# ShipGen: A Diffusion Model for Parametric Ship Hull Generation with Multiple
Objectives and Constraints

Noah J. Bagazinski
Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, MA 02139
<EMAIL_ADDRESS>
& Faez Ahmed
Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, MA 02139
<EMAIL_ADDRESS>Corresponding Author.

###### Abstract

Ship design is a years-long process that requires balancing complex design
trade-offs to create a ship that is efficient and effective. Finding new ways
to improve the ship design process can lead to significant cost savings in the
time and effort required to design a ship and cost savings in the procurement
and operation of a ship. One promising technology is generative artificial
intelligence, which has been shown to reduce design cycle time and create
novel, high-performing designs. In literature review, generative artificial
intelligence has been shown to generate ship hulls; however, ship design is
particularly difficult as the hull of a ship requires the consideration of
many objectives. This paper presents a study on the generation of parametric
ship hull designs using a parametric diffusion model that considers multiple
objectives and constraints for the hulls. This denoising diffusion
probabilistic model (DDPM) generates the tabular parametric design vectors of
a ship hull, which is then constructed into a point cloud and mesh for
performance evaluation. In addition to a tabular DDPM, this paper details
adding guidance to improve the quality of generated parametric ship hull
designs. By leveraging a classifier to guide sample generation, the DDPM
produced feasible parametric ship hulls that maintain the coverage of the
initial training dataset of ship hulls with a 99.5% rate, a 149x improvement
over random sampling of the design vector parameters across the design space.
Parametric ship hulls produced with performance guidance saw an average of
91.4% reduction in wave drag coefficients and an average of a 47.9x relative
increase in the total displaced volume of the hulls compared to the mean
performance of the hulls in the training dataset. The use of a DDPM to
generate parametric ship hulls can reduce design time by generating high-
performing hull designs for future analysis. These generated hulls have low
drag and high volume, which can reduce the cost of operating a ship and
increase its potential to generate revenue.

_Keywords_ Naval Architecture $\cdot$ Generative Artificial Intelligence
$\cdot$ Deep Generative Models $\cdot$ Denoising Diffusion Probabilistic Model
$\cdot$ DDPM $\cdot$ Multi-objective Design $\cdot$ Design Constraint
Satisfaction $\cdot$ Drag Reduction $\cdot$ Parametric Design $\cdot$ Ship
Design

##  1 Introduction

Generative artificial intelligence (AI) models produce new instances of
information that resemble the data used to train the model. While generative
AI is famously known for generating text and image information, it can also be
used to generate information to engineer products. Recent advances in
generative AI provide promising new avenues to quickly generate designs.
Including additional information in the training, such as a design’s
performance, can be leveraged to create designs with high performance. These
advances are especially useful in the design of ships. Ship design currently
requires a large team of naval architects to balance design trade-offs in a
single ship’s design. A generative AI model specifically trained to generate
ship hulls can improve this workflow. Training such a model successfully is
enabled by the availability of large datasets that include both design and
performance information for ship hulls [1]. Hull design was chosen as a
starting point for the generative model as the shape of the hull has a direct
impact on over 70% of the cost of a ship [2]. It is also one of the first
steps in the traditional workflow for ship design [3]. The hull shape affects
several key aspects of a ship’s performance, including the buoyancy, upright
stability, hydrodynamics, and general arrangements of the ship. With these
considerations, the design of ship hulls provides an impactful avenue for the
application of machine learning for engineering design.

A well-designed machine learning tool for ship design could learn design
trade-offs for ships through the continual design and evaluation of many ship
designs. This work demonstrates the use of a guided denoising diffusion
probabilistic model (DDPM), a type of deep generative model, to rapidly
generate high-performing and feasible parametric ship hull designs by
generating parameters in a tabular format. This model, called ShipGen,
generates early-stage hull designs considering seven performance metrics,
creating shapes with low drag and high cargo-carrying capacity. Figure 1 shows
an overview of the work presented, highlighting that the implementation of
classifier and performance guidance during the sampling process generates
hulls with high performance. This work features model training with a publicly
available dataset of parametric ship hulls, called ShipD [1]. The following
sections detail the literature review of previous work, the methodology for
creating and evaluating a tabular DDPM, the evaluation of ship hulls generated
by the DDPM, and a discussion on the impact of the work. The hulls generated
with the use of the guided DDPM are intended to be candidate designs for
future analysis. As such, these generated hulls may not necessarily look
exactly like realistic hull forms, but instead have design features that, in
combination, lead to high performance. Through the development of the
performance-guided DDPM for ship hull generation, the novel contributions of
this paper are:

  1. 1.

The first known use of denoising diffusion probabilistic models for generating
parametric tabular data for an engineering performance-focused design
application.

  2. 2.

Showcase that classifier guidance in the DDPM navigates complex design
feasibility constraints to generate feasible samples with over 99% success
while maintaining dataset design coverage.

  3. 3.

Use of guidance to improve ship hull performance, with samples having an
average 91.4% reduction in wave drag coefficient and 47.9x more displaced
volume compared to the mean performance of the hulls in the dataset.

Figure 1: Overview of utilizing a DDPM to generate parametric ship hull
designs. When leveraging classifier and performance guidance from pre-trained
neural networks, the DDPM is able to generate ship hull designs with high
performance.

##  2 Prior Work

Generative AI for ship hull design was influenced by research in computational
ship design and machine learning literature. The first subsection details
prior work in computational ship design, including ship hull design
representation, hull form design optimization, and the use of machine learning
in ship design. The second subsection details the development of diffusion
models and their applications in engineering design.

###  2.1 Computational Ship Design

Computational Ship Design refers to the application of computer-based
modeling, simulation, and optimization techniques in the design and analysis
of marine vessels, facilitating more efficient, innovative, and integrated
design solutions. Historically, computational ship design can be divided into
three categories: design representation, forward modeling which includes
surrogate models, and inverse design or synthesis, which includes optimization
methods. Recently, generative AI methods have emerged as a powerful technique,
which could be used for the representation and synthesis of ship hull designs.

In order to design a product with computational methods, the product needs to
be represented in a way that a computer can understand. For ship design, the
two most popular modes are parameterized vectors [4, 5, 6, 7, 8, 9, 10, 11,
12, 1], and free form deformation techniques [13, 14, 15, 16, 17, 18]. The
benefit of using parameterized design representations for a hull is that the
design is defined by a set of tunable parameters that both human designers and
computers can interpret. The ease of use of parametric design representations
has often limited the diversity of possible hull shapes. Conversely, FFD
techniques present a different landscape. They allow for the creation of a
broad array of shapes. Yet, these representations can be challenging for
humans to interpret without a visual representation of the hull form. The
works of Khan et al. [19, 20, 21], Shaeffer et al. [22, 23], and Bagazinski et
al. [1] have looked at various methods to create diverse design spaces and
design datasets for ship hull design. These efforts aim to harness machine
learning in ship hull design.

In addition to design representation, computational design often has metrics
for evaluating a design’s performance. Finding computationally efficient
methods for evaluating each generated design could lead to enhanced design
generation. Hydrodynamic drag stands out as the predominant performance metric
for ships in literature. Several rapid drag prediction techniques exist. Some,
like Hollenbach’s and Savitsky’s methods, rely on statistical regressions from
test data [24, 25, 26]. Other fast methods to predict wave drag are linear
wave solvers, which provide accurate drag measurements with reduced
computational effort relative to traditional computational fluid dynamics
techniques. These solvers use potential flow to simulate the waves produced by
a ship in a steady forward motion to estimate drag as a result of surface wave
propagation. Different linear wave solvers include Michell’s Integral [27,
28], Rankine Panel Methods [29], Neumann-Kelvin Theory (also called Dawson’s
Method) [30], and Neumann-Michell Theory [31, 32, 33]. These potential flow
solvers input the 3D geometry of a hull and provide estimates of drag at
typical operating speeds of a hull. The third method of creating a fast
prediction of drag is to build a small dataset of drag measures to train a
neural network to predict drag from a hull’s design representation [14, 15,
20, 19, 16, 6, 9, 5, 13, 34, 35].

Combining hull design representation with efficient drag prediction equips
designers with the tools needed for optimization algorithms. This enables hull
design creation tailored for specific scenarios. A common objective in
optimization literature is minimizing hull drag while adhering to geometric
constraints. More recently, computational hull design has also been attempted
using a tabular generative adversarial network (GAN) to quickly generate ship
hull instances, that could be used for seeding populations for design
optimization [21]. The next improvement for generative AI in ship design is to
implement a denoising diffusion probabilistic model (DDPM) for generating hull
designs. Diffusion models provide improvements over GANs for generative design
as DDPMs are more stable to train and provide superior sampling quality.
Additionally, diffusion models can implement guidance without retraining the
whole generative model. This way, new constraints or performance objectives
can be integrated into design generation simply, whereas a GAN would need to
be retrained for every new design consideration. [36]. For ship hull design,
this means that a single model can be trained to generate high quality hulls
that are tailored to specific user needs by integrating guidance models for
different design considerations. This is particularly useful for ship design
so that information from the design of many classes of ships can be considered
in designing a ship hull.

