# Discovering New Interpretable Conservation Laws as Sparse Invariants

Ziming Liu  Department of Physics, Institute of Artificial Intelligence and
Fundamental Interactions, Massachusetts Institute of Technology, Cambridge,
USA  Patrick Obin Sturm  Department of Earth Sciences, University of Southern
California, Los Angeles, USA  Saketh Bharadwaj  Department of Chemical
Engineering, Indian Institute of Technology, Hyderabad, India  Sam Silva
Department of Earth Sciences, University of Southern California, Los Angeles,
USA  Max Tegmark  Department of Physics, Institute of Artificial Intelligence
and Fundamental Interactions, Massachusetts Institute of Technology,
Cambridge, USA

###### Abstract

Discovering conservation laws for a given dynamical system is important but
challenging. In a theorist setup (differential equations and basis functions
are both known), we propose the Sparse Invariant Detector (SID), an algorithm
that auto-discovers conservation laws from differential equations. Its
algorithmic simplicity allows robustness and interpretability of the
discovered conserved quantities. We show that SID is able to rediscover known
and even discover new conservation laws in a variety of systems. For two
examples in fluid mechanics and atmospheric chemistry, SID discovers 14 and 3
conserved quantities, respectively, where only 12 and 2 were previously known
to domain experts.

##  I Introduction

Conservation laws are important concepts in physics, yet discovering them is
challenging. Ideally, the set of discovered conserved quantities should be
complete, independent and interpretable. Although several attempts have been
made to automate the discovery process with machine learning [1, 2, 3, 4, 5,
6, 7, 8], their complicated setups and blackbox nature make it hard to
guarantee all these desirable properties. This paper considers a simple yet
realistic setup where all these desirable properties can be met.

“Discovering conservation laws” can mean wildly different things for
experimentalists, computationalists and theorists, as shown in Table 1. Most
prior work [1, 2, 3, 4, 5, 6] takes on the experimentalist setup, assuming
knowledge of neither the differential equations nor the form of conservation
laws. [7] takes the computationalist setup, assuming knowledge of differential
equations. This work explores the theorist setup, where both differential
equations and basis functions of conservation laws are known. Admittedly, this
setup is simpler than the other two, but is still realistic when theorists
have the differential equations at hand and have educated guesses about the
basis functions that may span the conserved quantities.

We propose Sparse Invariant Detector (SID), an algorithm that reveals
conservation Laws. SID is incredibly simple in the sense that it only requires
linear algorithms (except for sparsification), so the results are much more
trustworthy and interpretable than blackbox machine learning methods. Note
that SID does not replace us human scientists, but rather acts as a helpful
assistant: while humans need to input basis functions (i.e., formulating
hypotheses) to SID, SID is good at computing conserved quantities (i.e.,
testing hypotheses) based on the given prompt. In this manner, human
scientists can focus on the more creative part of the job, while SID does the
technical and tedious work. This paper gives two examples where new conserved
quantities are successfully discovered by SID: one in fluid mechanics, and
another in atmospheric chemistry (see Table 2). In the former one, although
the new conserved quantities are somewhat expected in hindsight, humans alone
may need several more months to find them. In the latter one, a new conserved
quantity is found, which was unintended in the design of the model.

Table 1: Three setups of conservation law discovery Setup | Experimentalist | Computationalist | Theorist
---|---|---|---
Model-based | No | Yes | Yes
Known basis | No | No | Yes
Independence | Partial | Yes | Yes
Completeness | No | Partial | Yes
Interpretability | Partial | Partial | Yes
Reference |  [1, 2, 3, 4, 5, 6] |  [7] | This work

##  II Method

Figure 1: SID workflow: Inputs are differential equations, basis functions and sample points; Outputs are a set of conserved quantities which are complete, independent and interpretable. Table 2: The number of conserved quantities known to experts and discovered by SID | Fluid (2D) | Fluid (3D) | Atmosphere
---|---|---|---
Known | 8 | 12 | 2
SID |  8 (simpler) | 14 | 3

###  II.1 Problem setup

We consider a first-order differential equation
$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$, where
$\dot{\mathbf{x}}\equiv\frac{d\mathbf{x}}{dt}$,
$\mathbf{x}\equiv(x_{1},\cdots,x_{d})\in\mathbb{R}^{d}$ is the state vector
and $\mathbf{f}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is a vector field. This ODE
formulation is more general than it seems: (1) Hamiltonian systems are
subsumed as $\mathbf{x}\equiv(\mathbf{x}^{\prime},\mathbf{p}^{\prime})$; (2)
Higher-order differential equations (e.g.,
$\ddot{\mathbf{y}}=\mathbf{f}(\mathbf{y})$) are included as
$\mathbf{x}\equiv(\mathbf{y},\dot{\mathbf{y}},\cdots)$; (3) Partial
differential equations (PDEs) become ODEs once discretized.

A conserved quantity is a scalar function
$H(\mathbf{x}):\mathbb{R}^{d}\to\mathbb{R}$, such that its value remains
constant along any trajectory obeying
$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ 111The constant itself can be
different for different trajectories.. As proved in [7], a necessary and
sufficient condition for $H(\mathbf{x})$ being a conserved quantity is $\nabla
H(\mathbf{x})\cdot\mathbf{f}(\mathbf{x})=0$, since

$0=\dot{H}=\nabla H(\mathbf{x})\cdot\dot{\mathbf{x}}=\nabla
H(\mathbf{x})\cdot\mathbf{f}(\mathbf{x}).$ (1)

Given the differential equation $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$, we
hope to find a set of conserved quantities $\\{H_{1},\cdots,H_{c}\\}$ which
satisfies these three properties:

  * •

Independence: they are functionally independent, i.e.,
$g(H_{1},\cdots,H_{c})=0\Rightarrow g=0$.

  * •

Completeness: any conserved quantity $H$ (in the function space spanned by
basis functions) can be expressed by them, i.e., there exists $g$ such that
$H=g(H_{1},\cdots,H_{c})$.

  * •

Interpretability: conserved quantities can be written as (hopefully simple)
symbolic formulas.

###  II.2 Solving the linear equation and completeness

The prior work [7] parametrizes the conserved quantities
$H_{\bm{\theta}}(\mathbf{x})$ as neural networks and learns the parameters
$\bm{\theta}$ to make $|\nabla
H_{\bm{\theta}}(\mathbf{x})\cdot\mathbf{f}(\mathbf{x})|^{2}$ close to zero.
However, neural network training may get stuck at local minima, so the results
are not reliable. Moreover, the parameterized conserved quantities are not
immediately interpretable.

