# Data-driven discovery of dimensionless numbers and scaling laws from
experimental measurements

Xiaoyu Xie
Department of Mechanical Engineering
Northwestern University
Evanston, IL 60208
<EMAIL_ADDRESS>
&Wing Kam Liu
Department of Mechanical Engineering
Northwestern University
Evanston, IL 60208
<EMAIL_ADDRESS>
&Zhengtao Gan 11footnotemark: 1
Department of Mechanical Engineering
Northwestern University
Evanston, IL 60208
<EMAIL_ADDRESS>
Corresponding authors<EMAIL_ADDRESS>(W. Liu),
<EMAIL_ADDRESS>(Z. Gan)

###### Abstract

Dimensionless numbers and scaling laws provide elegant insights into the
characteristic properties of physical systems. Classical dimensional analysis
and similitude theory fail to identify a set of unique dimensionless numbers
for a highly-multivariable system with incomplete governing equations. In this
study, we embed the principle of dimensional invariance into a two-level
machine learning scheme to automatically discover dominant and unique
dimensionless numbers and scaling laws from data. The proposed methodology,
called dimensionless learning, can reduce high-dimensional parametric spaces
into descriptions involving just a few physically-interpretable dimensionless
parameters, which significantly simplifies the process design and optimization
of the system. We demonstrate the algorithm by solving several challenging
engineering problems with noisy experimental measurements (not synthetic data)
collected from the literature. The examples include turbulent Rayleigh-Bénard
convection, vapor depression dynamics in laser melting of metals, and porosity
formation in 3D printing. We also show that the proposed approach can identify
dimensionally-homogeneous differential equations with minimal parameters by
leveraging sparsity-promoting techniques.

_K_ eywords Dimensional analysis $\cdot$ Buckingham’s Pi theorem $\cdot$
Physics-informed machine learning $\cdot$ Fluid mechanics $\cdot$ Sparse
identification $\cdot$ Additive manufacturing

##  1 Introduction

All physical laws can be expressed as dimensionless relationships with fewer
dimensionless numbers and a more compact form [1]. A dimensionless number is a
power-law monomial of some physical quantities [2]. There is no physical
dimension (such as mass, length, or energy) assigned to a dimensionless
number, which provides a property of scale invariance, i.e., the dimensionless
numbers are invariant when length scale, time scale, or energy scale of the
system varies. More than 1200 dimensionless numbers have been discovered in an
extremely wide range of fields, including physics and physical chemistry,
fluid and solid mechanics, thermodynamics, electromagnetism, geophysics and
ecology, and engineering [3]. There are several significant advantages to
describe a physical process or system using dimensionless numbers. Using
dimensionless numbers can considerably simplify a problem by reducing the
number of variables that describe the physical process, thereby reducing the
amount of experiments (or simulations) required to understand and design the
physical system. For example, Reynolds number ($\mathrm{Re}$) is a well-known
dimensionless number in fluid mechanics, named after Osborne Reynolds who
investigated the fluid flow through pipes in 1883 [4]. The Reynolds number is
defined as a power law of four physical quantities, i.e., the fluid density,
the average fluid velocity, the diameter of the pipe, and the dynamic fluid
viscosity. The flow characteristics (laminar or turbulent) in a pipe is best
determined by the $\mathrm{Re}$, rather than the four individual dimensional
quantities. Moreover, the scale-invariance property of the dimensionless
numbers plays a critical role in the similitude theory [5]. Many small-scale
experiments have been designed to understand and predict the behaviors of the
full-scale applications in the engineering of aerospace [6], nuclear [7], and
marine [8], where the full-scale applications are typically extremely
expensive even dangerous. All the dimensionless numbers should be identical
between the small-scale and full-scale experiments which yields the perfect
similarities of geometry, dynamic, and kinematic between the two scales.
Furthermore, dimensionless numbers are ratios of two forces, energies, or
mechanisms. Thus, they are physically interpretable and can provide
fundamental insights into the behavior of complex systems. For example, The
Péclet number ($\mathrm{Pe}$) represents the ratio of the convection rate of a
physical quantity by the flow to the gradient-driven diffusion rate, which
enables the order-of-magnitudes analysis for transport phenomena of a process.