###  2.2 Generative Design with Diffusion Models

The transition from traditional design methods leads to a cutting-edge
generative AI model: the denoising diffusion probabilistic model (DDPM).
Gaining momentum in the machine learning domain, DDPMs iteratively modify a
noisy data vector over many specified steps, transforming random data to
mirror the statistics of training data [37]. The development of DDPMs in the
last few years has shown that they are capable of generating complex data and
already have applications for engineering design. For example, DDPMs were
shown to create higher quality images as compared to generative adversarial
networks [37], a particularly difficult task as images are comprised of large
patterns of pixels to visually represent something a human could see with
their eyes.

DDPMs work by training a neural network to predict small iterative denoising
steps. The algorithm for training a diffusion model as defined by Ho et al. is
in Table 1.

1: | repeat
---|---
2: | $X_{0}\sim q(X_{0})$
3: | $t\sim Uniform(\\{1,...,T\\})$
4: | $\epsilon\sim N(0,I)$
5: | Take gradient descent step on:
|
$\nabla_{\theta}||\epsilon-\epsilon_{\theta}(\sqrt{\bar{a_{t}}}X_{0}+\sqrt{1-\bar{a_{t}}}\epsilon,t)||^{2}$
6: |  until converged
Table 1: This is the training algorithm for a standard DDPM. The DDPM is
represented by the function $\epsilon_{\theta}(X_{0},\epsilon,t)$ in step 5.

In the algorithm, the generated sample (design parameters) are represented by
$X$, and noted with subscripts to indicate the denoising timestep. The DDPM
itself is represented by $\epsilon_{\theta}$, indicating that the DDPM is
trained to predict a small change in random noise across the vector. Once
trained, a DDPM generates samples by denoising a Gaussian noise vector over
the predetermined timesteps. This results in samples that are within the
training data’s statistical distribution. In the case of images, this could be
a “deep fake” that looks like the training data. In the case of ship hull
design, it could be a parameterized ship hull design. The sampling algorithm
defined by Ho et al. is defined in Table 2.

1: | $X_{T}\sim N(0,I)$
---|---
2: |  for $t=T,...,1$ do
3: |  $Z\sim N(0,I)$ if $t>1$, else $z=0$
4: | $X_{t-1}=\frac{1}{\sqrt{\alpha_{t}}}(X_{t}-\frac{1-\alpha_{t}}{\sqrt{1-\bar{\alpha_{t}}}}\epsilon_{\theta}(X_{t},t))+\sigma_{t}Z$
5: | end for
6: |  return $X_{0}$
Table 2: This is the sampling algorithm for a standard DDPM. The DDPM is
represented by the function $\epsilon_{\theta}(X_{t},t)$ in step 4.

Subsequent advancements in DDPMs introduced guidance, where gradients from a
classifier neural network guide image synthesis to match a specific image
classification label [36]. This evolution birthed text-to-image DDPMs that
employ text-based guidance to craft custom, lifelike images [38, 39]. Guided
DDPMs have found applications in generating 3D shapes from image data [40].

Guided diffusion can be applied to engineering design generation. For example,
guided diffusion has been used to create two-dimensional structures [41, 42,
43] and vehicles [44] using image data. In these instances, the guidance of
the design generation by image-based DDPMs is applied to constraint
satisfaction and improved performance. DDPMs can generate high-quality
designs, navigate complex constraints, and implement precise generation with
guidance, which makes them an excellent deep generative model for designing
ship hulls. The subsequent sections demonstrate a tabular DDPM to generate
parametric ship hull designs that give improved performance through the
implementation of guidance.

##  3 Methods

This section outlines the methodology behind developing a guided DDPM for ship
hull design. This section explores the ship hull dataset, delves into tabular
DDPMs, and introduces both classifier and performance guidance for sampling
ship hulls with a DDPM. A secondary methods section on conditional DDPMs is
included in the Appendix.

###  3.1 Ship-D Dataset and Hull Parameterization

The Ship-D dataset consists of 30,000 parameterized ship hulls. The hulls are
parameterized with 45 terms. These terms are applied to a set of algebraic
equations to define and characterize the surface of the hull. These terms were
construed through analyzing and characterizing the shape and curvature of many
different publicly available hull geometries. The parameters cover various
aspects:

  * •

Principal dimensions (e.g., overall length, beam at main deck)

  * •

Cross-section of the parallel midbody (e.g., deadrise angle, chine radius)

  * •

Geometry of bow and stern taper

  * •

Geometry of bulbs at bow and stern

These parameters, designed to capture a range of curvature and shapes,
encompass the features seen in a diverse variety of vessels from large ships
to smaller boats. Their dual design facilitates human understanding and
computer-generated input. Full documentation of the hull design parameters is
provided at https://decode.mit.edu/projects/ShipGen/. Additionally, Figure 20
in the Appendix lists the parameters and provides details so human designers
can create parametric hulls with this representation. A glimpse into the
Ship-D dataset is provided in Figure 2, showcasing the diverse shapes
achievable with the parametric design scheme. As these designs are randomly
sampled across the entire feasible design space, they do not necessarily look
like realistic hull designs. The performance of these hulls was not considered
in their random sampling. Many of these hulls are relatively low performing:
having high drag, low displacement volumes, and high surface area. The
feasibility criteria used to generate these hulls are described in the next
section.

Figure 2: A selection of hulls from the Ship-D dataset, showing the
variability possible with the hull parameterization. A random sampling from
the dataset may lead to unrealistic hulls, containing combinations of features
that do not resemble real-world ships and features that lead to poor
performance.

####  3.1.1 Feasibility Constraints for Hull Geometry

While the parameterization can define a large design space of hull geometries,
constraints on the parameterization are needed to ensure that a feasible hull
will be produced by a specific set of parameters. To satisfy a “feasible” hull
shape, the hull’s surface only needs to satisfy two criteria:

  1. 1.

The hull is watertight, meaning that there are no holes on its surface.

  2. 2.

The hull surface is not self-intersecting.

As the hull surface is defined by a set of equations with constants dictated
by the parameter values, conditions to determine whether a hull’s surface
satisfies the two main feasibility criteria can be solved algebraically. The
advantage to algebraically solving these conditions is significantly reduced
computational effort to check hull feasibility with the algebraic constraints
compared to feasibility checks with mesh generation. After searching through
the design space of the hull parameterization and examining the equations that
define the hull surface, a set of forty nine constraints were defined to
determine if a hull surface produced from a specific parameterization
satisfies the two feasibility criteria. Figure 21 in the Appendix lists the 49
algebraic constraints and provides information on each of their satisfaction
conditions.

Conversely, the two feasibility criteria can be checked by constructing the
mesh of a hull and analyzing its surface. Mesh generation and feasibility
checks are computed in $O(Nlog(N))$, where $N$ is the number of vertices on
the mesh. On an Intel Core i9-10980XE processor, the construction and check of
a hull mesh with approximately 80,000 vertices is 1.77 seconds. Comparatively,
the algebraic constraints check the design feasibility of a parametric hull in
0.000199 seconds. This is a ten-thousand-fold increase in speed for checking
design feasibility with the algebraic constraints. A uniform random sampling
of the design parameters leads to generation of a feasible hull in
approximately 1 per 150 tries. In addition to the 30,000 feasible hulls in the
Ship-D dataset, an additional 20,000 design vectors (called invalid samples)
that violate at least one feasibility constraint were generated. These invalid
samples are used to train models in classifying and distinguishing between
feasible and infeasible design vectors [41].

####  3.1.2 Hull Performance Measures

The Ship-D dataset already contains ten geometric measures and thirty-two wave
drag calculations for each hull. The ten geometric measures allow naval
architects to characterize a hull when designing a ship. The ten geometric
measures are calculated using trapezoidal integration at ten draft marks
spaced along the depth of the hull. These ten geometric measures are:

  1. 1.

Height of draftmark

  2. 2.

Length of the waterline

  3. 3.

Area of the waterplane

  4. 4.

Surface area of the hull below the specified draftmark (wetted surface)

  5. 5.

Longitudinal centers of flotation (waterplane centroid)

  6. 6.

Second moment of area about the longitudinal axis of the waterplane

  7. 7.