We consider a simpler setup. Assume that we know $H_{\bm{\theta}}(\mathbf{x})$
to be a linear combination of $K$ predefined basis functions
$b_{i}(\mathbf{x})\ (1\leq i\leq K)$ such that

$H_{\bm{\theta}}(\mathbf{x})=\sum_{i=1}^{K}\theta_{i}b_{i}(\mathbf{x})\equiv{\bm{\theta}}\cdot\mathbf{b}(\mathbf{x}),$
(2)

where only $\bm{\theta}\in\mathbb{R}^{K}$ are learnable parameters to be
determined and the vector $\mathbf{b}:\mathbb{R}^{d}\to\mathbb{R}^{K}$ defines
the basis functions. Since the number of conserved quantities can exceed one,
we define a set of parameters
$\bm{\Theta}\equiv\\{\bm{\theta}_{1},\bm{\theta}_{2},\cdots\\}$ and their
corresponding functions
$H_{\bm{\Theta}}\equiv\\{H_{\bm{\theta}}|\bm{\theta}\in\bm{\Theta}\\}$. As
shown in FIG. 1, Eq. (2) is equivalent to a neural network whose last linear
layer contains the only trainable parameters. With Eq. (2), the conservation
condition Eq. (1) becomes:

$\mathbf{g}(\mathbf{x})^{T}{\bm{\theta}}=0,\quad\mathbf{g}(\mathbf{x})\equiv(\nabla\mathbf{b}(\mathbf{x}))\mathbf{f}(\mathbf{x}),$
(3)

which is a linear equation of ${\bm{\theta}}$. Remember that in our setup,
both $\mathbf{b}(\mathbf{x})$ and $\mathbf{f}(\mathbf{x})$ are known, so
$\mathbf{g}(\mathbf{x})\equiv(\nabla\mathbf{b}(\mathbf{x}))\mathbf{f}(\mathbf{x})$
is known as well. In practice, we draw $P$ random points $\mathbf{x}_{i}\
(1\leq i\leq P)$ from phase space. A solution ${\bm{\theta}}$ should make Eq.
(3) hold for all $\mathbf{x}_{i}$, or more explicitly,

$\underbrace{\begin{pmatrix}g_{1}(\mathbf{x}_{1})&g_{2}(\mathbf{x}_{1})&\cdots&g_{K}(\mathbf{x}_{1})\\\
g_{1}(\mathbf{x}_{2})&g_{2}(\mathbf{x}_{2})&\cdots&g_{K}(\mathbf{x}_{2})\\\
\vdots&\vdots&&\vdots\\\
g_{1}(\mathbf{x}_{P})&g_{2}(\mathbf{x}_{P})&\cdots&g_{K}(\mathbf{x}_{P})\\\
\end{pmatrix}}_{\mathbf{G}}\underbrace{\begin{pmatrix}\theta_{1}\\\
\theta_{2}\\\ \vdots\\\ \theta_{K}\\\
\end{pmatrix}}_{\bm{\theta}}=\mathbf{0},$ (4)

which is simply linear regression. In practice, we apply singular value
decomposition to $\mathbf{G}=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^{T}$, where
$\mathbf{U}\in\mathbb{R}^{P\times P}$ and $\mathbf{V}\in\mathbb{R}^{K\times
K}$ are orthogonal matrices, $\mathbf{\Sigma}\in\mathbb{R}^{P\times K}$ is
diagonal with singular values $0\leq\sigma_{1}\leq\sigma_{2}\leq\cdots$. We
count $\sigma_{i}$ as effectively zero if $\sigma_{i}<\epsilon\equiv 10^{-8}$.
The number of vanishing singular values, denoted $M$, is equal to the
dimensionality of the solution space (null space), which is spanned by the
first $M$ columns of $\mathbf{V}^{T}$, denoted
${\bm{\Theta}}^{(1)}\equiv({\bm{\theta}}_{1}^{(1)},{\bm{\theta}}_{2}^{(1)},\cdots,{\bm{\theta}}_{M}^{(1)})\in\mathbb{R}^{K\times
M}$. The linear structure obviously gives completeness (in the space spanned
by basis functions), since any solution ${\bm{\theta}}$ can be expressed as a
linear combination of columns of ${\bm{\Theta}}^{(1)}$.

###  II.3 Interpretability

In order to gain more interpretability, we want $\bm{\Theta}^{(1)}$ to be
sparse. Note that if $\mathbf{R}\in\mathbb{R}^{M\times M}$ is an orthogonal
matrix, the columns of $\bm{\Theta}^{(2)}=\bm{\Theta}^{(1)}\mathbf{R}$ also
form a set of complete and orthogonal solutions. Therefore we can encourage
sparsity by finding and applying the orthogonal matrix that minimizes the
following:

$\mathbf{R}^{*}=\underset{\mathbf{R}^{T}\mathbf{R}=\mathbf{I}}{\rm argmin}\
||{\bm{\Theta}^{(1)}}\mathbf{R}||_{1},\quad{\bm{\Theta}^{(2)}}={\bm{\Theta}^{(1)}}\mathbf{R}^{*},$
(5)

where $||\mathbf{M}||_{1}\equiv\sum_{ij}|M_{ij}|$ denotes the $L_{1}$-norm of
a matrix $\mathbf{M}$, encouraging sparsity.

###  II.4 Independence

Although columns of $\bm{\Theta}^{(2)}$ are linearly independent,
$H_{\bm{\Theta}^{(2)}}$ are not guaranteed to be functionally independent.
Take the 1D harmonic oscillator $\mathbf{x}=(x,p)$, for example. Restricting
basis functions to be polynomials in $x$ and $p$ up to the 4th order, there
are two solutions:

$H_{\bm{\theta}_{1}}=x^{2}+p^{2},H_{\bm{\theta}_{2}}=H_{\bm{\theta}_{1}}^{2}=x^{4}+2x^{2}p^{2}+p^{2},$
(6)

where $\bm{\theta}_{1}$ and $\bm{\theta}_{2}$ are orthogonal (hence
independent), but $H_{\bm{\theta}_{2}}=H_{\bm{\theta}_{1}}^{2}$, so they are
not functionally independent. Consequently, we want a subset of
$\bm{\Theta}^{(2)}$, denoted $\bm{\Theta}^{(3)}$, such that
$H_{\bm{\Theta}^{(3)}}$ is both independent and complete (i.e., can generate
$H_{\bm{\Theta}^{(2)}}$). The first question is: how many elements, denoted
$c$, does $\bm{\Theta}^{(3)}$ have? As shown in [7], $c$ is equal to the rank
of the following matrix:

$\mathbf{A}=\begin{pmatrix}\frac{\partial H_{\bm{\theta}_{1}}}{\partial
x_{1}}&\frac{\partial H_{\bm{\theta}_{2}}}{\partial
x_{1}}&\cdots&\frac{\partial H_{\bm{\theta}_{M}}}{\partial x_{1}}\\\
\frac{\partial H_{\bm{\theta}_{1}}}{\partial x_{2}}&\frac{\partial
H_{\bm{\theta}_{2}}}{\partial x_{2}}&\cdots&\frac{\partial
H_{\bm{\theta}_{M}}}{\partial x_{2}}\\\ \vdots&\vdots&&\vdots\\\
\frac{\partial H_{\bm{\theta}_{1}}}{\partial x_{d}}&\frac{\partial
H_{\bm{\theta}_{2}}}{\partial x_{d}}&\cdots&\frac{\partial
H_{\bm{\theta}_{M}}}{\partial x_{d}}\\\ \end{pmatrix}$ (7)

which hinges on the fact that gradients of functionally dependent functions
are linearly dependent 222Suppose $H_{1}$ is dependent on $H_{2},H_{3}$, then
there exists a function $g$ such that $H_{1}=g(H_{2},H_{3})$. Taking gradients
on both es gives $\nabla H_{1},\frac{\partial g}{\partial H_{2}}\nabla
H_{2}+\frac{\partial g}{\partial H_{3}}\nabla H_{3}$, which means $\nabla
H_{1}$ is linear combination of $\nabla H_{2}$ and $\nabla H_{3}$. So
$\mathbf{A}=[\nabla H_{1},\nabla H_{2},\nabla H_{3}]$ has rank two (if $H_{2}$
and $H_{3}$ are independent). In practice, applying singular value
decomposition to $\mathbf{A}$ gives
$\mathbf{A}=\mathbf{U}^{\prime}\mathbf{\Sigma}^{\prime}\mathbf{V}^{\prime T}$,
where $\mathbf{U}^{\prime}\in\mathbb{R}^{d\times d}$ and
$\mathbf{V}^{\prime}\in\mathbb{R}^{M\times M}$ are orthogonal matrices, and
$\mathbf{\Sigma}^{\prime}$ is a diagonal matrix with singular values
$s_{1}\geq s_{2}\geq\cdots\geq 0$. We count $s_{i}$ as effectively non-zero if
$s_{i}>\epsilon=10^{-8}$. The number of non-zero singular values is equal to
${\rm rank}(\mathbf{A})$, which is in turn equal to $c$. After determining
$c$, we aim to obtain $\bm{\Theta}^{(3)}$ by selecting $c$ elements from
$\bm{\Theta}^{(2)}$. The selection process is as follows: (1) We assign each
conserved quantity a complexity score (based on entropy 333For a vector
${\bm{\theta}}$, we can associate it to a probability distribution
$p_{i}\equiv|\theta_{i}|/(\sum_{i}|\theta_{j}|)$, whose entropy is
$S=-\sum_{i}p_{i}{\rm log}p_{i}$.) and sort them from the simplest to the most
complex. (2) Starting from an empty set $\bm{\Theta}^{(3)}$, looping over
element $\bm{\theta}\in\bm{\Theta}^{(2)}$, we add $\bm{\theta}$ to
$\bm{\Theta}^{(3)}$ if $H_{\bm{\theta}}$ is independent of
$H_{\bm{\Theta}^{(3)}}$ (functions already added), until $\bm{\Theta}^{(3)}$
contains $c$ elements.

##  III Results

To better illustrate SID, we apply it to three dynamical systems.

(a)

(b)

Figure 2: Three-species Lokta-Volterra equation. SID correctly discovers that:
(a) there are $4$ CQs (top) in polynomials up to 3rd order, yet only 2 of them
(bottom) are independent. (b) coefficients of conserved quantities
$\bm{\Theta}^{(i)}(i=1,2,3)$, with more interpretability. Figure 3: (a)(b) SID
discovers 3 independent conserved quantities in the ozone photochemical
production model. (c) The coefficients of these linear conserved quantities.
The first two correspond to the known carbon and nitrogen conservation, while
$CQ_{3}$ is identified for the first time. Figure 4: The evolution of
concentrations in simulation. Under both conditions, $CQ_{3}$ is well
conserved.

###  III.1 Systems biology

Our first application is from systems biology. The Lotka–Volterra equations
(LV hereafter) describe how population of many species evolve in time via
inter-species interactions. We study this particular equation:

$\dot{x}=x(y-z),\dot{y}=y(z-x),\dot{z}=z(x-y)$ (8)

with two known conserved quantities $H_{1}=x+y+z$ and $H_{2}=xyz$. We define
basis functions to be all polynomials up to 3rd order (including $K=19$ terms,
shown in FIG. 2(b)). We draw $P=100$ data points from the standard Gaussian
distribution, i.e.,
$\mathbf{x}_{i}\equiv(x_{i},y_{i},z_{i})\sim\mathcal{N}(0,\mathbf{I}_{3\times
3})\ (1\leq i\leq P)$.

Within the function space spanned by the basis functions, $M=4$ conserved
quantities are found, since FIG. 2(a) top shows that there are 4 vanishing
singular values $\sigma_{i}$. The coefficients of four CQs are somewhat mixed,
shown in FIG. 2(b) top. After sparsification (Eq. (5)), the coefficients
become less entangled, shown in FIG. 2(b) middle, although their represented
conserved quantities are still dependent. Among the 4 CQs, only $c=2$ are
independent, since FIG. 2(a) bottom shows that there are 2 non-vanishing
singular values $s_{i}$. FIG. 2(b) bottom shows the final outputs: the two
conserved quantities agree with our prior knowledge.

What will happen if we choose the set of basis functions to be smaller or
larger? (1) smaller: if we include polynomials up to the 1st or the 2nd order
($K=3$ or $K=9$) only, then only $H_{1}$ can be discovered by SID, while
$H_{2}$ is not discovered. (2) larger: if we instead include polynomials up to
order 4,5,6 ($K=34,55,83$), then both $H_{1}$ and $H_{2}$ are still
discovered, although the sparsification and independence process may take
longer than with only 3rd order polynomials. Detailed results are included in
Appendix A. The take-home message is that SID does not replace human
scientists since it requires the input of basis functions (formulate
hypothesis) from human scientists. SID is good at testing hypotheses (which
could be technical and tedious), however, it is human scientists who formulate
hypotheses (which requires creativity).

###  III.2 Fluid mechanics

Arguably the biggest puzzle in fluid mechanics is turbulence [12, 13].
Turbulence, and chaos in general, are due to lack of sufficient conserved
quantities. Therefore, studying conserved quantities of fluid systems is
relevant to understanding turbulence. As a preliminary step, we study
conserved quantities of a fluid element in ideal fluid (zero viscosity and
incompressible). In 2D (3D), The fluid element is a triangle (tetrahedron),
which is represented by its 3 (4) vertices [14]. Effectively, we can view the
system as 3 (4) “free” particles, with the only constraint being that the area
(volume) of the triangle (tetrahedron) should remain unchanged. The equations
of motion are included in Appendix C, which appear a bit intimidating
(especially for 3D).

