameters to construct the BBN meta-model.

##### Step 4: Build a BBN meta-model

Building a BBN meta-model is a process of associating the probabilistic
relationships of inputs and outputs. These relationships may be characterized
by a high degree of non-linearity and possibly multiple interactions and
correlations between model parameters. Consequently, there are two main steps
in this process: (1) learning the network topology as a DAG structure and (2)
estimating the CPD attributed to the neuron-like nodes of the network.

In this demonstration, we adhere to a particular network topology to ensure
the reversibility of the trained model [9]. Accordingly, this BBN has only two
layers: one corresponding to the input and one corresponding to the output.
However, effectively only a single layer of neurons operates in the middle of
the two layers. In this case, the selected parameters from the sensitivity
analysis results are represented by the input nodes, and the BENG 1, BENG 2,
and BENG 3 parameters are the output nodes (see Figure 8). Therefore, we skip
the topology learning step in the conventional BBN modelling; because the
topology of this particular network is assumed to be a complete bipartite DAG.
In particular, we use the _pgmpy_ Python package to model the network topology
[1].

The next step is to estimate the nodes’ CPDs from the input-output dataset.
The CPDs for the nodes can be learnt directly from the input-output simulation
data generated in steps 1 and 2, using the Maximum Likelihood algorithm.
Additionally, we discretize each variable range into a fixed number of
intervals. All numerical input distributions generated using a space-filling
approach, like Sobol’s sequence or Latin Hypercube, are sampled based on
continuous ranges, and should therefore be discretized. Discretization is done
by dividing the interval of the parameter over a fixed number of ranges
between the minimum and maximum values.

Figure 8: The single layer BBN; Right: BENG 1, BENG 2, and BENG 3 parameters
are the output nodes; Left: Most sensitive input parameters as input nodes.

##### Step 5: Validating the Meta-Model

To assess how our trained BBN approximates the original NTA 8800 model, we use
a cross-validation approach in combination with Normalized Root Mean Square
Error (NRMSE) and Mean Absolute Percentage Error (MAPE) [19]. Cross-validation
splits the generated input-output dataset (step 2) into a _training set_ and
_testing set_ , before building the BBN (step 3). The BBN is trained on the
_training set_ and assessed based on the _testing set_. However, to obtain a
more reliable estimate of the model’s performance, the dataset is split into
several subsets or folds, with each fold used as both a training set and a
testing set. This research adopts a k-fold cross-validation technique, where
$k$ refers to the number of groups that the data set is split into. We set
$k=10$ based on experimentation to ensure a low bias and a modest variance.
The model is then trained on $k-1$ of the folds, and the remaining fold is
used for testing. This process is repeated $k$ times, with each fold used for
testing once. The performance of the model is then evaluated by averaging the
performance across all $k$ runs.

Following the approach suggested by [9] [9], we computed the mean difference
of the predicted and actual output values and normalize the RMSE values
through division by standard deviation to achieve NRMSE.

To calculate the accuracy of the meta-model, it is recommended to use multiple
metrics to get a comprehensive evaluation of the model’s accuracy. Hence,
NRMSE is combined with the MAPE metric. MAPE measures the average absolute
percentage difference between the predicted and actual values. It is a measure
of the magnitude of the errors in the model’s predictions. Lower NRMSE and
MAPE values indicate better model performance. The larger the error between
the two, the higher the NRMSE and MAPE values will become. Therefore, the
NRMSE and MAPE results will indicate how dispersed the prediction data is
compared to the actual model response.

###  4.3 Results

This section presents the numerical results obtained from the experiment of
NTA 8800 meta-model.

####  4.3.1 BBN Validation Results

Here we elaborate on the results of the cross-validation technique in
combination with NMSRE and MAPE based on the test data set (s=1100). The
interpretation of what is considered an acceptable NRMSE and MAPE score
depends on the specific problem and the context in which the meta-model is
being used. In general, it is recommended to compare the NRMSE and MAPE scores
of the meta-model with the baseline models and the state-of-the-art models in
the field. This can provide a benchmark for what is considered acceptable
performance in the specific context of the problem.