Despite the scientific significance and the wide use of dimensionless numbers,
it is still challenging to discover new dimensionless numbers and their
relationships (i.e., scaling laws) from experiments, especially for a complex
physical system without complete governing equations. A traditional solution
is dimensional analysis [2] based on Buckingham $\pi$ theorem [9], which
provides a systematic approach to examine the units of a physical system and
form a set of dimensionless numbers that satisfy the principle of dimensional
invariance [10]. However, the dimensional analysis has several well-known
limitations. First, the derived dimensionless numbers are not unique. The
Buckingham $\pi$ theorem [9], from the viewpoint of mathematics, provides a
linear subspace of exponents that produces dimensionless numbers. Any basis
for the subspace is equally valid. Thus, it fails to identify the
dimensionless numbers, given a particular choice of basis, that are dominant
for the physical system. Second, the mathematical relation between
dimensionless numbers (i.e., scaling law) remains unknown by using dimensional
analysis alone. A common approach to establish the scaling law is to leverage
the results of the dimensional analysis with experimental measurements of the
physical system. The experimental measurements are transformed into the
dimensionless numbers obtained by dimensional analysis and fitted a high-
dimensional response surface to represent the scale-invariant relationship.
However, since the dimensional analysis does not provide unique dimensionless
numbers, this procedure is very time-consuming and highly relies on the
experience of domain experts to select a set of appropriate dimensionless
numbers by a lot of trial-and-error.

Theses limitations could be overcame by integrating the dimensional analysis
with advanced data science and artificial intelligence (AI). Mendez and
Ordonez proposed an algorithm called SLAW (i.e., Scaling LAWs) to identify the
form of a power law from experimental data (or simulation data) [11]. The
proposed SLAW combines dimensional analysis with multivariate linear
regressions. This approach has been applied to some engineering areas, such as
ceramic-to-metal joining [11] and plasma confinement in Tokamaks [12].
However, for the sake of simplification, the proposed algorithm assumes the
relationship between the dimensionless numbers obeys a power law, which is
invalid in many applications. Constantine, Rosario, and Iaccarino proposed a
rigours mathematical framework to estimate unique and relevant dimensionless
groups [13, 14]. Active subspace methods are connected to dimensional
analysis, which reveals that all physical laws are ridge functions [14]. Their
method is applicable to idealized physical system meaning that the experiments
can be conducted for arbitrary values of the independent input variables (or
dependent input variables with a known probability density function), and
noises or errors in the input and output are negligible.

In this study, we propose a mechanistic data-driven approach, called
dimensionless learning, in which the principle of dimensional invariance
(i.e., physical laws cannot depend on an arbitrary choice of basic units of
measurements [1]) is embedded into a two-level machine learning scheme to
automatically discover dominant dimensionless numbers and scaling laws from
noisy experimental measurements of complex physical systems. We add a physical
constraint of the dimensional invariance into the learning algorithm, which
steers the learning toward scale-invariant and physically-interpretable low-
dimensional patterns of complex high-dimensional systems. We demonstrate the
proposed approach by solve multiple challenging problems in science and
engineering. The examples include turbulent Rayleigh-Benard convection, vapor
depression dynamics, and porosity formation in 3D printing. The experimental
datasets are collected from the literature. Furthermore, we demonstrate the
potential of the proposed approach by combining it with the sparsity-promoting
techniques (such as SINDy [15]) to identify dimensionless differential
equations from data, which involves natural coordinates of spatial and
temporal variables.

##  2 Results

###  2.1 Turbulent Rayleigh-Bénard convection

In this section, we demonstrate the proposed dimensionless learning using the
example of a classical fluid mechanics problem: turbulent Rayleigh-Bénard
convection. The goal is to rediscover the Rayleigh number (Ra) directly from
experimental measurements. The Ra is named after Lord Rayleigh who
investigated a non-isothermal buoyancy-driven flow in 1916 [16], which is
named Rayleigh-Bénard convection now. The turbulent Rayleigh-Bénard convection
is a paradigm system to study turbulent thermal flow, occurring in a planar
horizontal layer of fluid in a container heated from below. The internal fluid
could develop complex turbulent dynamics due to the effects of buoyancy, fluid
viscosity, and gravity (Fig. 1a).

Figure 1: The proposed dimensionless learning demonstrated on the turbulent
Rayleigh-Benard convection. (a) A schematic of the Rayleigh-Benard convection
[17] with related physical quantities. (b) Collected experimental
measurements. (c) Constructed dimension matrix $\boldsymbol{D}$ of the input
variables. (d) First-level of the two-level optimization scheme for training
the coefficients $\boldsymbol{\gamma}$ with respect to the computed basis
vectors. (e) Second-level of the two-level optimization scheme for optimizing
the unknown coefficients $\boldsymbol{\beta}$ in the representation learning.
(f) Explored dimensionless space with a measure of $R^{2}$. The location with
the maximum $R^{2}$ marked as a yellow star is related to the classical
Rayleigh number. (g) Identified one-dimensional scaling law between Nu and Ra.
(h) Discovered linear scaling law between two identified dimensionless
numbers.