Fluid dynamics experts (including some authors of this paper) have attempted
to find the conserved quantities with pencil and paper. Given the complexity
of the calculations, it is impressive that they found 8 (12) conserved
quantities for 2D (3D). However, they were unsure whether there were more
undiscovered conserved quantities, and whether the discovered ones were in
their simplest. So we turn to SID for help. The results below are for the
basis function set selected to be polynomials up to 2nd (3rd) order for 2D
(3D), but more polynomial orders are also tried in Appendix C.

For the 2D case, SID finds 8 conserved quantities, agreeing with experts’
expectation. Interestingly, the conserved quantities found by SID appear to be
simpler. In fact, all the conserved quantities discovered by SID are 1st or
2nd order polynomials, while experts found a 4th order polynomial, which we
find to b a combination of two 2nd order are conserved quantities discovered
by SID.

For the 3D case, SID finds 14 conserved quantities, while experts found only
12. The two new conserved quantities can be interpreted as the angular
momentum in the center of mass (COM) frame. They are non-trivial because it is
easy to (falsely) think that the COM angular momentum is dependent on the
angular momentum and the linear momentum 444Also, the COM angular momentum has
three components in 3D, so why are there only two new conserved quantities,
not three? It turns out that the third component is not independent of the 14
conserved quantities.. Although humans alone will probably get the results
right at the end of the day without SID, SID can take care of subtle details
automatically, thus saving human experts’ mental labor to a great extent.

###  III.3 Atmospheric chemistry

We next apply SID to a truncated atmospheric chemistry model of photochemical
ozone production [16], where an exotic new conserved quantity is found.
Although simplified, this dynamical system contains 11 species and 10
reactions involved in ozone formation, including NOx, organic, and radical
chemistry [17]. A key characteristic of this system is conservation of carbon
and nitrogen atoms, $H_{C}$ and $H_{N}$, respectively. Though species in this
model contain two other elements, hydrogen and oxygen, neither are conserved,
as $H_{2}O$ molecules are not one of the 11 species whose concentrations are
tracked, and diatomic oxygen $O_{2}$ is treated as an infinite source and sink
due to its abundance. $H_{C}$ and $H_{N}$ are implied in the coefficients of a
stoichiometric matrix $\mathbf{B}\in\mathbb{Z}^{11,10}$ used in prior work to
enforce conservation of atoms in machine learning surrogate models [16].
$H_{C}$ and $H_{N}$ can be represented by linear combinations of species
concentrations, the coefficients of which form a basis for the null space of
$\mathbf{B}^{T}$. Further details are provided in Appendix B.

We applied SID to simulation trajectories expecting to discover up to 2
conserved quantities which are linear combinations of concentrations. The
training data are points on simulation trajectories at pressure $P=0.95$ atm
and temperature $T=20.0^{\circ}$C. As shown in FIG. 3, besides $H_{C}$ and
$H_{N}$, SID surprisingly discovers a third conserved quantity $CQ_{3}$ that
is a linear combination of species concentrations ($C_{X}$ means the
concentration of $X$):

$\displaystyle CQ_{3}\approx$ $\displaystyle\
6C_{CO_{3}}-5C_{NO}+C_{NO_{2}}+3C_{HCNO}$ (9)
$\displaystyle+9C_{HO_{2}}+6C_{HO_{2}H}+2C_{OH}+6C_{O}$
$\displaystyle+4C_{HNO_{3}}-3C_{CO}-2.21C_{H_{2}}$

This additional quantity is linearly independent of $H_{C}$ and $H_{N}$ and is
not in the null space of $\mathbf{B}^{T}$. $CQ_{3}$ has a relative variation
of less than 0.1% in 995 of 1000 simulated cases. Two representative
simulation trajectories are shown in Figure 4, where $CQ_{3}$ holds under
different chemical and meteorological conditions. The evolving concentrations
of $O_{3}$, $NO$, and $NO_{2}$ are included as contrasts to the invariance of
$CQ_{3}$. We have not yet identified the underlying cause of $CQ_{3}$, and
whether it is physically exact or numerically approximate. We have ruled out
symmetry corresponding to hydrogen conservation: when explicitly incorporating
production of $H_{2}O$ as an additional buildup species, SID identifies
approximate hydrogen conservation as well as a fourth conserved quantity (see
Appendix B.3). This implies that $CQ_{3}$ might be a non-trivial conserved
quantity that is worth thorough study in future work.

##  IV Conclusions

We have presented an algorithm SID to automatically discover conserved
quantities from dynamical equations. In constrast to previous blackbox models,
SID is guaranteed to be robust and interpretable thanks to its algorithmic
simplicity. We demonstrate the power of SID on two examples in atmospheric
chemistry and fluid mechanics, revealing new conserved quantities hitherto
unknown to human experts. Although SID does not replace human scientists, it
is a helpful assistant that can facilitate the discovery process. Promising
future directions include applying SID to a broader range of applications and
explicitly deal with symmetries that users may want to impose.

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##  Appendix A More Lotka-Volterra Results

We study the dependence of SID’s outputs on the selection of basis functions.
For the Lotka-Volterra example, there are two conserved quantities
$H_{1}=x+y+z$ (1st order polynomial) and $H_{2}=xyz$ (3rd order polynomial).
We choose the set of basis functions to be polynomials up to
$n=\\{1,2,3,4,5,6\\}$ order. As shown in Figure 5, we find that when $n=1$ or
$n=2$, only $H_{1}$ is discovered. For $n=3,4,5,6$, both $H_{1}$ and $H_{2}$
are discovered by SID.

Figure 5: SID’s results of the three-species Lokta-Volterra equation with
different selections of basis functions. The number of conserved quantities is
correct (2) if the basis set is large enough (polynomial order $n\geq 3$).

##  Appendix B Atmospheric chemistry

###  B.1 Background

The simplified model of ozone photochemistry is written in the Julia computing
language and represents the broad temporal scales and variability in the
prediction of ozone and its precursors. This model relates 11 chemical species
via 10 reactions [16]. The state of the system is described by the
concentration vector

$\mathbf{x}\equiv[C_{O_{3}},C_{NO},C_{NO_{2}},C_{HCHO},C_{HO_{2}},C_{HO_{2}H},C_{OH},C_{O},C_{HNO_{3}},C_{CO},C_{H_{2}}]\equiv[C_{1},C_{2},\dots].$
(10)

Species and reactions form two distinct sets in a bipartite network [18] and
can be related via a weighted biadjacency matrix
$\mathbf{B}\in\mathbb{Z}^{11,10}$ that specifies the stoichiometry of the
chemical system. Conservation of carbon and nitrogen atoms are implicit in the
coefficients of $\mathbf{B}$.