In our case, the BBN does not compete with other models but rather competes
with consulting building energy specialists in estimating the building energy
performance in the early design stages. However, to assess the proficiency of
our model in capturing the underlying relationships using solely the 15
selected parameters, we employ the following benchmarks: The NRMSE values
should be in the range of $(0.20\%,0.60\%)$ as baseline, and in the range of
$(0.10\%,0.30\%)$ as state-of-the-art [7]; The MAPE values should be in the
range of $(0.10,0.30)$ as baseline, and in the range of $(0.05,0.15)$ as
state-of-the-art [21].

The NRMSE of the trained BBN for BENG 1, BENG 2, and BENG 3 respectively is
$0.82\%$, $1.52\%$, and $0.47\%$. This indicates that except for the BENG 3
indicator, the prediction of the model is not accurate enough. The MAPE of the
trained BBN for BENG 1, BENG 2, and BENG 3 respectively is $0.35$, $0.28$, and
$0.33$. This indicates that the predictions of the model are on the upper
threshold of being acceptable as baseline models. Absolute prediction
difference can be seen in Figure 9.

The prediction accuracy is calculated by
$PA=\frac{100y^{\prime}}{100*y-y^{\prime}}$ Where $PA$ is the prediction
accuracy, $y^{\prime}$ is the prediction of the meta-model and $y$ is the NTA
8800 estimation.

Figure 9: Histogram of BBN prediction and NTA 8800 outputs (top) and their
comparison (bottom) for BENG1, BENG2, and BENG3, in order from lft to right.

####  4.3.2 Toy Problem

Here we present a test case that demonstrates the effectiveness of the BBN
meta-model in building design. The study involves two toy problems that
showcase the advantages and utilization of the meta-model. The toy problems
regard two common design challenges that cannot be solved using the currently
available tools, such as the NTA 8800. The first problem involves predicting
the BENG 1 energy performance of a typical Dutch dwelling during the early
design stage. The spatial characteristics of the building are fed to the meta-
model. As output, the meta-model returns a range and the confidence level of
that range.

In this toy problem, the meta-model predicts the BENG 1 value to be within the
range of $(0-50)kWh/m^{2}.y$, with a confidence level of $100\%$ (see Figure
10). To validate this result, we cross-checked the predicted result with the
final configuration of the dwelling using the original NTA 8800 model. The NTA
8800 model returns a value of $39.8kWh/m^{2}.y$, confirming the prediction
capability of the meta-model.

The second problem reverses the first problem and involves the ex-ante
determination of the most probable design configuration that satisfies a
specific energy performance goal. In this example, the BENG 3 value of a
typical Dutch dwelling design $(35\%)$ does not satisfy the minimal
requirements $(50\%)$. Since this problem arises in the final design stage,
some input parameters cannot be changed anymore. In this case, the architects
and engineers are limited to modifying only the area (AreaPV) and power (PPV)
of the PV panels. Since the minimum required performance goal for BENG 3 is
$50\%$, we set the goal value to a range of $60-80\%$. Given the binning
approach employed, it should be noted that the AreaPV value of 5 depicted in
the figure corresponds to a range of $[40,50]m^{2}$, while the PPV value of 5
corresponds to a PV Power range of $[200,250]W/m^{2}$. Accordingly, the meta-
model advises increasing the PV area to $[40,50]m^{2}$, and the PV power to
$[200,250]W/m^{2}$ (See Figure 10).

This discretisation allows for a clearer representation of the recommended
parameter values within the specified ranges, facilitating the interpretation
and practical implementation of the BBN meta-model outputs. Since these ranges
are the maximum of both scales: the meta-models advise can be interpreted as
maximizing the PV area and PV power to reach the goal BENG 3 value of $60\%$
to $80\%$.

Figure 10: Output Recommendations of the BBN Meta-model for Achieving BENG 3
Compliance.

###  4.4 Discussion

In the end, to validate this result, we finish the loop by calculating the
final configuration of the dwelling with the original NTA 8800 model. The NTA
8800 returns a value of $71\%$, confirming the reverse inference capability of
the meta-model. These results illustrate in a simple and digestible example
how the BBN meta-model is capable of providing valuable insights and assisting
architects and engineers in navigating the multidimensional decision space.