Experimentally, the heat flux through the container, $q$, can be measured,
which depends on the height of the container $h$, the temperature difference
between the top and bottom surfaces $\Delta T$, the gravitational acceleration
$g$, and the properties of the fluid including the thermal conductivity
$\lambda$, the thermal expansion coefficient $\alpha$, the viscosity $\nu$,
and the thermal diffusivity $\kappa$. To obtain a causal relationship, we need
to specify the dependent (i.e., output) and independent (i.e., input)
variables from the physical quantities describing the system. To simplify the
demonstrate, we assume the form of the output variable as the Nusselt number
$\mathrm{Nu}=\frac{qh}{\lambda\Delta T}$ (a more general case using $q$ as the
output will be presented later), and a list of physical quantities
$\boldsymbol{p}$ as input variables. The causal relationship to be determined
can be represented as

$\mathrm{Nu}=\frac{qh}{\lambda\Delta T}=f(h,\Delta
T,\lambda,g,\alpha,\nu,\kappa)=f(\boldsymbol{p})$ (1)

It is a high-dimensional parametric space. To explore it, we collect an
experimental dataset of the turbulent Rayleigh-Bénard convection from two
different articles [18, 19], including 182 experiments with various input
variables and corresponding output measurements (Fig. 1b). Tons of machine
learning models can fit the data. However, a lot of those are black box
models, such as neural networks, without good interpretability and physical
insights. Alternatively, we aim to identify a low-dimensional scale-invariant
scaling law that best represent the dataset. In the scaling law, the products
of powers of the input variables $\boldsymbol{p}$ form a dimensionless number
$\mathrm{\Pi}$. Thus, the causal relationship can be rewritten as

$\mathrm{Nu}=f_{1}(\mathrm{\Pi})$ (2) $\mathrm{\Pi}=h^{w_{1}}\Delta
T^{w_{2}}\lambda^{w_{3}}g^{w_{4}}\alpha^{w_{5}}\nu^{w_{6}}\kappa^{w_{7}}$ (3)

where $\boldsymbol{w}=[w_{1},...,w_{7}]^{T}$ are the powers that produce the
dimensionless number and are to be determined. In this example, we assume that
there is only one input dimensionless number governing the process. An
algorithm for determining the number of the dimensionless numbers required
from data is provided in Supplementary Information Section 1.3.

To embed the physical constraint of dimensional invariance, we conduct the
dimensional analysis, i.e., the powers $\boldsymbol{w}=[w_{1},...,w_{7}]^{T}$
need to satisfy the a linear system of equations

$\boldsymbol{Dw}=0$ (4)

where $\boldsymbol{D}$ is the dimension matrix of the input variables (Fig.
1c). Each column of the dimension matrix is the dimension vector of the
corresponding variable. The dimension vector represents the exponents of the
physical quantity with respect to the fundamental dimensions. It is worthy
noting that there are only seven fundamental dimensions in nature: mass
[$\mathrm{M}$], length [$\mathrm{L}$], time [$\mathrm{T}$], temperature
[$\mathrm{\Theta}$], electric current [$\mathrm{I}$], luminous intensity
[$\mathrm{J}$], and amount of substance [$\mathrm{N}$] [20]. All the other
dimensions are power-law monomials of the fundamental dimensions [1]. In this
example, we use four fundamental dimensions, i.e., [$\mathrm{M}$],
[$\mathrm{L}$], [$\mathrm{T}$], and [$\mathrm{\Theta}$] (Fig. 1c). The
dimension matrix includes the physical dimensions of the input variables. The
linear system of equations $\boldsymbol{Dw}=0$ guarantees the power-law
monomial of the input variables (Eqn. 3) is dimensionless [21]. Since the
linear system is under-determined (i.e., the number of unknown variables is
more than the number of equations), there are infinitely many solutions,
indicating that there are infinitely many forms of dimensionless numbers
obtained from the dimensional analysis. Furthermore, we can represent the
solutions of the linear system (Eqn.4) as linear combinations of three basis
vectors $\boldsymbol{w_{b1}}$, $\boldsymbol{w_{b2}}$, and
$\boldsymbol{w_{b3}}$