$\mathbf{B}=\setcounter{MaxMatrixCols}{11}\begin{bmatrix}[c|rrrrrrrrrr]&R1&R2&R3&R4&R5&R6&R7&R8&R9&R10\\\
\hline\cr O_{3}&0&1&-1&0&0&0&0&0&0&0\\\ NO&1&0&-1&0&0&0&-1&0&0&0\\\
{NO}_{2}&-1&0&1&0&0&0&1&-1&0&0\\\ HCHO&0&0&0&-1&-1&-1&0&0&0&0\\\
{HO}_{2}&0&0&0&2&0&1&-1&0&0&1\\\ {HO}_{2}H&0&0&0&0&0&0&0&0&-1&-1\\\
OH&0&0&0&0&0&-1&1&-1&2&-1\\\ O&1&-1&0&0&0&0&0&0&0&0\\\
HNO_{3}&0&0&0&0&0&0&0&1&0&0\\\ CO&0&0&0&1&1&1&0&0&0&0\\\
H_{2}&0&0&0&0&1&0&0&0&0&0\\\ \end{bmatrix}$ (11)

A system of coupled ODEs represents the time evolution of the concentrations
of species. In this system, each reaction $j$ has a rate $r_{j}$ defined by
the law of mass action

$r_{j}=k_{j}\prod_{i|B_{ij}<0}C_{i}^{\absolutevalue{B_{ij}}}$ (12)

where the reaction rate constant $k_{j}$ is determined by an Arrhenius
relation. The rate of change of concentration for each species $i$ is the sum
of reaction rates describing its production and loss:

$\frac{dC_{i}}{dt}=\sum_{j}B_{ij}r_{j},$ (13)

Substituting Eq. (12) for $r_{j}$ in Eq. (13), the full set of ODEs for the
system in Eq. (11) can be written as

$\displaystyle\frac{dC_{O_{3}}}{dt}$ $\displaystyle=k_{2}C_{O}\ -{\
k}_{3}C_{O_{3}}C_{NO},$ (14) $\displaystyle\frac{dC_{NO}}{dt}$
$\displaystyle=k_{1}C_{NO_{2}}\ -{\ k}_{3}C_{O_{3}}C_{NO}-{\
k}_{7}{C_{NO}C}_{HO_{2}},$ $\displaystyle\frac{dC_{NO_{2}}}{dt}$
$\displaystyle={-\ k}_{1}C_{NO_{2}}\ +{\
k}_{3}C_{O_{3}}C_{NO}+k_{7}C_{NO}C_{HO_{2}}-{\ k}_{8}C_{NO_{2}}C_{OH},$
$\displaystyle\frac{dC_{HCHO}}{dt}$ $\displaystyle={-\ k}_{4}C_{HCHO}\ -{\
k}_{5}C_{HCHO}-{\ k}_{6}C_{HCHO}C_{OH},$ $\displaystyle\frac{dC_{HO_{2}}}{dt}$
$\displaystyle={2k}_{4}C_{HCHO}+\ {\ k}_{6}C_{HCHO}C_{OH}-{\
k}_{7}{C_{NO}C}_{HO_{2}}+k_{10}C_{HO_{2}H}C_{OH},$
$\displaystyle\frac{dC_{HO_{2}H}}{dt}$
$\displaystyle=-k_{9}C_{HO_{2}H}-k_{10}C_{HO_{2}H}C_{OH},$
$\displaystyle\frac{dC_{OH}}{dt}$ $\displaystyle=-{\ k}_{6}C_{HCHO}C_{OH}+{\
k}_{7}{C_{NO}C}_{HO2}-{\
k}_{8}C_{NO2}C_{OH}+{2k}_{9}C_{HO2H}-k_{10}C_{HO2H}C_{OH},$
$\displaystyle\frac{dC_{O}}{dt}$ $\displaystyle=k_{1}C_{NO2}-k_{2}C_{O},$
$\displaystyle\frac{dC_{HNO3}}{dt}$ $\displaystyle={\ k}_{8}C_{NO2}C_{OH},$
$\displaystyle\frac{dC_{CO}}{dt}$ $\displaystyle={\ k}_{4}C_{HCHO}\ +{\
k}_{5}C_{HCHO}+{\ k}_{6}C_{HCHO}C_{OH},$ $\displaystyle\frac{dC_{H2}}{dt}$
$\displaystyle={\ k}_{5}C_{HCHO}.$

The table below gives example values for $k$ at T=298K and P=1 atm, with
photolytic constants representative of mid-day conditions in Southern
California. In the model and the testing of the conserved quantities, rate
constants are dependent on reaction-specific parameters, as well as
temperature and pressure, determined by a dry adiabatic lapse rate and the
hypsometric equation whose values are determined by an altitude $z$ drawn from
a uniform distribution on the interval $[0,2000]$ meters.

rate constant | $k_{1}$ | $k_{2}$ | $k_{3}$ | $k_{4}$ | $k_{5}$ | $k_{6}$ | $k_{7}$ | $k_{8}$ | $k_{9}$ | $k_{10}$
---|---|---|---|---|---|---|---|---|---|---
value | 0.5 | 22.179 | 26.937 | 0.015 | 0.022 | 13844.97 | 12652.43 | 15454.98 | 0.0003 | 2492.71

To reduce the numerical stiffness of this system, the pseudo-steady state
approximation is applied to two species (hydroxyl radical $OH$ and atomic
oxygen $O$) whose rates of change are approximated as zero and concentrations
rewritten as algebraic expressions of the other species concentrations. The
concentration vector $C$ is restricted to its semi-positive half space
(negative concentrations are unphysical) initialized from uniform
distributions where ozone is selected to range from 0-100 ppb and other ranges
informed by initialization ranges in prior work [19].

###  B.2 Conserved quantity results

Known conserved quantities We expect $H_{C}=C_{HCHO}+C_{CO}$ and
$H_{N}=C_{NO}+C_{NO_{2}}+C_{HNO_{3}}$ to be conserved, as this is built into
the model.

SID SID identifies three conserved quantities, two of which are $H_{C}$ and
$H_{N}$. To validate the results of SID, we initialize one thousand 20-minute
simulations under the above conditions. The known conservation properties hold
in the model to very high precision, with coefficients of variation (mean-
normalized standard deviation) of less than $10^{-14}$ for all simulations.
The rediscovered conservation laws for carbon and nitrogen atom closure also
display high accuracy, with maximum coefficients of variation over all 1000
cases on the order of $10^{-11}$ and $10^{-13}$ respectively. $CQ_{3}$ is not
conserved to such high precision in the 1000 validation simulations: its
maximum coefficient of variation is $0.96\%$. However, the coefficient of
variation of $CQ_{3}$ is within 0.1% in all but 5 of 1000 cases, and has a 95
percentile over all cases of 0.02%.