Second moment of area about the transverse axis of the waterplane

  8. 8.

Displaced volume below the draftmark

  9. 9.

Longitudinal center of buoyancy

  10. 10.

Vertical center of buoyancy

As these metrics have the units of length to some power $L^{n}$, they are
normalized by the first term in the parameterization, $LOA$, to its respective
power. For example, lengths are normalized by $LOA$, areas by $LOA^{2}$,
volumes by $LOA^{3}$, and area moments of inertia by $LOA^{4}$. This allows
computational analysis on the geometry of the hull to be performed
independently of the hull’s scale.

In addition to the geometric measures, the Ship-D dataset has thirty-two wave
drag coefficients for each hull across four different drafts and eight
velocity conditions. The four drafts are 25%, 33%, 50%, and 67% of the hull’s
total depth. The eight velocity conditions are normalized using Froude
scaling. The eight velocities are between $F_{n}=0.10$ and $F_{n}=0.45$ in
increments of 0.05, corresponding to typical operating conditions of
traditional displacement hulls [45, 46]. The Froude number is the relative
scaling between inertial and gravitational forces described in the equation
below:

$F_{n}=\frac{U}{\sqrt{gL}}$ (1)

Where $U$ is the hull speed, $g$ is gravity and $L$ is a length scale. The
length used in simulating the 32 speed-draft conditions of the hulls was the
length of the waterline at the tested draft mark. This way, thirty two unique
conditions were measured. Wave drag is both a function of the hulls geometry,
and the hydrodynamics of waves propagating off of the hull from it’s forward
motion. Including a full spectrum of speed-draft conditions in the dataset
allows a machine learning model to learn the effects of drag due to changing
submerged geometry with draft and speed. This provides significantly more
information relating to the geometry and performance of a hull than available
by measuring a single operating condition. This allows a generative model
using the Ship-D dataset to generalize wave drag in the design process. As the
generative model is intended to produce conceptual hull designs, it is
imperative that the exact speed-draft condition be unknown so that the model
generates hulls that generally have low drag. The future work section will
detail goals for generating hull designs tailored to specific use cases, which
could include specific speed-draft conditions.

The Michell Integral was chosen to simulate wave drag over other linear wave
methods for its relative computational efficiency and the accuracy it
provides. The Michell integral is a linear estimate of the wave drag of a
slender ship in forward motion. It is defined by the following equation [27,
28]:

$R_{w}=\frac{A\rho g^{2}}{\pi
U^{2}}\int_{1}^{\infty}(I^{2}+J^{2})\frac{\lambda^{2}}{\sqrt{\lambda^{2}-1}}\,d\lambda$
(2)

where $\rho$ is the density of water, $g$ is gravitational acceleration, $U$
is the ship’s speed, and $A,I,andJ$ are integrated terms relating to the
surface normal across the hull and the direction of wave propagation. Further
insight into these terms is in Michell’s paper form 1898 [27].

In addition to scaling the relative speed and draft conditions for the hulls,
the wave drag is also scaled to ensure consistency across the dataset:

$C_{w}=\frac{R_{w}}{\frac{1}{2}\rho U^{2}\cdot\text{LOA}^{2}}$ (3)

Typical drag coefficients of hulls are scaled by the wetted surface area of
the hull. Within the dataset, however, the wetted surface area of the hulls
can vary greatly. Instead, the Length-Overall (LOA) is used instead as this is
the first term in the parameterization. For the purposes of applying machine
learning using the dataset, the wave drag coefficient can be characterized by
the remaining 44 terms in the parameterization and the hull’s relative speed
and draft.

An additional two measures of the hulls are included in this paper and will be
added to the Ship-D dataset. The first measure is the Gaussian curvature of
the hull’s surface. The second metric is a measure of the largest rectangular
prism that can be vertically lowered into the hull, referred to as MaxBox for
the remainder of this paper. Gaussian curvature quantifies the double
curvature of a surface. The average Gaussian curvature is calculated for these
hulls to assess the manufacturing complexity of the hull’s surface. As most
large ships are constructed from welded sheet steel or aluminum, bending a
sheet along two principal axes of curvature is a difficult task for both the
sheet forming process and for welding the edge of a complex surface to
another. By measuring the average double curvature of each hull, an
understanding of the difficulty of manufacturing the hull surface is gained
for the dataset. The Gaussian curvature is calculated for the hulls using a
finite difference method to measure the principal curvature of the hull in the
YZ plane and in the XY plane for a uniform grid of points on the hull [47].
Equation 4 calculates the average Gaussian curvature over the surface of the
hull. The terms $R_{XY}$ and $R_{YZ}$ are the radii calculated using the
finite difference method along the two principal axes of hull’s surface.
Gaussian curvature has units $1/L^{2}$ and is hence normalized by $LOA^{2}$

$GC=\frac{\oiint_{S}\frac{\mathrm{d}A}{R_{XY}(x,y,z)\cdot
R_{YZ}(x,y,z)}}{\text{Total Surface Area}}$ (4)

The MaxBox measures the box with maximum volume that is completely inscribed
by the hull that can be vertically lowered into the hull through the
waterplane at the hull’s top deck. This provides a measure for evaluating a
candidate region within the ship hull for allocating cargo holds.
Additionally, as the MaxBox is open at the deck of the ship, a crane can
service this entire volume within each hull. The MaxBox for each hull was
optimized with a Nelder-Mead simplex optimization algorithm to maximize the
volume of the box while constrained by the surface of the hull and the
waterplane of the top deck [48]. Included in the dataset is the forward (X)
position of the box, its length, width, depth, and volume. These results are
normalized by their length dimensionality, $1/LOA$ and $1/LOA^{3}$.

Among the available performance measures in the Ship-D dataset, seven were
selected to be implemented in the performance-driven design generation of ship
hulls. The goal of selecting the seven performance metrics that generally
describe the quality of a hull. These metrics provide an avenue to compare
hulls directly to each other with useful characteristics that consider the
hulls’ hydrodynamics, hydrostatics, and manufacturability. These seven
performance metrics are:

  1. 1.

Aggregated sum of wave drag coefficients.

  2. 2.

Surface area of the hull up to 50% of its total depth

  3. 3.

Total surface area of the hull

  4. 4.

Displaced volume of the hull up to 50% of its total depth

  5. 5.

Total displaced volume of the hull

  6. 6.

Volume of the MaxBox

  7. 7.

Gaussian curvature

The aggregated sum of wave drag coefficients was selected as a way to quickly
characterize the general wave drag of a given hull. In large ships, wave drag
is the primary component in the ship’s total drag. By learning how a ship’s
hull shape affects drag, a generative AI model could generate hulls with low
wave drag, saving ship operation costs through reduced fuel consumption. The
aggregated sum of the wave drag coefficients is defined in Equation 5. It is
important to note that this performance metric and five of the other metrics
are all represented on a logarithmic scale. Due to the geometry of the hull
designs these performance metrics span several orders of magnitude across the
Ship-D dataset. The distribution of the logarithmic scaled performance metrics
is normal, a desired quality for machine learning.

$C_{w*}=\sum_{i=1}^{32}\log_{10}(C_{w_{i}})$ (5)

The surface area of the lower half of the hull was selected as a performance
measure as this is the portion of a hull’s surface that is most likely to be
submerged when the hull is in water. The wetted surface of the ship affects
the viscous drag acting on the hull. Reducing the wetted surface area of a
hull can reduce the total drag of a ship, saving operation costs through
reduced fuel consumption. Additionally, the total surface area of the hull was
selected as this can consider the amount of material needed to manufacture the
surface of the ship hull. By reducing the total surface area of the ship,
manufacturing costs can be reduced by generating hull designs with less total
surface area. The two measures of surface area are provided in Equation 6 and
Equation 7.

$SA_{50\%*}=\log_{10}\left(\frac{\int_{0}^{T/D_{d}=0.5}\delta SA(z)\,\delta
z}{LOA^{2}}\right)$ (6)

$SA_{100\%*}=\log_{10}\left(\frac{\int_{0}^{T/D_{d}=1.0}\delta SA(z)\,\delta
z}{LOA^{2}}\right)$ (7)

The displaced volume of the bottom half of the hull was selected as it
characterizes the portion of the hull that contains much of the displaced
volume for buoyant forces. This performance metric therefore characterizes the
relative total weight of the ship and its cargo. Learning how the hull design
parameters affect the displaced volume of the hull can lead to generating hull
designs that can carry more weight. The total displaced volume of the hull was
also selected as this measure characterizes the total volume capacity
available for cargo, outfitting, and other systems on the ship. Learning how
the ship hull design parameters affect its total displaced volume, ship hulls
with greater total volume can be generated. These two measures of volume
affect the ability of ships to generate revenue through the shipment of cargo.
The two volume measures are calculated with Equation 8 and Equation 9. With
the intention of maximizing the volume metrics in hull generation, the volume
measures are multiplied by -1. This conforms the volume maximization problem
to a “minimization” problem akin to the other performance objectives.