$\boldsymbol{w}=\gamma_{1}\boldsymbol{w_{b1}}+\gamma_{2}\boldsymbol{w_{b2}}+\gamma_{3}\boldsymbol{w_{b3}}$
(5)

where $\boldsymbol{\gamma}=[\gamma_{1},\gamma_{2},\gamma_{2}]^{T}$ are the
coefficients with respect to the three basis vectors in this case. The number
of basis vectors is equal to the number of the input variables (seven in this
case) minus the rank of the dimension matrix (four in this case). This formula
aligns with the Buckingham $\pi$ theorem [9]. Since the basis vectors can be
computed based on Eqn. 4 (an algorithm for computing basis vectors is provided
in Supplementary Information Section 1.5), the basis vectors’ coefficients (or
just called basis coefficients) are the unknowns to be determined. A set of
computed basis vectors for this case are

$\boldsymbol{w_{b1}}=[0,0,0,0,0,1,-1]^{T}$ (6)
$\boldsymbol{w_{b2}}=[0,1,0,0,1,0,0]^{T}$ (7)
$\boldsymbol{w_{b3}}=[3,0,0,1,0,-2,0]^{T}$ (8)

Once the basis coefficients $\gamma_{1}$, $\gamma_{2}$, and $\gamma_{3}$ are
obtained, the form of the dimensionless number $\mathrm{\Pi}$ could be
determined by Eqns. 3 and 5 (Fig. 1d).

To determine the values of basis coefficients using the collected dataset, a
model representing the scaling relation between the input and output
dimensionless numbers is required, which introduces another set of unknown
parameters $\boldsymbol{\beta}$ (i.e., the representation learning shown in
Fig. 1e). We use a 5-ordered polynomial model in this case (more advanced
models such as tree-based models and deep neural networks are optional
depending on the complexity of the problem to be solved, see Section 2.3 and
Supplementary Information Section Section 3 for more demonstrations). The
polynomial model can be expressed as

$\mathrm{Nu}=\beta_{0}+\beta_{1}\Pi+\beta_{2}\Pi^{2}+...+\beta_{5}\Pi^{5}$ (9)

where $\boldsymbol{\beta}=[\beta_{0},\beta_{1},...,\beta_{5}]^{T}$ are
polynomial coefficients that represents the scaling relation.

We design an iterative two-level optimization scheme to determine the two sets
of unknown parameters in the regression problem, i.e., the basis coefficients
$\boldsymbol{\gamma}$ and polynomial coefficients $\boldsymbol{\beta}$. The
optimization scheme includes multiple interactive steps. At each step, we
adjust the first-level basis coefficients $\boldsymbol{\gamma}$ while fix the
second-level polynomial coefficients $\boldsymbol{\beta}$, and then optimize
the second-level polynomial coefficients $\boldsymbol{\beta}$ while fix the
first-level basis coefficients $\boldsymbol{\gamma}$. This process is repeated
until the result is converged, i.e., the values of $\boldsymbol{\gamma}$ and
$\boldsymbol{\beta}$ are unchanged. There are several advantages of the
proposed two-level approach over a single-level approach combining the two
sets of unknowns together during the optimization. We can use different
optimization methods and parameters (such as the learning rate) for these two-
level models to significantly improve the efficiency of the optimization. More
importantly, we can utilize the physical insights to inform the learning
process. The first-level basis coefficients $\boldsymbol{\gamma}$ have clear
physical meaning, which is related to the powers that produce the
dimensionless number. Thus, those values have to be rational numbers to
maintain the dimensional invariance. Moreover, the range of them are typically
not large. Note that absolute values of coefficients in most of the
dimensionless numbers and scaling laws are less than four [1]. To leverage
those physical insights or constraints, we design several methods for
optimizing the first-level basis coefficients, including a simple grid search
(used in this section) and a pattern search that is much more efficient
(Supplementary Information Section 3.2). For the second-level coefficients, we
conduct multiple standard representation learning methods, including
polynomial regression used in this section, tree-based extreme gradient
boosting (XGBoost [22]) used in Section 2.3, and general gradient descent
method (Supplementary Information Section 3.1). Details of the two-level
optimization framework is provided in Supplementary Information Section 3.