The coefficients discovered by SID are:

$\mathbf{B}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}[c|ccccccccccc]&O_{3}&NO&NO_{2}&HCHO&HO_{2}&HO_{2}H&OH&O&HNO_{3}&CO&H_{2}\\\
\hline\cr CQ_{1}&0&0&0&0.707&0&0&0&0&0&0.707&0\\\
CQ_{2}&0&0.577&0.577&0&0&0&0&0&0.577&0&0\\\
CQ_{3}&0.370&-0.310&0.061&0.185&0.555&0.370&0.185&0.370&0.247&-0.185&-0.135\\\
\end{pmatrix}$ (15)

Note that the ratios of non-zero coefficients are very close to rational
numbers, so if we divide rows by $[0.707,0.577,0.062]$, we approximately get

$\mathbf{B}^{\prime}=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}[c|ccccccccccc]&O_{3}&NO&NO_{2}&HCHO&HO_{2}&HO_{2}H&OH&O&HNO_{3}&CO&H_{2}\\\
\hline\cr CQ_{1}&0&0&0&1&0&0&0&0&0&1&0\\\ CQ_{2}&0&1&1&0&0&0&0&0&1&0&0\\\
CQ_{3}&6&-5&1&3&9&6&3&6&4&-3&-2\\\ \end{pmatrix}$ (16)

The snapped $CQ_{3}$ is not quite as conserved: in 910 out of 1000 cases,
$CQ_{3}$ is conserved within 0.1%. However, if we do not snap the coefficient
for $H_{2}$ as 2 (instead set 2.213115), $CQ_{3}$ holds to within 0.1% for 995
out of 1000 cases.

###  B.3 $CQ_{3}$ is independent of hydrogen atom conservation

In the 11-species model, the number of hydrogen atoms is not conserved because
the net production of $H_{2}O$ is ignored. The atomic composition of each
species is not explicitly represented in the model but rather implied by the
stoichiometric coefficients of the reactions, which ensure carbon and nitrogen
conservation. We find that $CQ_{3}$ is balanced in 8 out of 10 individual
reactions. This motivates construction of an experiment to assess whether SID
could be inferring some incorrect composition of hydrogen in each species
(which is conserved as represented by $CQ_{3}$). To assess this, we augment
the model to track net production of $H_{2}O$ as a 12th species so that the
number of hydrogen atoms becomes:

$H_{H}=2C_{HCHO}+C_{HO_{2}}+2C_{HO_{2}H}+C_{OH}+C_{HNO_{3}}+2C_{H_{2}}+2C_{H_{2}O}.$
(17)

We note that (17) is not fully conserved in the 12-species model due to the
pseudo-steady state approximation of $OH$, but is approximately conserved,
with a maximum mean-normalized standard deviation of 4.2% over all cases, and
holding in 972 out of 1000 cases to within 0.1%. With $H_{2}O$ represented in
the 12-species model, there are three known conserved quantities: conservation
of carbon and nitrogen and approximate conservation of hydrogen. If $CQ_{3}$
indeed arises from an incorrectly inferred composition of hydrogen in the set
of 11 species, we would expect three conserved quantities for the 12-species
model, with the third quantity representing the known hydrogen balance in
(17). However, SID finds 4 conserved quantities, demonstrating that there is
indeed an additional conserved quantity independent of hydrogen atom
conservation.

##  Appendix C Fluid mechanics

###  C.1 Background

Fluid mechanics systems can be very complicated, especially when exhibiting
turbulence [12, 13]. Turbulence, and chaos in general, emerge due to lack of
sufficient conserved quantities. To understand turbulence, it is thus relevant
to study conserved quantities of fluid systems. We will deal with ideal fluid,
and partition the continuum of fluid into small elements (triangles in 2D or
irregular tetrahedra in 3D) [14]. For ideal incompressible fluid, the only
internal interaction between elements is via pressure, which may perform work
on individual element (pressure gradients as external forces), but the total
work for the whole fluid is zero. Since we care about global conserved
quantities of the whole fluid, without loss of generality, we can assume zero
external forces on each individual element. Therefore, we can first study
local conserved quantities (for each element), and the global conserved
quantities of the whole fluid is the summation (integral) of conserved
quantities of individual elements. This additivity property allows us to study
a fluid element first. A fluid element is described by its vertices (3 in 2D,
4 in 3D), illustrated in Figure 6. Every vertex $i$ is in turn described by
its position $(x_{i},y_{i},z_{i})$ and velocity
$(u_{i},v_{i},w_{i})\equiv(\dot{x}_{i},\dot{y}_{i},\dot{z}_{i})$. The motion
of the fluid element can thus be described by the motion of its vertices.
Because the fluid element is small, one can imagine unit masses put on the
vertices, with no mass elsewhere. So vertices move like free particles, with
the only interaction (constraint) being that the area/volume is conserved.
Note that since we partition the fluid to tetrahedra (which lack knots or
linkages), we may miss topological invariants (e.g., helicity, which is the
dot product of the velocity and vorticity).

Figure 6: Illustration of fluid element motion. Left: 2D, right: 3D.

####  C.1.1 2D

Next, we use Lagrangian mechanics to obtain equations of motion in 2D, which
is straightforwardly generalized to 3D. The phase space is 12D:
$\mathbf{x}=(x_{1},y_{1},u_{1},v_{1},x_{2},y_{2},u_{2},v_{2},x_{3},y_{3},u_{3},v_{3})$.
The area of the triangle is

$A=x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}-x_{2}y_{1}-x_{3}y_{2}-x_{1}y_{3}.$ (18)

Since the area is conserved (we are assuming an incompressible fluid), we
impose that

$\frac{dA}{dt}=u_{1}(y_{2}-y_{3})+v_{1}(x_{3}-x_{2})+u_{2}(y_{3}-y_{1})+v_{2}(x_{1}-x_{3})+u_{3}(y_{1}-y_{2})+v_{3}(x_{2}-x_{1})=0$
(19)

The Lagrangian is the kinetic energy of three particles, plus the area
constraint ($\lambda$ is the Lagrange multiplier):

$\mathcal{L}=\frac{1}{2}(u_{1}^{2}+v_{1}^{2}+u_{2}^{2}+v_{2}^{2}+u_{3}^{2}+v_{3}^{2})+\lambda(x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}-x_{2}y_{1}-x_{3}y_{2}-x_{1}y_{3})$
(20)