$V_{50\%*}=-\log_{10}\left(\frac{\int_{0}^{T/D_{d}=0.5}\delta V(z)\,\delta
z}{\text{LOA}^{3}}\right)$ (8)

$V_{100\%*}=-\log_{10}\left(\frac{\int_{0}^{T/D_{d}=1.0}\delta V(z)\,\delta
z}{\text{LOA}^{3}}\right)$ (9)

An additional measure of volume is the MaxBox volume. As described earlier in
this section, the MaxBox metric measures the ratio of the most useful cargo-
carrying volume of the ship compared to the hull’s total displaced volume.
Learning how the design parameters affect MaxBox can lead to the generation of
hulls with more useful cargo-carrying capacity. This can also lead to greater
revenue through a ship’s operation. MaxBox is not on a logarithmic scale like
the other measures and it is calculated with Equation 10.

$\text{MaxBox}_{*}=-\frac{\text{Volume}_{\text{MaxBox}}}{\text{Volume}_{T/D_{d}=1.0}}$
(10)

The final performance metric selected is the average Gaussian curvature of the
hull. Since Gaussian curvature is a measure of a hull’s surface complexity, it
affects the manufacturing costs of a ship. Reducing the average Gaussian
curvature of a ship hull can lead to reduced manufacturing costs, making it a
critical metric for a ship hull. The average Gaussian curvature is normalized
for machine learning using Equation 11.

$GC_{*}=\log_{10}(GC\cdot LOA^{2})$ (11)

While the aforementioned seven metrics were chosen to demonstrate the efficacy
of the proposed methodology, it is crucial to highlight that this is not an
exhaustive list of performance measures for ship hull evaluation. Indeed, a
significant strength of the proposed diffusion model lies in its adaptability.
It allows users to integrate additional performance metrics without
necessitating retraining. This flexibility underscores the model’s robustness
and its potential to be tailored to various specific needs, optimizing designs
based on a myriad of performance criteria.

###  3.2 Dataset Coverage and Generated Sample Evaluation

In order to characterize the DDPMs’ abilities to cover the total parametric
dataset space and generate feasible designs, two measures are utilized
throughout the remainder of the paper. To visually characterize how a set of
generated hull designs covers the dataset space of the Ship-D hulls, a two-
dimensional principal component analysis is trained with the Ship-D parametric
design vectors. When evaluating the designs generated with DDPMs, the PCA of
the generated samples is plotted against the PCA of a random selection of the
Ship-D dataset hulls. This shows the relative spread of the generated designs
compared to the dataset hulls. In addition to visualization, coverage and
realism quantify a model’s ability to generate samples similar to the training
dataset.

  * •

Coverage is quantified as the mean chamfer distance of each dataset instance
from its nearest neighbor among the generated samples.

  * •

Realism, on the other hand, measures the mean chamfer distance of each
generated sample instance from its closest match within the dataset [49].

Chamfer distance is the Euclidean distance between two hull design vectors.
For two sets of parameterized hull designs, $A$ and $B$, the Chamfer distance
finds the distances from a design vector in $A$ to its nearest neighbor in
$B$. The distance metric used is the squared Euclidean distance between the
two vectors. The formula for this evaluation metric is:

$CD=\|A_{n}-B_{n*}\|^{2}$ (12)

where $B_{n*}$ is the nearest neighbor of the design vector $A_{n}$ in $B$.
Chamfer distance is normalized for coverage and realism with the following
equation:

$CD_{*}=\frac{\frac{1}{N_{A}}\sum_{n=1}^{N_{A}}CD_{n}-CD_{\text{worst-
case}}}{CD_{\text{best-case}}-CD_{\text{worst-case}}}$ (13)

where $N_{A}$ is the number of points in set $A$ and $CD_{*}$ is a normalized
Chamfer distance, being coverage or realism. As a baseline, coverage is
linearly normalized between the best- and worst-case scenarios derived from an
analysis of the dataset. The best-case coverage is the mean nearest neighbor
distance of each hull in the dataset (100% coverage), which is equal to 4.315.
The worst-case coverage is the mean chamfer distance between the centroid of
the dataset hulls and each hull in the dataset, (0% coverage), which is equal
to 26.930. realism is also linearly normalized to be between the minimum and
maximum chamfer distance between the dataset hulls and their nearest neighbor.

To benchmark the DDPMs’ ability to generate feasible samples, two studies on
the feasibility of hulls generated by interpolating between hull design
vectors were conducted. The first study generated sample hulls by finding the
midpoint between two random hull design vectors from the Ship-D dataset. The
second study generated sample hulls by finding the midpoint between a random
hull and its nearest neighbor in the Ship-D dataset. The results of these
studies are provided in the Results Section.

An additional benchmark study was conducted using a tabular generative
adversarial network called CTGAN [50]. The CTGAN was trained to generate
feasible hull designs, only gathering information from the 30,000 feasible
hull designs in the Ship-D dataset. The goal of this benchmark study is to
compare the ability of the CTGAN and the DDPM to generate feasible hulls and
cover the dataset without explicitly identifying feasible or infeasible hull
designs for the model. In training, the CTGAN learns the parametric
information encoded in the design vectors and generates samples to match the
distribution of the dataset samples. The results of this study are included in
Table 3 and Table 4.

When leveraging classifier guidance with a DDPM to generate feasible samples,
the dataset coverage is greatly affected by the same hyperparameter that
influences the feasibility of generated samples.

###  3.3 Denoising Diffusion Probabilistic Models

A denoising diffusion probabilistic model (DDPM) is a generative AI model that
generates new instances of data by denoising random information over many
steps, so that the generated sample falls within the statistical distribution
of the training dataset samples. A tabular DDPM was built and trained on the
ship hull parametric design information from the Ship-D dataset. The DDPM used
to create ShipGen was inspired by the work of Kotelnikov et.al, called TabDDPM
[51]. Unlike popular image-focused diffusion models, this DDPM is trained on
tabular information to generate tabular information. Prior to training, the
parametric design vectors were transformed with a quantile normalizer to re-
scale the distribution of the design parameters to have a normal distribution
with the same mean and variance as the parameters in the dataset. A second
linear transformation re-scaled the bounds so that the range of each parameter
exists between -1 and 1. These transformations ensure that the parametric
design data is fit for the tabular DDPM. Training this model provided a
baseline to verify that the tabular DDPM produces ship hulls with parametric
information within the relative distribution of the Ship-D dataset. The
Results Section provides the results of both the parameter distribution and
feasibility constraint satisfaction of these generated samples.

####  3.3.1 Standard Diffusion Model

The standard DDPM follows the training and sampling algorithms defined in
Table 1 and Table 2. The standard DDPM implicitly learns the statistical
relationships between the parameters in each sample. In the sampling process,
the trained DDPM generates samples that are statistically similar to the
designs in the dataset. There is no extra consideration for design feasibility
or design performance. Since the Ship-D dataset is comprised of randomly
sampled hulls that meet the feasibility criteria, any increase in the DDPMs
ability to produce feasible hulls compared to pure random sampling is due to
the DDPM implicitly learning the relationships between the design parameters
that lead to feasible hull designs.

During training, a feasible design vector is quantile normalized and partially
noised according to the training algorithm in Table 1. Then the DDPM predicts
a noised vector given the timestep embedding and the partially noised vector.
The mean squared difference between the predicted noise vector and a pure
noise vector is the loss of this prediction. The mean squared loss then back-
propagates through the DDPM to update its weights and biases. This process is
repeated for the 30,000 feasible design vectors across one thousand denoising
timesteps in random batches to train the DDPM. Figure 3 illustrates the
training process for one design vector at one timestep.

Figure 3: During training, the DDPM predicts a denoising step, given a
timestep embedding and a partially noised sample design vector.

After training, the standard DDPM can sample new design vectors. The initial
seed for sampling is a Gaussian noise vector of size N, where N is the number
of design parameters. The DDPM denoises this vector one thousand times, taking
into account the timestep embedding at each iteration. After the denoising
process, the final denoised vector is reverse-quantile normalized so that it
becomes a design vector fitting the Ship-D parametric design scheme. This
generated design vector can then be checked for feasibility constraint
satisfaction. If the design is feasible, a point cloud and mesh of the hull is
generated to evaluate the sample’s performance. Figure 4 illustrates the
sampling process for a single design vector.