We illustrate the first-level grid search for $\gamma_{2}$ and $\gamma_{3}$
ranging from -2 to 2 with a interval of 1/100 (Fig. 1f). We fix $\gamma_{1}$
as one to avoid too large or too small powers of the dimensionless number
$\mathrm{\Pi}$. For each $\boldsymbol{\gamma}$ in the dimensionless space, the
polynomial coefficients $\boldsymbol{\beta}$ are trained based on the
collected data. The data is separated into 80% training set and 20% test set.
The coefficient of determination ($R^{2}$) of test set is shown in Fig. 1f as
a measure of learning performance. We can identify an unique point with the
maximum $R^{2}$ (0.999) from Fig. 1f (marked as a yellow star), where
$\gamma_{1}$=$\gamma_{2}$=$\gamma_{3}$ = 1. Using these optimized basis
coefficients, the expression of the dominant dimensionless number can be
identified as

$\mathrm{\Pi}=\frac{g\alpha\Delta Th^{3}}{\nu\kappa}$ (10)

This form is identical as the classical Rayleigh number, indicating that the
proposed dimensionless learning can rediscover the well-known dimensionless
number directly from data. Moreover, we demonstrate that for the given
parameter list the Rayleigh number is the unique dimensionless number to best
fit the dataset because there is only one global maximum of $R^{2}$ within the
dimensionless space (Fig. 1f). The log-log scaling relation between Ra and Nu
is a simple one-dimensional pattern where all the datapoints collapse onto a
single curve (Fig. 1g).

The proposed dimensionless learning can deal with dimensional output variable
as well. A combination of the input variables with the same dimension as the
output variable can be searched to non-dimensionalize the output variable (the
detailed algorithm for output non-dimensionalization is provided in
Supplementary Information Section 1.3). Using the heat flux $q$ as the output
variable (instead of Nu used in the previous example), the dimensionless space
is expanded and thus more dominant dimensionless numbers and scaling laws can
be found. We discover a new set of dimensionless numbers to best represent
($R^{2}$ = 0.999) the collected experimental measurements. More interestingly,
the identified log-log scaling relation between the dimensionless numbers is
almost linear (Fig. 1h). This finding could lead to new physical insights into
the complex turbulent Rayleigh-Bénard dynamics.

###  2.2 Vapor depression dynamics in laser-metal interaction

Another challenging problem in the application of dimensionless learning is
the laser-metal interaction dynamics. People have been curious about the
physical responses of a metallic material due to a high-power laser
irradiation since 1964 when Patel invited an electric discharge
$\mathrm{CO_{2}}$ laser [23] that was dramatically scaled up in power shortly
after. During the laser-metal interaction, a vapor-filled depression (termed a
keyhole) frequently forms on a puddle of liquid metal melted by the laser. The
keyhole is caused by vaporization-induced recoil pressure, and its dynamics is
inherently difficult to understand because of its complex dependence upon many
physical mechanisms but important to be able to quantify because it is highly
related to energy absorption and defect formation in many industrial and
military applications, such as laser-based materials processing and
manufacturing [24], high-energy laser weapons [25], and aerospace laser-
propulsion engine [26].

Recently, high-quality in situ experimental data on keyhole dynamics became
available via high-speed X-ray imaging [27]. Using X-ray pulses, images of the
keyhole region inside the metals can be recorded with micrometer spatial
resolution [28]. The keyhole depth $e$ can be measured from the X-ray images
(Fig. 2a), and it depends on different materials and several process
parameters such as the effective laser power $\eta P$, the laser scan speed
$V_{s}$, and laser beam radius $r_{0}$. We collect a dataset of keyhole X-ray
images from literatures, including ninety-one experiments with various process
parameters and three different materials: a titanium alloy (Ti6Al4V), an
aluminium alloy (Al6061), and a stainless steel (SS316) [29, 24]. We represent
a material using a set of material properties: the thermal diffusivity
$\alpha$, the material density $\rho$, the heat capacity $C_{p}$, the
difference between melting temperature and ambient temperature $T_{l}-T_{0}$.
Therefore, the casual relationship can be expressed as

$e=f(\eta P,V_{s},r_{0},\alpha,\rho,C_{p},T_{l}-T_{0})$ (11)

Figure 2: Discover dimensionless numbers governing keyhole dynamics in laser-
metal interaction. (a) A illustrative X-ray image of keyhole morphology [29].
The dataset including X-ray imaging experiments on three different materials.
(b) Global optimum of the dimensionless space, which represents the scaling
law between the keyhole aspect ratio and the identified keyhole number using
dimensionless learning. (c) Local optimum of the dimensionless space. (d)
Dimensionless space using $\gamma_{2}$ and $\gamma_{3}$ as coordinates. The
values of $R^{2}$ indicate the learning performance for the corresponding
dimensionless number in the dimensionless space. The values of $R^{2}$ less
than -1 are shown as -1.