The Euler-Lagrange equations give

$\displaystyle\dot{x_{1}}=u_{1},\quad\dot{u_{1}}=\lambda(y_{2}-y_{3}),\quad\dot{y_{1}}=v_{1},\quad\dot{v_{1}}=\lambda(x_{3}-x_{2}),$
(21)
$\displaystyle\dot{x_{2}}=u_{2},\quad\dot{u_{2}}=\lambda(y_{3}-y_{1}),\quad\dot{y_{2}}=v_{2},\quad\dot{v_{2}}=\lambda(x_{1}-x_{3}),$
$\displaystyle\dot{x_{3}}=u_{3},\quad\dot{u_{3}}=\lambda(y_{1}-y_{2}),\quad\dot{y_{3}}=v_{3},\quad\dot{v_{3}}=\lambda(x_{2}-x_{1}),$

where $\lambda$ can be solved by noticing $\frac{d^{2}A}{dt^{2}}=0$:

$\lambda=\frac{(u_{1}v_{2}-u_{1}v_{3}-u_{2}v_{1}+u_{2}v_{3}+u_{3}v_{1}-u_{3}v_{2})}{(-x_{1}^{2}+x_{1}x_{2}+x_{1}x_{3}-x_{2}^{2}+x_{2}x_{3}-x_{3}^{2}-y_{1}^{2}+y_{1}y_{2}+y_{1}y_{3}-y_{2}^{2}+y_{2}y_{3}-y_{3}^{2})}.$
(22)

####  C.1.2 3D

The derivation of equations of motion is similar to 2D, only with area
switched to volume. The phase space is 24D:
$\mathbf{x}=(x_{i},y_{i},z_{i},u_{i},v_{i},w_{i}),i=1,2,3,4$. We list the
equations here:

$\displaystyle\dot{x_{1}}=u_{1},\dot{u_{1}}=\lambda(-y_{2}z_{3}+y_{2}z_{4}+y_{3}z_{2}-y_{3}z_{4}-y_{4}z_{2}+y_{4}z_{3}),$
(23)
$\displaystyle\dot{y_{1}}=v_{1},\dot{v_{1}}=\lambda(x_{2}z_{3}-x_{2}z_{4}-x_{3}z_{2}+x_{3}z_{4}+x_{4}z_{2}-x_{4}z_{3}),$
$\displaystyle\dot{z_{1}}=w_{1},\dot{w_{1}}=\lambda(-x_{2}y_{3}+x_{2}y_{4}+x_{3}y_{2}-x_{3}y_{4}-x_{4}y_{2}+x_{4}y_{3}),$
$\displaystyle\dot{x_{2}}=u_{2},\dot{u_{2}}=\lambda(y_{1}z_{3}-y_{1}z_{4}-y_{3}z_{1}+y_{3}z_{4}+y_{4}z_{1}-y_{4}z_{3}),$
$\displaystyle\dot{y_{2}}=v_{2},\dot{v_{2}}=\lambda(-x_{1}z_{3}+x_{1}z_{4}+x_{3}z_{1}-x_{3}z_{4}-x_{4}z_{1}+x_{4}z_{3}),$
$\displaystyle\dot{z_{2}}=w_{2},\dot{w_{2}}=\lambda(x_{1}y_{3}-x_{1}y_{4}-x_{3}y_{1}+x_{3}y_{4}+x_{4}y_{1}-x_{4}y_{3}),$
$\displaystyle\dot{x_{3}}=u_{3},\dot{u_{3}}=\lambda(-y_{1}z_{2}+y_{1}z_{4}+y_{2}z_{1}-y_{2}z_{4}-y_{4}z_{1}+y_{4}z_{2}),$
$\displaystyle\dot{y_{3}}=v_{3},\dot{v_{3}}=\lambda(x_{1}z_{2}-x_{1}z_{4}-x_{2}z_{1}+x_{2}z_{4}+x_{4}z_{1}-x_{4}z_{2}),$
$\displaystyle\dot{z_{3}}=w_{3},\dot{w_{3}}=\lambda(-x_{1}y_{2}+x_{1}y_{4}+x_{2}y_{1}-x_{2}y_{4}-x_{4}y_{1}+x_{4}y_{2}),$
$\displaystyle\dot{x_{4}}=u_{4},\dot{u_{4}}=\lambda\cdot(y_{1}z_{2}-y_{1}z_{3}-y_{2}z_{1}+y_{2}z_{3}+y_{3}z_{1}-y_{3}z_{2}),$
$\displaystyle\dot{y_{4}}=v_{4},\dot{v_{4}}=\lambda(-x_{1}z_{2}+x_{1}z_{3}+x_{2}z_{1}-x_{2}z_{3}-x_{3}z_{1}+x_{3}z_{2}),$
$\displaystyle\dot{z_{4}}=w_{4},\dot{w_{4}}=\lambda(x_{1}y_{2}-x_{1}y_{3}-x_{2}y_{1}+x_{2}y_{3}+x_{3}y_{1}-x_{3}y_{2}),$