Figure 4: During sampling, the standard DDPM denoises a vector over one
thousand timesteps, generating a sample design vector that statistically
aligns with the training data.

####  3.3.2 Classifier Guidance for Diffusion Models

An additional method of influencing sample generation towards feasibility
constraint satisfaction is with classifier guidance [36]. Classifier guidance
leverages the gradients of a trained design classifier during the standard
DDPM’s sampling process to influence a design to meet a certain classifier
label. In this case, the classifier label characterizes whether a design is
feasible or infeasible. Here, the 30,000 Ship-D design vectors and the 20,000
infeasible design vectors trained a classifier to predict design feasibility.
At each timestep in the sampling process, the gradient of the trained
classifier for a target class, $f_{\phi}(y|X_{t})$, with respect to the
parameterized design vector, $X_{t}$, is calculated. This gradient is
multiplied by a hyperparameter, $\gamma$, and is added to the sample during
Step 4 of the DDPM sampling algorithm defined in Table 2. A classifier guided
DDPM is created by replacing Step 4 with Equation 14.

$X_{t-1}=\frac{1}{\sqrt{\alpha_{t}}}\left(X_{t}-\frac{1-\alpha_{t}}{\sqrt{1-\bar{\alpha}_{t}}}\epsilon_{\theta}(X_{t},t)\right)+\sigma_{t}(Z(1-\gamma))+\gamma\nabla_{X_{t}}f_{\phi}(y|X_{t})$
(14)

The Results Section provides data from tuning $\gamma$, as it will be shown
that the hyperparameter has an effect on both the likelihood of producing
feasible hull design vectors and on the distribution of these vectors relative
to the Ship-D dataset. Without the need to additionally train the standard
DDPM itself, adding guidance gradients in the sampling process can be
accomplished easily. Figure 5 and Figure 6 illustrate classifier guidance used
in conjunction with performance guidance to generate parameterized hull
designs with high performance. The next subsection details the addition of
more guidance models to generate hulls while considering the hull’s
performance.

####  3.3.3 Performance Guidance for Diffusion Models

Similar to classifier guidance, a neural network trained to predict the
performance of a hull can also be used to guide sample generation. Seven
residual neural networks were trained to predict the normalized performance of
a hull given its parametric design vector. The 30,000 feasible hull designs in
the ShipD dataset were used for the training data. The performance prediction
neural networks all have the same structure: 4 hidden layers with 256 nodes,
where the first hidden layer is added as a residual to the final hidden layer.
The normalization of the performance metrics distributes them over a Gaussian,
improving the prediction accuracy of the neural network.

During sampling, the gradients of the normalized performance prediction from
the neural networks is used to guide the DDPM’s sampling process. The
performance gradient of each of the objectives, $\nabla_{X_{t}}P_{i}(X_{t})$
is multiplied by a weight, $\lambda_{i}$. While generating samples, the
weights of the $\lambda$ values for each performance objective are normalized
so that they are positive and sum to 1.0 for each sample. This way, a broad
spectrum of samples is generated with unique combinations of weighted
influences from the seven performance objectives. Performance guidance is
achieved by replacing Step 4 in the DDPM sampling algorithm with Equation 15.

$X_{t-1}=\frac{1}{\sqrt{\alpha_{t}}}\left(X_{t}-\frac{1-\alpha_{t}}{\sqrt{1-\bar{\alpha}_{t}}}\epsilon_{\theta}(X_{t},t)\right)+\sigma_{t}(Z(1-\gamma))+\gamma\nabla_{X_{t}}f_{\phi}(y|X_{t})-\sum_{i=1}^{7}{\lambda_{i}\nabla_{X_{t}}P_{i}(X_{t})}$
(15)

During sampling, the gradients of both the classifier and performance
prediction models are calculated at each timestep. Weighting these gradients
with $\gamma$ and $\lambda$ influences the impact each individual model has on
the sampling process. The classifier guidance weight, $\gamma$, is set equal
to 0.5 so that both a high degree of sample diversity and sample feasibility
are maintained. Figure 5 shows how guidance from the classifier and
performance prediction models are implemented into the denoising process.

Figure 5: For guided sampling, generated vectors are denoised with the
standard DDPM at each timestep and then influenced with guidance gradients in
each denoisng iteration.

Figure 6 highlights the contributions of both the feasibility classifier and
the performance prediction neural networks in guidance. For performance
guidance, the gradients of the seven performance prediction networks are
calculated for each sampling timestep for $X_{t}$. Then, the gradients are
weighted by their respective $\lambda$ value. The sum of the weighted
gradients is subtracted from the output of the standard DDPM to create the
next partially denoised vector, $X_{t-1}$, in the sampling process. The
gradients are subtracted to follow the scheme of “minimizing” the performance
objectives in generated samples.

Figure 6: Classifier and performance guidance is the result of leveraging
gradients of pre-trained models to influence the denoising process of the
DDPM. The figure highlights the models implemented for this experiment.

After sampling, the generated design vectors are checked for feasibility. The
performance of the feasible generated hulls are then calculated with the same
simulations used to create the original dataset. In the Results Section, Table
5 will showcase the mean normalized performance between the dataset hulls and
the generated hulls, as well as a scale factor between the true performance of
the two sets of hulls. The Results Section also provides data on the fit of
these performance regression networks.

##  4 Results

This section contains the results of the studies described in the Methods
Section. The first subsection provides results on the feasibility and design
spread of parameterized hulls generated with interpolation between existing
hulls in the Ship-D dataset. The second subsection gives the results of
generating feasible hulls using a standard tabular DDPM and with a guided
DDPM. The third subsection provides the results on generating hulls using
performance guidance, including the results of the performance prediction
residual neural networks. The Appendix contains the results of training a
conditional DDPM with both feasible and invalid hulls.

###  4.1 Benchmark Feasibility Constraint Satisfaction Studies

An initial study generating hulls using interpolation methods was conducted to
measure the success rate of generating feasible hulls using the Ship-D
dataset. The first study generated thirty thousand parametric hulls by
interpolating the parameters halfway between two random hulls belonging to the
Ship-D dataset. This interpolation method generated feasible hulls at a rate
of 93.1%, listed in Table 4. The second interpolation method generated hulls
by interpolating between a dataset hull and its nearest neighbor hull. The
second interpolation method generated feasible hulls, with a success rate of
93.8%. Table 3 lists the dataset coverage of these two interpolation methods.
The first interpolation method maintains a normalized coverage ratio of 0.965
compared to the baseline coverage, while the second interpolation method
exceeds the baseline coverage, having a ratio of 1.059. These two
interpolation methods will serve as benchmarks for the feasibility and dataset
coverage analysis on hulls generated with the DDPM.

The CTGAN benchmark study provided a baseline for a trained generative model
to generate feasible hulls implicitly. The CTGAN was trained on the 30,000
feasible hull designs Ship-D dataset to implicitly learn the combinations of
parameter values that define “feasibility”. This study generated thirty
thousand samples of hulls and measured the dataset coverage and feasibility of
these samples. The sample coverage was decreased with the CTGAN, having a
normalized coverage ratio of 0.94 compared to the baseline. This is slightly
reduced from the interpolation benchmark coverage ratios of 0.965 and 1.059.
The coverage measures for the CTGAN study are included in Table 3. Of the
CTGAN generated samples, only 0.7% satisfy all feasibility constraints. This
finding is included in Table 4. The CTGAN is only marginally better than
randomly sampling the design space to create a feasible parametric hull
design. Further analysis of the CTGAN benchmark study is included in the
Discussion Section.

###  4.2 Feasibility Constraint Satisfaction with Tabular Denoising Diffusion
Probabilistic Models

This subsection provides the design feasibility and dataset coverage of
samples generated with different types of DDPMs. The types of DDPMs considered
for sample generation are the standard DDPM and guided DDPM. The following
subsections provide the results for each type of DDPM.

####  4.2.1 Standard DDPM Leads to Good Feasibility and Coverage

A standard DDPM is trained only on the parametric design information from the
dataset. Samples generated from a standard DDPM are made up of the implicit
statistical relationships learned from the parameters in feasible hulls. The
standard DDPM produces feasible hulls 51.1% of the time, as seen in Table 4.
Throughout the Results Section, a two-dimensional principal component analysis
(PCA) is used to illustrate the spread of generated sample hulls compared to
the Ship-D dataset hulls. The PCA is trained on the parametric hull design
data from the Ship-D dataset and is used to transform generated samples into
the two-dimensional PCA for visualization. Figure 7 shows that the standard
DDPM generates samples that maintain most of the dataset coverage, maintaining
a normalized coverage ratio of 0.984. The dataset coverage and feasibility of
samples created with the standard diffusion model are included in Table 3 and
Table 4.