We can use the dimensionless learning described in the previous section to
extract a low-dimensional scale-free relation from the parameter list. The
dimension matrix $\boldsymbol{D}$ and computed basis vectors
$\boldsymbol{w_{b1}}$, $\boldsymbol{w_{b2}}$, and $\boldsymbol{w_{b3}}$ of
this example are provided in Supplementary Information Section 1.6. We firstly
demonstrate the grid search ranging from -2 to 2 with a interval of 1/100 for
the first-level optimization, and 5-ordered polynomial regression for the
second-level optimization. We fix $\gamma_{1}$ as 0.5 and normalize the output
variable as the keyhole aspect ratio $e^{*}=\frac{e}{r_{0}}$, which is a
widely used dimensionless parameter to represent the keyhole characteristic
[30]. By searching the dimensionless parametric space, we can find one local
optimum in term of the $R^{2}$ criteria marked as a blue star ($R^{2}=0.64$)
in Fig. 2d. The expression of the dimensionless number $\Pi=\frac{\rho
C_{p}(T_{l}-T_{0})V_{s}^{1.5}r_{0}^{2.5}}{\sqrt{\alpha}\eta P}$ is computed
based on the basis coefficients $\gamma_{2}=\gamma_{3}=-1$. However, the
datapoints are scattered shown in Fig. 2c, indicating the dimensionless number
located at the local maximum of the dimensionless space is not a good scaling
parameter for this problem. The global optimum of the dimensionless space,
where $\gamma_{2}=\gamma_{3}=1$, provides a much better scaling behavior with
a 0.98 $R^{2}$ score (Fig. 2b). The dominant dimensionless number emerged in
the keyhole dynamics is

$\Pi=\frac{\eta P}{(T_{l}-T_{0})\pi\rho C_{p}\sqrt{\alpha V_{s}r_{0}^{3}}}$
(12)

This dimensionless number is identified directly from data, and it has the
same form as the newly discovered keyhole number Ke [29] (or sometimes called
normalized enthalpy [31]), which can be derived from heat transfer theory.
Even if we use dimensional variable $e$ as the output, the dimensionless
learning algorithm still confirms that the form of the keyhole number (i.e.,
Eqn. 12) is the unique and dominant for controlling the value of keyhole
aspect ratio. Details of the procedure and results are provided in
Supplementary Information Section 2.1. Using the identified dimensionless
number, a simple scaling law emerges to control the keyhole aspect ratio,
which simplifies the original high-dimensional problem into an univariate
scaling law as

$e^{*}=0.12\mathrm{Ke}-0.30$ (13)

Providing a sufficient parameter list is critical for the dimensionless
learning. If one or more important quantities are omitted, it is impossible to
achieve a high $R^{2}$ for the learning and identify the right form of the
dimensionless number(s). We demonstrate in Supplementary Information Section
3.1 that if we assume a parameter list omitting the thermal diffusivity
$\alpha$, the maximum $R^{2}$ is below 0.80 over the dimensionless space,
which is much less than the value for the sufficient parameter list (i.e.,
Eqn. 11). Another scenario often happened in the applications is that we
consider more quantities than sufficient, including some irrelevant or
unimportant quantities. We demonstrate this scenario in Supplementary
Information Section 3.2 by considering one more quantity, e.g., the latent
heat of melting $L_{m}$ or the difference between boiling temperature and
ambient temperature $T_{l}-T_{0}$, in the parameter list. The form of the
keyhole number still can be identified in the scenario. Moreover, there are a
few more dimensionless numbers, which consist of the added quantity, providing
as high $R^{2}$ as the keyhole number. It implies that more experiments are
needed to select the distinguished one from the identified dimensionless
numbers.

We provide two efficient algorithms, i.e., gradient-based and pattern search-
based two-level optimization schemes, in Supplementary Information Section 4
to improve the efficiency of the optimization used in this section. These
algorithms are especially helpful to explore a high-dimensional parametric
space including a lot of parameters to describe the physical system and
several dimensionless numbers to construct the low-dimensional pattern.

###  2.3 Porosity formation in 3D printing of metals.

Three-dimensional (3D) printing or additive manufacturing is a disruptive
technology of making three-dimensional solid objects from a digital file,
which provides a new paradigm to change manufacturing [32]. In metal 3D
printing, metallic parts are built layer by layer via local melting and
(re)solidification of metallic powders by a laser or electron beam. 3D
printing enables remarkable freedom for designing local geometrical and
compositional features. However, this process has large numbers of parameters
to be considered when making a part, and trends to produce defects such as
internal porosity during the process if inappropriate process parameters are
used (Fig. 3a).