By using the numerical Design of Experiments and Sensitivity Analysis we are
effectively conducting a dimensionality reduction task similar to the low rank
SVD as introduced before. As mentioned earlier, the mention of this particular
approach of ACD was to illustrate the utility of the framework with a concrete
example in a societally relevant context where a Machine Learning approach to
modelling can help make an otherwise very complicated simulation procedure to
be approximately scaled up massively for policy analysis. Here we discuss the
potentials and the shortcomings of the model and note the issues with this
large-scale black-box approximation that require further investigation.

The existence of categorical variables in the inputs of the BBN limits the
general applicability of ACD as it affects the smoothness and
differentiability of the underlying function that is being approximated.
However, for pragmatic reasons, we have ignored this issue to demonstrate the
idea in a large-scale case.

##### Validation results

Compared to the NTA 8800, the Bayesian meta-model is capable of capturing the
most important relationships between inputs and outputs. However, the
difference between the meta-model’s predictions and the NTA 8800 predictions
can be rather high. This means that there is a large difference between the
output of the meta-model and the NTA 8800. The NRMSEs of BENG 1 ($0.82\%$),
and BENG 2 ($1.52\%$) show that the BBN is able to follow the NTA 8800 to some
extent, but is far from accurate as NRMSE is greater than $0.5\%$. On the
other hand, the model could predict BENG 3 ($0.47$) relatively accurately.
This insufficient accuracy was expected as we have dictated a particular
topology to the BBN while learning the network structure is an important step
in constructing BBNs. This decision was made to enable the model to function
in a bidirectional way: inference and reverse inference (i.e. evaluation and
derivation in the terminology of our ACD framework).

##### Foreward Inference

The trained BBN is now capable of inferring the outputs of interest given
certain input configurations. This inference uses the learned CPDs to predict
the most likely values for the outputs. In this way, we can predict the energy
performance of buildings, in a quick and intuitive way for ex-ante assesment,
based on a certain design configuration. In particular, inference demonstrates
the potentials of a mapping described in Section 3.

##### Backward Inference

Since our BBN had only two layers in its network, it can be reversed. This
means that instead of presenting evidence to it, we can present the desired
performance values and ask for the derivation of the particular configuration
of inputs that will produce such an output. This can be done through the
_Variable Elimination_ module of the _pgmpy_ [1]. The same is also true for a
combination of given input-outputs; meaning that evidence can be given for
both inputs and outputs of the BBN. In such cases, the given inputs can also
function as the design constraints. The reverse inference demonstrates how we
can utilize probabilistic models to navigate a decision space as explained in
Section 3.

##### Augmenting

The Bayesian meta-model is capable of representing the input-output
relationships in a bidirectional and probabilistic format, illustrating a
complete example of mapping and navigating processes. However, the use of a
subset of the most influential variables of the NTA 8800 limits the navigation
to decision space made out of the selected variables. Nevertheless, this
selection was necessary to manage the computational resource-intensive task of
learning. Therefore, BBN does not compete with, or mimic the NTA 8800 model,
but rather complements it by increasing its accessibility and providing
navigation capabilities. The result is a model that can augment the designers’
intuition or experience and enhance the level of accuracy even in otherwise
vague processes of policy formulation, e.g. in assessing the potential
efficacy of alternative subsidies and incentives for building renovation aimed
at sustainable energy transition.

##### Binning

The variables are divided into a specified number of bins based on the
frequency of the values. In equal frequency binning, the data is first sorted
in ascending order and then divided into the specified number of bins, with
each bin containing an equal number of observations. Equal frequency binning
can be useful for analyzing data that has a skewed distribution or contains
outliers. By dividing the data into equal-frequency bins, the impact of
outliers may be reduced, and the distribution of the data may be more balanced
across the bins. Considering the prediction robustness of the discretized
values, each bin should contain a fair amount of data points. A fair amount is
a bit vague description because there is no method or rule of thumb for
deciding the number of bins. However, we need to keep in mind that the number
of bins corresponds with the number of states a parameter can have, and there
with the number of parameters that are used to learn the network. Increasing
the number of bins, results in an exponential increase of computational
demand. Decreasing the number of bins, however, results in too few states to
gain the desired insight accuracy in the input-output probabilities. This
means that the number of bins should be chosen carefully, keeping in mind the
trade-offs, and satisfying the research accuracy and the robustness of the
Bayesian meta-model.