Figure 7: Two-dimensional Principal Component Analysis of the hull
parameterization shows that hulls generated with a standard DDPM maintain most
of the dataset coverage.

####  4.2.2 Guided Denoising Diffusion Probabilistic Model for Enhanced
Feasibility

While the standard DDPM generates feasible hulls with a relatively high
success rate, feasibility can be improved by leveraging guidance from a pre-
trained classifier neural network. The classifier identifies hulls as
satisfying all the constraints or violating at least one of the forty nine
constraints. This classifier network was implemented in the denoising step of
generating samples with a standard DDPM to guide the generation of hulls
towards satisfying the feasibility criteria. As mentioned in the Methods
Section, the degree to which the guidance influences sample denoising is tied
to a hyperparameter, $\gamma$. Figure 8 shows the percentage of generated
feasible samples among generated samples versus $\gamma$. Note that when
$\gamma=0$, the denoising process is the same as the standard DDPM. In
addition to design feasibility, $\gamma$ also affects the dataset coverage of
the generated samples, as shown in Figure 9. As defined in the Methods
Section, generated samples have increased realism with a generated sample by
decreasing the Chamfer distance to its nearest neighbor belonging to the
dataset of designs. Similarly, the generated samples have increasing dataset
coverage with decreasing distance of every dataset point to its nearest
neighbor belonging to the generated samples. Realism and coverage are measured
as the mean normalized chamfer distance between the generated hulls and the
dataset hulls. Table 3 quantifies coverage, showing that increasing $\gamma$
reduces the dataset coverage substantially. To maintain dataset coverage
similar to the interpolation studies, $\gamma$ should be less than or equal to
0.35. To balance both design feasibility and dataset coverage among generated
samples, $\gamma$ is set to 0.5. This way, feasible samples are generated
99.5% of the time and maintain a dataset coverage ratio greater than 0.9. The
remaining plots in this subsection capture a snapshot of samples generated
with guided diffusion with $\gamma$ set to 0.2, 0.35, 0.5, 0.65, 0.80, and
1.0. Table 4 shows the trend of both increasing success in generating feasible
hulls. The PCA charts in Figure 10 illustrate the reduction in coverage with
increasing $\gamma$. Figure 11 shows that classifier guidance has a
significant influence on the feasibility of generated samples throughout the
denoising process.

Figure 8: Hull design feasibility is highly dependent on $\gamma$ in guided diffusion. The percentage of feasible generated hulls is above 90% when $\gamma$ is greater than 0.3 Figure 9: Realism and coverage of the generated samples is strongly affected by $\gamma$. When $\gamma$ is approximately 0.5, the sum of realism and coverage is maximized. Generation Method | Chamfer Distance | Normalized Coverage
---|---|---
(Lower is Better) | (Higher is Better)
Random Dataset Sample | 4.315 | (Baseline) 1.000
Interpolation Study 1 | 5.099 | 0.965
Interpolation Study 2 | 2.976 | 1.059
CTGAN | 5.660 | 0.940
Standard DDPM | 4.672 | 0.984
Guidance: $\gamma=0.2$ | 4.731 | 0.982
Guidance: $\gamma=0.35$ | 5.067 | 0.967
Guidance: $\gamma=0.5$ | 6.002 | 0.925
Guidance: $\gamma=0.65$ | 8.453 | 0.817
Guidance :$\gamma=0.8$ | 13.611 | 0.589
Guidance: $\gamma=1.0$ | 27.054 | -0.005
Table 3: The table provides the dataset coverage for the different sampling methods. These values are normalized between the best- and worst-case scenarios found in the dataset. The standard DDPM covers the dataset better than the CTGAN. By adding guidance to the DDPM, dataset coverage is maintained when $\gamma\leq 0.5$. Figure 10: Two-dimensional Principal Component Analysis of the hull parameterization shows that the dataset coverage is reduced by increasing the hyperparameter, $\gamma$ Generation Method | Feasibility Rate
---|---
Interpolation Study 1 | 0.931
Interpolation Study 2 | 0.938
CTGAN | 0.007
Standard DDPM | 0.511
Guidance: $\gamma=0.2$ | 0.839
Guidance: $\gamma=0.35$ | 0.962
Guidance: $\gamma=0.5$ | 0.995
Guidance: $\gamma=0.65$ | 1.000
Guidance: $\gamma=0.8$ | 1.000
Guidance: $\gamma=1.0$ | 1.000
Table 4: The table shows the fraction of generated samples that are feasible.
The standard DDPM generates feasible samples 73x more often than CTGAN.
Increasing $\gamma$ increases the proportion of feasible samples. All
generated samples are feasible when $\gamma\geq 0.65$ Figure 11: Leveraging
classifier guidance improves the feasibility of generated samples throughout
the denoising process compared to the standard DDPM (no guidance). The
classifier guidance is tuned to $\gamma=0.5$.

###  4.3 Hull Generation with Performance Guided Denoising Diffusion
Probabilistic Model

In addition to generating feasible samples, guidance can also generate high-
performing parametric hull designs. The following subsections provide the
results from training performance prediction neural networks on seven
objectives and the results from measuring and simulating hulls generated using
multi-objective performance guidance.

####  4.3.1 Performance Prediction Training

Using the performance data from the Ship-D dataset, seven residual neural
networks were trained to predict the performance of the hulls given the
parameterized design vector. Table 5 summarizes the results of the training,
using $R^{2}$ as a measure of the goodness of fit for these neural networks.
Figure 12 shows the plot of the regression prediction versus the simulation
calculation for the aggregate wave drag measurement. Figure 13 shows the same
plots for the remaining six performance metrics. The blue dashed line in these
figures represents the perfect regression prediction, exactly aligning with
the simulation calculation. The wave drag coefficient, surface area, and
volume prediction neural networks have high $R^{2}$ fits and hug the blue
dashed line closely. The MaxBox and Gaussian Curvature predictions have lower
$R^{2}$ values, however, they are still sufficient for use with performance-
guided DDPM sampling [44].

Performance Objective | Training Fit: [$R^{2}$]
---|---
Wave Drag $C_{w}$ | 0.973
Surface Area50% | 0.983
Surface Area100% | 0.982
Volume50% | 0.988
Volume100% | 0.986
Volume${}_{\text{MaxBox}}$ | 0.784
Gaussian Curvature | 0.765
Table 5: The performance prediction neural networks have high goodness-of-fits
to the training data, which enables performance guidance in DDPM sampling with
these objectives. Figure 12: Comparison of the neural network prediction to
the simulation value (ground truth) across the dataset for aggregate wave
drag. This regression had a $R^{2}$ equal to 0.973. A perfect prediction
($R^{2}=1$) is shown by the blue dashed line. Figure 13: Comparison of the
neural network prediction to the simulation value (ground truth) across the
dataset for the remaining six performance metrics. All of these performance
metric regressions are well resolved. A perfect prediction ($R^{2}=1$) is
shown by the blue dashed line.

####  4.3.2 Multi-objective Guided Performance Hull Generation

The seven performance prediction neural networks were implemented with the
guided DDPM to generate 1000 hulls. Each objective in these samples was
randomly weighted so that the influence of each of the performance metrics
varied among the samples. The feasibility classifier guidance was tuned to
$\gamma$ = 0.5 to maintain some variability and dataset coverage among the
samples and to not overpower the performance guidance. The samples generated
with performance guidance were feasible 83.9% of the time. The PCA plot of
these generated samples is shown in Figure 14. These samples do not cover the
sample range of the design space as the Ship-D dataset hulls.

After sampling, the 839 feasible hull designs were simulated and measured with
the seven performance objectives. The mean and standard deviation of the
performance metrics among the Ship-D dataset and the generated samples are
provided in Table 6. These metrics are scaled according to Equations 5\- 11,
so it is important to note that these values exist on a logarithmic scale.
Among these samples, the wave drag coefficients and displaced volumes showed
significant improvements in their performance. These improvements were at the
expense of a relative increase in the surface area and Gaussian curvature. The
generated samples have wave drag coefficients for any single speed/draft
condition that is, on average, 91.4% lower than the average wave drag
coefficients of the Ship-D dataset hulls. For the displaced volumes, these
generated hulls have an average 114x increase in displaced volume in the
bottom 50% of the hull depth and an average 47.9x increase in the total
displaced volume of the hull. The generated hulls have, on average, 2.1x more
total surface area, 4.4x more surface area in the bottom 50% of the hull, and
1.51x more double curvature compared to the Ship-D hulls. This is not
desirable. The MaxBox metric saw a small, but negligible decrease in the
volume ratio of the hull belonging to the MaxBox, where the generated samples
have an average 5.2% reduction in the MaxBox volume ratio compared to the
hulls in the dataset. This result, however, is far overshadowed by the
substantial increase in total available volume in the hull.