To extract elegant insights into the complex behavior of the porosity
formation in 3D printing, we collect an experimental dataset collected from
six independent studies [33, 34, 35, 36, 37, 38], including ninety-three 3D
printed parts with measured porosity volume fraction and various process
parameters. Three kinds of materials were used: a titanium alloy (Ti6Al4V), a
nickel-based alloy (Inconel 718), and a stainless steel (SS316L). The porosity
volume fraction $\Phi$ depends on many process parameters and materials used
in the experiments, which can be expressed as

$\Phi=f(\eta_{m}P,V_{s},d,\rho,C_{p},\alpha,T_{l}-T_{0},H,L)$ (14)

where $\eta_{m}P$ is the effective laser power input, $V_{s}$ is the laser
scan speed, $d$ is the laser beam diameter, $\rho$ is the material density,
$C_{p}$ is the material heat capacity, $\alpha$ is the thermal diffusivity,
$T_{l}-T_{0}$ is the difference between melting temperature and ambient
temperature, $H$ is the hatch spacing between the two adjacent laser scans,
$L$ is the layer thickness of the metallic powders. It is a high-dimensional
relation, and thus hard to understand and visualize. Traditionally, some
combined parameters, such as energy density $\frac{\eta_{m}P}{V_{s}d^{2}}$,
are used to simplify this relation. However, the $R^{2}$ score of a polynomial
model using energy density as input is very low (0.13) as shown in Fig. 3b,
indicating that an universal physical relation, which is valid for different
materials and processing conditions, cannot be built by using the energy
density alone because it is not a scale-free parameter. The form of the
relation has to be modified when the energy scale is changed in the
experiments with varying process parameters or materials.

Figure 3: Discover dimensionless numbers governing porosity formation in 3D
printing. (a) A schematic of a 3D printed metal part with internal porosity
defect [39]. The dataset including X-ray micro-computed tomography (micro-CT)
measurements on three different materials. (b) Porosity measurements with
various energy density, a traditional combined parameter to correlate porosity
with process parameters. (c) Identified 2D scaling relation combining both
lack of fusion and keyhole porosity with two discovered dimensionless numbers
(d) Another identified 2D scaling relation with higher $R^{2}$ score. The
reduced parametric space can be easily visualized and interpreted, while the
original high-dimensional problem is hard since the porosity is govern by nine
parameters.

We apply the dimensionless learning to this challenging engineering problem
and discover some dominant dimensionless numbers that provide an universal
physical relation remained accurate for all the experimental conditions. The
dimension matrix and computed basis vectors of this example are provided in
Supplementary Information Section 1.6. The two-level optimization applied in
this example includes a pattern search for the first-level and a XGBoost
method to capture the second-level relationships (Supplementary Information
Section 2.2). We find that two dimensionless numbers are necessary to
represent the dataset since there is no high value of the $R^{2}$ score (e.g.,
greater than 0.5) can be achieve if we set only one dimensionless number in
the training. A systematic algorithm that determines the number of
dimensionless numbers required to govern a physical system is provided in
Supplementary Information Section 1.3.

We identify several low-dimensional patterns with the propriety of the scale-
free from data. They can achieve high $R^{2}$ for both the training and test
sets (a table summarizing the identified dimensionless numbers are provided in
Supplementary Information Section 2.2). Interestingly, we identify another
dimensionless number (besides the keyhole number), which has been discovered
by theory-driven approach [33, 29]: the normalized energy density NED (Fig.
3c). It can be expressed as

$\mathrm{NED}=\frac{\eta_{m}P}{V_{s}\rho C_{p}(T_{l}-T_{0})HL}$ (15)

The normalized energy density NED represents the ratio of laser energy input
within the powder layer to sensible heat of melting. This dimensionless number
is govern the lack of fusion porosity in metal 3D printing, which is a well-
know porosity mechanism due to insufficient laser energy input to fully melt
the powder material [40]. The other dimensionless number
$\frac{\eta_{m}PH}{V_{s}\rho C_{p}(T_{l}-T_{0})d^{3}}$ in Fig. 3c is related
to another porosity mechanism, i.e., keyhole porosity, caused by bubbles of
gas gotten trapped underneath the surface during the fluctuation of a unstable
keyhole [28]. This dimensionless number is a modified normalized enthalpy
product, i.e., $\mathrm{NEP}\cdot\frac{H}{d}$, where the normalized enthalpy
product NEP is proven to be related to the keyhole instability, and a unstable
keyhole with high NEP could lead to keyhole pores [31]. The NEP can be
expressed as