##### Discrete, Categorical, and Numerical Variables

The outcome of the inference exercises helped to reveal the relationship
between architectural language and engineering behaviour and increases the
designer’s creative intuition to guide the design process. The design goals of
the research were therefore (1) to develop a methodology to gain general and
explorative knowledge about the association between spatial and technical
building configurations and energy performance in dwellings, and (2) to
develop a proposal for an adapted representation of the NTA 8800 model tool,
embedded in a computational model, to support intelligent decision making. The
research workflow that has been described is a valid approach to reaching the
stated design goals.

##### Toy problem

The results of the toy problem show the implementation possibilities, and
show, of course, simplified, the possibilities for professionals to utilize
the tool in the practical field. As shown in the previous chapter, the tool
works as expected, and is capable of making preliminary predictions with very
few input variables. It appeals to the imagination when the tool is connected
to spatial configurations instead of unmeaning variables in a computer script.

##  5 Outlook

The ACD framework and its constituent concepts can be best positioned within
the context of performance-driven computational design and generative design.
In particular, the idea of approximating complex and non-linear functions for
estimating measurable performance indicators from configurations of decision
variables, even if referring to non-spatial decision variables, is
generalisable to all areas of computer-aided design. However, such surrogate
models are not to replace simulation models based on first principles as they
can not match their transparency and explainability. Nevertheless, in cases
where one needs to estimate the effects of design decisions on human factors,
ergonomics, or combinations of many different types of governing equations, an
estimation model trained from actual data can be of utility in that it
provides a basis for comparisons in the absence of analytical knowledge. In
other words, the utility of ANNs for _mapping_ the associations between
decision data points and performance data points is apparent.

The _navigation_ problem, on the other hand, is much harder, philosophically,
technically, and mathematically for being solved in any sense. The real
advantage of an AI framework in dealing with a design space navigation problem
can be attained if the latent space of the model reveals interpretable
information or if it is at least coupled with a sensible low-dimensional
space. If the latent space of e.g. an Auto Encoder [27] is understandable as a
low-dimensional vector space (as an endpoint of the _mapping_ and the start
point of the _navigating_ processes) then it can be used not only to guide the
navigation process but also to gain insight as to which design variables are
more important in determining the attainment levels of outcomes of interest.
In other words, even though it appears that in the mapping process, the
information content of the decision data points is reducing gradually, one can
think of this process as a distillation of an elixir from a large data point
that makes the information richer from a human perspective.

In this light, the major advantages of the proposed framework are twofold:
Firstly, providing an elegant framework for applying AI in computational
design in the presence of many complex quality criteria; and secondly,
providing an elegant framework for designing spatial manifolds very much like
the methodology of electrical engineering in designing electronic circuits and
systems for signal processing. The latter point requires much more space for
discussing the theoretical minimum for such an approach to design from a
signal processing standpoint. In short, however, we can briefly mention that
the idea of defining a central representation of a configuration as a discrete
manifold provides for directly modelling the functionality of the spatial
manifold w.r.t. the flows of walks, light rays, or forces not only from the
point of view of spatial movement but also much more elegantly and efficiently
in the frequency or spectral domain (which can be attained using Discrete
Fourier Transform or Spectral Mesh Analysis). One fundamental idea of analogue
circuit design from a signal processing point of view is that of designing
passive “filters” whose properties can much better be understood in the so-
called frequency domain analyses put forward by Fourier and Laplace transforms
of the so-called transfer functions of the RLC (Resistor, Self-Induction Loop,
Capacitor) circuits. This approach to circuit design can be traced back to the
ideas and propositions of Oliver Heaviside (1850-1925), a self-educated
pioneer of electrical engineering. Arguably, this frequency-based outlook
(relating to the spectrum of eigen frequencies of vibration of shapes, thus
also identifiable as a spectral approach), has revolutionized the formation of
the field of electronics and thus contributed significantly to the development
of AI as we know it today. Identifying spectral latent spaces and associating
them with low-dimensional performance spaces and latent spaces of ANNs is a
topic that calls for further theoretical research and computational
experimentation.