In addition to measuring the performance of these hulls, a .stl mesh and 5
images of each hull were created for visual analysis. Figure 15 shows nine of
these hulls. A major difference in these generated hulls is their higher
length-to-beam ratios compared to the Ship-D hulls seen in Figure 2.

Performance Objective | Ship-D Dataset | Generated Samples | Scaled Factor
---|---|---|---
($Y_{*gen}$) | ($Y_{*DS}$) | ($Y_{gen}/Y_{DS}$)
Mean | Std. | Mean | Std.
Wave Drag $C_{w}$ | -73.40 | 17.38 | -107.45 | 23.90 | 0.086
Surface Area50% | -1.71 | 0.53 | -1.07 | 0.19 | 4.365
Surface Area100% | -1.09 | 0.45 | -0.76 | 0.19 | 2.138
Volume50% | 4.78 | 0.81 | 2.72 | 0.59 | 114.815
Volume100% | 3.80 | 0.62 | 2.12 | 0.43 | 47.863
Volume${}_{\text{MaxBox}}$ | -0.407 | 0.010 | -0.384 | 0.072 | 0.948
Gaussian Curvature | 2.43 | 0.529 | 2.61 | 0.24 | 1.514
Table 6: The table shows the mean and standard deviation of the performance
metrics across the Ship-D dataset hulls and the feasible generated hulls. The
generated hulls saw a 47.9x increase in total volume and a 91.4% relative
decrease in wave drag coefficient across all speeds. Figure 14: Two-
dimensional Principal Component Analysis of the hull parameterization shows
that the performance-guided DDPM with $\gamma=0.5$ leads to sample coverage
that is skewed relative to the distribution of the Ship-D dataset as a result
of the performance guidance. Figure 15: A selection of hulls generated with
multi-objective guided performance generation. Notice the relative slenderness
of the hulls leading to drastically reduced drag coefficients relative to the
dataset hulls.

##  5 Discussion

The following subsections provide insight into the results of the studies
presented. The first subsection reviews the successful generation of feasible
designs with the different DDPMs. The second subsection provides an analysis
of the dataset coverage of the DDPMs, with special attention made to the
$\gamma$ hyperparameter used in the classifier-guided DDPM. The third
subsection analyzes the performance of the hulls generated with performance
guidance. The Appendix contains a discussion on the conditional DDPM study.

###  5.1 Feasibility Constraint Satisfaction

Of the different DDPMs, only the classifier-guided DDPM showed to successfully
sample feasible parameterized hull designs with the same success rate as the
interpolation study. The standard DDPM, while only producing a feasible hull
approximately half of the time, was able to do so only by implicitly learning
the statistical relationships between design parameters in feasible hulls. The
standard DDPM’s feasibility success rate of 51.1% success rate is
significantly higher than the success rate of 0.66% seen by randomly sampling
the design space. In the comparison between CTGAN and DDPM, a significant
performance gap was identified, particularly in constraint modeling. The CTGAN
benchmark study revealed that the standard DDPM is two orders of magnitude
more successful at generating feasible hulls than the CTGAN model. While the
dataset coverage study demonstrated CTGAN’s ability to produce parameterized
vectors representing the dataset statistics, leading to high coverage, it
struggled to generate these design vectors with combinations of parameter
values that result in high feasibility. This suggests that CTGAN may face
challenges in implicitly learning the statistical correlations between the
parameters to the extent that DDPM does, underlying the need for a
comprehensive examination. Such an examination, backed by empirical and
theoretical analysis, is essential to delve deeper into the observed
challenges and understand the inherent model characteristics or learning
behaviors causing the performance disparity. Without this thorough analysis,
making definitive claims regarding the observed differences remains
speculative. A deeper comparative study on constraint satisfaction across
different deep generative models is needed to make such claims.

Finally, the guided classifier guidance showed that by tuning the $\gamma$
hyperparameter, the rate of feasible hull generation varied. In order to meet
the feasibility benchmark of 93% feasible hulls, the $\gamma$ should be set
between 0.35 and 1. Table 4 also shows that simply by including a small
influence of guidance ($\gamma=0.2$), the success of generating feasible hulls
improves significantly compared to the standard DDPM. For performance
guidance, the success rate of generating feasible hulls was 83.9%, which is
lower than the benchmark target, but this comes at the benefit of producing
high-performing hulls, even with $\gamma=0.5$. This reduction in the
feasibility satisfaction rate is due to the added influence of the performance
guidance, which does not consider feasibility when generating samples. Further
work in hyperparameter tuning can lead to higher success rates in feasible and
high-performing hull generation. Overall, a DDPM with classifier guidance can
be used to generate feasible design vectors with a high degree of success in
an extremely complicated design space.

###  5.2 Dataset Coverage

Among the DDPMs, there were varying degrees of dataset coverage. Visually
comparing the design space coverage of the Ship-D dataset hulls and the
generated samples proved to be reasonably effective at analyzing the dataset
coverage. Among the two interpolation methods, interpolation between a random
design vector and its nearest neighbor was the best benchmark for dataset
coverage as the diversity in the generated samples relied on the diversity of
the randomly selected design vectors. The standard DDPM was also effective at
maintaining dataset coverage as it was trained to generate sample hulls that
are representative of the hulls in the dataset, as seen in Figure 7. The
standard DDPM also maintains dataset coverage better than CTGAN. Finally, the
DDPM with classifier guidance heavily relied on $\gamma$ for maintaining
dataset coverage. As $\gamma$ is increased to produce feasible hulls with a
higher success rate, the dataset coverage of the feasible samples decreases.
Figure 9 was made to quantify the coverage and realism of the samples in
addition to visually inspecting the PCA distribution of the generated samples.
The best balance of maintaining a high feasibility success rate and dataset
coverage was at $\gamma\approx 0.35-0.5$, as shown in Table 3. With increasing
$\gamma$ the generated samples lose diversity and cluster around the center of
the PCA distribution. While the performance-guided DDPM was not intended to
generate designs that cover the dataset, Figure 14 suggests that these
generated samples do maintain some diversity and dataset coverage. Overall,
the classifier-guided DDPM is shown the maintain a large breadth of dataset
coverage with careful tuning of its hyperparameters.

###  5.3 Performance Guidance

The DDPM with performance guidance produced hulls with mixed results. The
performance guidance created hulls with an average 91.4% lower drag and 47.9x
higher displaced volumes than the hulls from the original dataset. This is a
highly desirable outcome of the performance guided sample generation. This
outcome, however, came at the expense of the generated hulls having increased
surface area and double curvature, which is not ultimately desirable. Further
in-depth analysis, such as life cycle costs assessments, is needed to weigh
the impact these results would have on a real, scaled-up ship instead of a
non-dimensionalized parametric hull shape. Future work can also consider
different tuning of the $\gamma$ and $\lambda$ hyperparameters in the
performance guidance of the hulls. The random $\lambda$ weights used for this
study were likely not scaled appropriately for the magnitude of the gradients
of the different performance metrics. This could explain why the aggregate
wave drag coefficient and displaced volume metrics were improved drastically;
while MaxBox was relatively unaffected, and surface area and Gaussian
curvature were increased. As these performance gradients were calculated using
the weights of the regression neural networks, it is possible that the
magnitudes of gradients between the different regression models
disproportionately affected the net influence of guidance on design
generation. This is rather apparent when comparing the significant, yet
desired, increase in hull volume and the undesired increase in surface area.
These two competing performance objectives should have maintained some balance
of improvement across the DDPM generated samples; however, the displaced
volume performance objectives saw overwhelming improvement that included a
relative detriment to the surface area objectives. Nonetheless, the
significant improvements in wave drag coefficient and displaced volume have
strong economic prospects on the cost of operating a ship: the cost of fuel
(drag) and the ability to generate revenue (carry cargo). Leveraging a DDPM
with performance guidance has been shown to generate hull shapes considering
multiple objectives that can lead to huge cost savings to ship operators.
Future work will explore generating hulls with specific performance
requirements to find explicit applicability of guided DDPMs to generate hulls
tied to real cost savings in ship design.

In addition to the performance of the generated hulls, these generated samples
share more semblance of real ship hulls than do the Ship-D hulls. Figure 15
shows a sample of the generated hulls for visual inspection. These hulls have
higher length-to-beam ratios than the Ship-D hulls and have streamlines that
are more akin to real ship hulls.