$\mathrm{NEP}=\frac{\eta_{m}P}{V_{s}\rho C_{p}(T_{l}-T_{0})d^{2}}$ (16)

Since the NEP is derived from the single-track laser scan condition [31], the
modified term $\frac{H}{d}$ emerges to consider the effect of multiple-track
scanning. Another identified low-dimensional pattern that can achieve even
higher $R^{2}$ (0.95) is shown in Fig. 3d. It is also based on NED and NEP,
but with different modification terms. By reducing high-dimensional parameter
space, much less experiments would be required to determine optimal processing
conditions and parameters for new materials and thus ease the Edisonian burden
endemic among current metal 3D printing practitioners.

##  3 Discussions

The two-level optimization scheme makes the dimensionless learning very
flexible. The first-level scheme guarantees the dimensional invariance (or
called dimensional homogeneity), and thus many representation learning methods
can be used for the second-level scheme to capture scale-free relationships.
We demonstrate polynomial and tree-based XGBoost [22] methods in the previous
sections. However, the capability of the dimensionless learning can be
improved by leveraging more methods, including deep neural networks [41],
symbolic regression [42], and Bayesian machine learning [43]. Another highly
promising candidate is the sparse identification of nonlinear dynamics (SINDy)
[15], which can identify ordinary differential equation (ODE) or partial
differential equation (PDE) from data. By integrating the proposed
dimensionless learning with SINDy, the forms of dimensionally-homogeneous
differential equations can be discovered directly from data, which provides
even more insights and interpretations of the physical system. We demonstrate
this using a simple example of a spring-mass-damper system in Supplementary
Information Section 5. Using the proposed two-level optimization with SINDy to
analysis the time series of the spring-mass-damper system, we can obtain not
only the governing equation, but several key physical parameters such as the
natural time scale (i.e., the inverse of natural frequency), natural length
scale and dimensionless damping coefficient (Fig. 4). The dimensionless form
of the governing equation involves the reduced number of variables from six to
three, which is the minimalistic representation of the system.

Figure 4: Recover dimensionless numbers and the governing equation of a
spring-mass-damping system. In the first step, we generate three sets of data
based on the governing equation with different combinations of parameters
(spring constant $k$, mass $m$, initial displacement $\delta$, and damping
coefficient $c$). After analyzing the parametric space, we can obtain the
dimension matrix $\boldsymbol{D}$. In the second step, a two-level
optimization is developed to find the best basis coefficients and sparse
coefficients sequentially. In third step, we can obtain the best dimensionless
numbers and dimensionless governing equation after model selection using
$R^{2}$ score and Bayesian information criterion (BIC) score. Detailed
information is shown in Supplementary Information Section 5.

One important benefit for embedding dimensional invariance as a physical
constraint into machine learning algorithms is that it reduces the learning
space by eliminating the strong dependence of the dimension for different
variables. The models can represent features as dimensionless numbers and
transform the datapoints into a low-dimensional pattern that is insensitive to
units and scales. It significantly improve the interpretability of the
representation learning since the dimensionless numbers are physically-
interpretable, which enable qualitative and quantitative analysis of the
systems of interest. It is worthy noting that other kinds of invariances have
been embedded into machine learning algorithms and achieved prominent
successes, such as translation invariance in convolutional neural network
(CNN), time translation invariance in recurrent neural network (RNN), and
permutation invariance in graph neural network (GNN).

The proposed dimensionless learning enables a systematic and automatic
learning of scale-free low-dimensional laws from high-dimensional data. It can
be applied to a vast number of physical, chemical, and biological systems to
discover new or modify existing dimensionless numbers. Furthermore, it can be
combined with other data-driven methods such as SINDy to discover
dimensionless differential equations and scaling laws. In material science,
the identified compact mathematical expressions provide simple transition
rules that translate optimal process parameters from one material (or existing
materials) to another (or new materials). The dimensionless learning is able
to reduce complex, highly multivariate problem spaces into descriptions
involving just a few dimensionless parameters with clear physical meanings,
which is particularly useful for the engineering problems including many
adjustable parameters with various dimensions or units, such as advanced
materials processing and manufacturing, microfluidic flow control for precise
drug delivery, solar energy systems design, and modern financial market
analysis.

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