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##  Appendix A Acronyms

Table 1: Acronyms Acronym |  Term
---|---
ACD |  Augmented Computational Design
AEC |  Architecture, Engineering, and Construction
AI |  Artificial Intelligence
ANN |  Artificial Neural Networks
BBN |  Bayesian Belief Networks
BEM |  Building Energy Modelling
BENG |  Bijna Energie Neutrale Gebouwen: Nearly Zero-Energy Buildings
BIM |  Building Information Model
CAD |  Computer-Aided Design
CEN |  Comité Européen de Normalisation: European Committee of Normalization
CPD |  Conditional Probability Distribution
DAG |  Directed Acyclic Graph
DoE |  Design of Experiment
EPBD |  European Energy Performance of Buildings Directive
JPD |  Joint Probability Distributions
MAPE |  Mean Absolute Percentage Error
MAGMA |  Multi-Attribute Gradient-Driven Mass Aggregation
ML |  Machine Learning
NEN |  Nederlandse Norm: Royal Dutch Standardization Institute
NRMSE |  Normalized Root Mean Square Error
NTA 8800 |  Nederlandse Technische Afspraak (Dutch Technical Agreement)
PGM |  Probabilistic Graphical Models
SVD |  Singular Value Decomposition
VAE |  Variational Auto-Encoders

##  Appendix B Notation

Table 2: Notations Notation |  Name |  Definition
---|---|---
$\mathbf{x}$ |  design/decision space |  $\mathbf{x}\in(0,1]^{n}$; each $x_{i}$ corresponds to a single spatial decision variable
$\mathbf{o}$ |  performance space |  $\mathbf{o}\in(0,1]^{q}$; each $o_{k}$ corresponds to an objective or outcome of interest
$\mathbf{o}=f(\mathbf{x}):=[f_{k}(\mathbf{x})]_{q\times 1}$ |  map from design to performance |  $f:(0,1]^{n}\mapsto[0,1]^{q}$; representing a meta-model that approximately maps the decision space to the performance space
$\mathbf{x}=f^{-1}(\mathbf{o})$ |  map from performance to design |  $f^{-1}:[0,1]^{q}\mapsto(0,1]^{n}$; pseudo-inverse of a meta-model that approximately maps the performance space to the decision space
$\mathbf{J}:=[J_{k,i}]_{q\times n}$ |  Jacobian matrix of $f$ |  $[J_{k,i}]_{q\times n}=[\frac{\partial f_{k}}{\partial x_{i}}]_{q\times n}=[\nabla^{T}f_{q}]_{q\times 1}$
$\mathbf{U}_{q\times q}:=[\mathbf{u}_{k}]_{1\times q}$ |  matrix of left singular vectors |  $\mathbf{U}\mathbf{U}^{T}=\mathbf{U}^{T}\mathbf{U}=\mathbf{I}_{q\times q}$; ordered by importance
$\mathbf{V}_{n\times n}:=[\mathbf{v}_{i}]_{1\times n}$ |  matrix of right singular vectors |  $\mathbf{V}\mathbf{V}^{T}=\mathbf{V}^{T}\mathbf{V}=\mathbf{I}_{n\times n}$; ordered by importance
$\boldsymbol{\Sigma}_{q\times n}:=[\mathbf{u}_{k}]_{1\times q}$ |  matrix of singular value |  $\boldsymbol{\Sigma}$ is an ${q\times n}$ rectangular diagonal matrix with non-negative real numbers on the diagonal ordered by importance, i.e. singular values $\sigma_{c},c\in[0,\min\\{q,n\\})$