##  6 Conclusion

The goal of this work was to generate ship hulls using a denoising diffusion
probabilistic model that considers the performance of the hull as part of the
design generation. First, by training a DDPM on a dataset of randomly
generated feasible hull designs, the DDPM was able to implicitly learn
statistical relationships between the design parameters to generate feasible
parametric hulls. Then, by incorporating guidance from performance prediction
models trained on the same dataset of hulls, the DDPM was able to generate
high-performing hulls with only information learned from the low-performing
hulls in the dataset.

One critical aspect of leveraging generative AI on parametric design
information is to generate feasible designs. A standard diffusion model can
generate feasible hulls approximately 51.1% of the time. While this is much
better than the success rate seen by randomly generating hull
parameterizations ($\sim 0.66\%$), standard diffusion models do not yield
feasible hulls at rates similar to interpolation methods ($\sim 93\%$). By
leveraging the gradients of a classifier during the sampling process, the
standard DDPM saw increased feasibility among generated hull designs. The
classifier guidance in the denoising process was influenced by a tunable
parameter, $\gamma$. By varying $\gamma$, the guided DDPM was able to generate
hulls with different success rates of feasibility at the expense of design
coverage across the dataset. It was found that $\gamma$ = 0.5 led to high hull
feasibility (99.5%) with limited detriment to dataset coverage.

As guidance was shown to be the most successful and versatile method of
producing feasible hull designs, guidance was also used to generate high-
performing designs. Seven neural networks were trained to predict the
different performance metrics given a hull’s design vector. The gradients of
these performance prediction neural networks were implemented for performance
guidance in the DDPM’s sampling process. The aggregate wave drag coefficients
of the generated hulls had a 91.4% mean reduction in drag coefficient compared
to the Ship-D hulls. The total displaced volume of the generated samples was
on average 47.9x larger than the mean displaced volume of the Ship-D dataset
hulls. However, surface area, Gaussian curvature, and MaxBox of the generated
samples did not improve compared to the hulls in the dataset. Overall, the
significantly reduced drag coefficients and increased displaced volume are
extremely beneficial to ship design. These performance metrics dictate how
expensive a voyage is (fuel costs due to drag) and how much cargo the ship can
carry (how much money can be made on a voyage). With this work, the economic
prospect of leveraging generative AI to design ship hulls is shown.

###  6.1 Future Work

Future work will focus on the continued study of generative AI to generate
ship hulls and other systems on a ship. Immediate future work will look at
continued tuning of the $\lambda$ weights during performance guidance to
generate hulls that have improved performance in all of the objectives. To
accomplish this, a study on hyperparameter tuning and guidance gradients will
be performed. Further, work in leveraging guided diffusion to generate high-
performing ship hulls with specific performance targets will be explored. The
goal of this future work is to generate hulls that consider specific user-
defined constraints (such as dimensions, volume, speed, etc.) with high
performance. This way, the design of ship hulls using DDPMs could be analogous
to similarly structured online text-to-image DDPMs, such as Dall-E [38] and
Stable Diffusion [39]. In addition to generating ship hulls, further work in
DDPMs to generate other aspects of ship design will be explored, such as
structural design generation, packing arrangements, machinery, and outfitting
on a ship.

##  7 Acknowledgements

This research is funded by the United States’ Department of Defense, Office of
Naval Research, via the National Defense Science and Engineering Graduate
(NDSEG) Fellowship program. The authors would like to thank MIT Supercloud for
providing some of the computational resources needed to perform this work
[52]. The data and code used for the ShipGen project are available at
https://decode.mit.edu/projects/ShipGen/

##  Appendix A APPENDIX: Conditional Diffusion Model

The appendix contains an additional study performed by training a conditional
DDPM. Conditional DDPMs are similar to the standard DDPM, however, their
structure includes extra layers that embed extra information in the training
and sampling process. The extra information in this study is a sample’s
classification of being feasible or invalid. The following subsections detail
the Methods, Results, and a Discussion on leveraging a conditional DDPM to
generate hull designs.

###  A.1 Methods

A tabular DDPM was built with additional conditioning embedding layers to
influence the model to produce designs that satisfy the feasibility
constraints. In addition to the 30,000 parametric hulls in the Ship-D dataset,
20,000 design vectors were randomly generated that do not meet at least one of
the feasibility constraints. The feasible ship hull design vectors and the
infeasible vectors were labeled respectively. While training the tabular DDPM,
the feasibility label was provided to the conditional embedding layer with its
respective design vector in training. The goal of the conditioning is to use
that additional label to influence the sampling process to guide the tabular
DDPM to produce designs that satisfy the feasibility constraints. To modify
the standard DDPM to become a conditional DDPM modify the gradient step (Step
5) in the DDPM training algorithm with Equation 16, where C is the conditional
embedding layer that is concatenated to the first layer of the standard DDPM.
In sampling, replacing Step 4 of the sampling algorithm with Equation 17,
where C is the same conditional embedding layer concatenated to the first
layer of the DDPM. The DDPM training and sampling algorithms are provided in
Table 1 and Table 2, respectively. The Results Section provides the sample
distribution and constraint satisfaction of ship hull design vectors generated
with the conditioned tabular DDPM.

$\nabla_{\theta}||\epsilon-\epsilon_{\theta}(\sqrt{\bar{a_{t}}}X_{0}+\sqrt{1-\bar{a_{t}}}\epsilon,t,C)||^{2}$
(16)

$X_{t-1}=\frac{1}{\sqrt{\alpha_{t}}}(X_{t}-\frac{1-\alpha_{t}}{\sqrt{1-\bar{\alpha_{t}}}}\epsilon_{\theta}(X_{t},t,C))+\sigma_{t}Z$
(17)

###  A.2 Results

The conditional DDPM was trained on the thirty thousand Ship-D hulls that
satisfy all forty nine constraints and twenty thousand invalid samples that
violate varying numbers of the forty nine constraints. Two separate sample
generations were performed with the conditional DDPM. The first study tried to
intentionally generate feasible hulls. Figure 16 shows that the conditional
DDPM produces feasible hulls 39.8% of the time. The PCA plot in Figure 17
shows that the spread of the generated samples is within the bounds of the
dataset. This conditional DDPM can also intentionally generate invalid
samples. Although this is not useful in design work, the generation of invalid
samples with the conditional DDPM shows that the model can distinguish between
“positive” and “invalid” samples in sample generation. Figures 18 and 19
showcase the results of hulls that were intentionally created to violate the
hull parameterization’s feasibility criteria.

Figure 16: Bar graph showing the number of individuals generated with an
increasing number of constraint violations. Leveraging a conditional DDPM to
generate hull parameterization leads to hull parameterizations that satisfy
all constraints 39.8% of the time Figure 17: Two-dimensional Principal
Component Analysis of the hull parameterization shows that hulls generated
with a conditional DDPM maintain most of the dataset coverage; however, there
is less design feasibility among the samples than desired. Figure 18: Bar
graph showing the number of individuals generated with an increasing number of
constraint violations. Leveraging a conditional DDPM to generate invalid hull
parameterization leads to the generation of samples that violate a large
spread of a number of design feasibility constraints. Figure 19: Two-
dimensional Principal Component Analysis of the hull parameterization shows
that invalid hull samples cover the dataset space and more, as some of the
parameters are intentionally sampled outside of the feasible range of some
parameters.

###  A.3 Discussion

The conditional DDPM was not able to produce feasible hulls at a rate that
meets the interpolation study benchmarks. It was surprising to see how poorly
the conditional DDPM performed given that the model was trained with
distinction between feasible and infeasible design vectors. The conditional
model, however, could intentionally create infeasible hulls at will, which
further adds to the surprise of the poor feasible hull generation. It seems
that the conditional DDPM struggled to distinguish the statistical
relationships between the parameters in the feasible and infeasible hulls
during training. This result did not yield improvement over the standard DDPM
in generating feasible hulls, but it was excellent in generating a large
diversity of hulls with at least one constraint violation. This could be
useful in future studies concerning design feasibility.

##  Appendix B APPENDIX: Parametric Hull Design: Parameters and Constraints

This appendix provides documentation for the 45 hull design parameters and the
49 algebraic feasibility constraints. Figure 20 lists the design parameters,
describes the features, provides the ranges for each parameter in the Ship-D
Dataset. Figure 21 lists each of the 49 algebraic constraints, and describes
the conditions to satisfy each constraint. By satisfying all 49 algebraic
constraints, the hull will satisfy the two feasibility criteria:

  1. 1.

The hull is watertight, meaning that there are no holes on its surface.

  2. 2.

The hull surface is not self-intersecting.

Figure 20: List of the hull design parameters, their scaling, and their value
ranges within the dataset. Figure 21: List of the parametric hull design
constraints and a description of their satisfaction criteria

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