where

$\displaystyle\lambda=$ $\displaystyle\ -2p/q,$ (24) $\displaystyle p=$
$\displaystyle\
x_{1}(-v_{2}w_{3}+v_{2}w_{4}+v_{3}w_{2}-v_{3}w_{4}-v_{4}w_{2}+v_{4}w_{3})+y_{1}(u_{2}w_{3}-u_{2}w_{4}-u_{3}w_{2}+u_{3}w_{4}+u_{4}w_{2}-u_{4}w_{3})$
$\displaystyle+z_{1}(-u_{2}v_{3}+u_{2}v_{4}+u_{3}v_{2}-u_{3}v_{4}-u_{4}v_{2}+u_{4}v_{3})+x_{2}(v_{1}w_{3}-v_{1}w_{4}-v_{3}w_{1}+v_{3}w_{4}+v_{4}w_{1}-v_{4}w_{3})$
$\displaystyle+y_{2}(-u_{1}w_{3}+u_{1}w_{4}+u_{3}w_{1}-u_{3}w_{4}-u_{4}w_{1}+u_{4}w_{3})+z_{2}(u_{1}v_{3}-u_{1}v_{4}-u_{3}v_{1}+u_{3}v_{4}+u_{4}v_{1}-u_{4}v_{3})$
$\displaystyle+x_{3}(-v_{1}w_{2}+v_{1}w_{4}+v_{2}w_{1}-v_{2}w_{4}-v_{4}w_{1}+v_{4}w_{2})+y_{3}(u_{1}w_{2}-u_{1}w_{4}-u_{2}w_{1}+u_{2}w_{4}+u_{4}w_{1}-u_{4}w_{2})$
$\displaystyle+z_{3}(-u_{1}v_{2}+u_{1}v_{4}+u_{2}v_{1}-u_{2}v_{4}-u_{4}v_{1}+u_{4}v_{2})+x_{4}(v_{1}w_{2}-v_{1}w_{3}-v_{2}w_{1}+v_{2}w_{3}+v_{3}w_{1}-v_{3}w_{2})$
$\displaystyle+y_{4}(-u_{1}w_{2}+u_{1}w_{3}+u_{2}w_{1}-u_{2}w_{3}-u_{3}w_{1}+u_{3}w_{2})+z_{4}(u_{1}v_{2}-u_{1}v_{3}-u_{2}v_{1}+u_{2}v_{3}+u_{3}v_{1}-u_{3}v_{2})$
$\displaystyle q=$ $\displaystyle\
(-y_{2}z_{3}+y_{2}z_{4}+y_{3}z_{2}-y_{3}z_{4}-y_{4}z_{2}+y_{4}z_{3})^{2}+(x_{2}z_{3}-x_{2}z_{4}-x_{3}z_{2}+x_{3}z_{4}+x_{4}z_{2}-x_{4}z_{3})^{2}$
$\displaystyle+(-x_{2}y_{3}+x_{2}y_{4}+x_{3}y_{2}-x_{3}y_{4}-x_{4}y_{2}+x_{4}y_{3})^{2}+(y_{1}z_{3}-y_{1}z_{4}-y_{3}z_{1}+y_{3}z_{4}+y_{4}z_{1}-y_{4}z_{3})^{2}$
$\displaystyle+(-x_{1}z_{3}+x_{1}z_{4}+x_{3}z_{1}-x_{3}z_{4}-x_{4}z_{1}+x_{4}z_{3})^{2}+(x_{1}y_{3}-x_{1}y_{4}-x_{3}y_{1}+x_{3}y_{4}+x_{4}y_{1}-x_{4}y_{3})^{2}$
$\displaystyle+(-y_{1}z_{2}+y_{1}z_{4}+y_{2}z_{1}-y_{2}z_{4}-y_{4}z_{1}+y_{4}z_{2})^{2}+(x_{1}z_{2}-x_{1}z_{4}-x_{2}z_{1}+x_{2}z_{4}+x_{4}z_{1}-x_{4}z_{2})^{2}$
$\displaystyle+(-x_{1}y_{2}+x_{1}y_{4}+x_{2}y_{1}-x_{2}y_{4}-x_{4}y_{1}+x_{4}y_{2})^{2}+(y_{1}z_{2}-y_{1}z_{3}-y_{2}z_{1}+y_{2}z_{3}+y_{3}z_{1}-y_{3}z_{2})^{2}$
$\displaystyle+(-x_{1}z_{2}+x_{1}z_{3}+x_{2}z_{1}-x_{2}z_{3}-x_{3}z_{1}+x_{3}z_{2})^{2}+(x_{1}y_{2}-x_{1}y_{3}-x_{2}y_{1}+x_{2}y_{3}+x_{3}y_{1}-x_{3}y_{2})^{2}$

and with the constant-volume constraint:

$\displaystyle 0=$ $\displaystyle\
(-y_{2}z_{3}+y_{2}z_{4}+y_{3}z_{2}-y_{3}z_{4}-y_{4}z_{2}+y_{4}z_{3})u_{1}+(x_{2}z_{3}-x_{2}z_{4}-x_{3}z_{2}+x_{3}z_{4}+x_{4}z_{2}-x_{4}z_{3})v_{1}$
(25)
$\displaystyle+(-x_{2}y_{3}+x_{2}y_{4}+x_{3}y_{2}-x_{3}y_{4}-x_{4}y_{2}+x_{4}y_{3})w_{1}+(y_{1}z_{3}-y_{1}z_{4}-y_{3}z_{1}+y_{3}z_{4}+y_{4}z_{1}-y_{4}z_{3})u_{2}$
$\displaystyle+(-x_{1}z_{3}+x_{1}z_{4}+x_{3}z_{1}-x_{3}z_{4}-x_{4}z_{1}+x_{4}z_{3})v_{2}+(x_{1}y_{3}-x_{1}y_{4}-x_{3}y_{1}+x_{3}y_{4}+x_{4}y_{1}-x_{4}y_{3})w_{2}$
$\displaystyle+(-y_{1}z_{2}+y_{1}z_{4}+y_{2}z_{1}-y_{2}z_{4}-y_{4}z_{1}+y_{4}z_{2})u_{3}+(x_{1}z_{2}-x_{1}z_{4}-x_{2}z_{1}+x_{2}z_{4}+x_{4}z_{1}-x_{4}z_{2})v_{3}$
$\displaystyle+(-x_{1}y_{2}+x_{1}y_{4}+x_{2}y_{1}-x_{2}y_{4}-x_{4}y_{1}+x_{4}y_{2})w_{3}+(y_{1}z_{2}-y_{1}z_{3}-y_{2}z_{1}+y_{2}z_{3}+y_{3}z_{1}-y_{3}z_{2})u_{4}$
$\displaystyle+(-x_{1}z_{2}+x_{1}z_{3}+x_{2}z_{1}-x_{2}z_{3}-x_{3}z_{1}+x_{3}z_{2})v_{4}+(x_{1}y_{2}-x_{1}y_{3}-x_{2}y_{1}+x_{2}y_{3}+x_{3}y_{1}-x_{3}y_{2})w_{4}$

###  C.2 Conserved quantity results

Known conserved quantities Human experts obtained 8 conserved quantities for
2D and 12 conserved quantities for 3D. This is quite impressive already, given
the complexity of the required calculations. However, they were unsure whether
there are undiscovered conserved quantities, and whether the conserved
quantities they found are in the simplest form.

SID We run SID to search for conserved quantities. For the 2D case, SID finds
there are indeed 8 conserved quantities, agreeing with human expertise.
However, the conserved quantities found by SID are simpler. For example, all
the conserved quantities discovered by SID are first or second-order
polynomials, while the human expert finds a 4th order polynomial, which is
later found (by human experts) to be derived from second-order conserved
quantities discovered by SID. For the 3D case, SID finds there are 14
conserved quantities, i.e., two more than human experts discovered. Below we
present SID’s results given various basis sets (polynomials up to order
$n=\\{1,2,3,4\\}$).

####  C.2.1 2D

As shown in Figure 7, when $n=1,2,3,4$, SID discovers 2, 8, 8, 8 conserved
quantities, respectively. This means that there is no new 3rd order conserved
quantity. Therefore we set $n=2$, and visualize the discovered conserved
quantities (coefficients) in Figure 8, which appear to be sparse (hence
interpretable).

Figure 7: SID’s results of the a simplified fluid system (2D) with various
basis set (polynomials up to order $n$). When $n=1,2,3,4$, SID discovers 2, 8,
8, 8 conserved quantities respectively. Figure 8: SID’s discovered conserved
quantities (coefficients) for the simplified fluid system (2D) with the basis
set chosen to be polynomials up to order 2.

####  C.2.2 3D

As shown in Figure 9, when $n=1,2,3,4$, SID discovers 3, 12, 14, 14 conserved
quantities, respectively. For $n=3$, we visualize SID’s discovered conserved
quantities (coefficients) in Figure 10, which appear to be sparse (hence
interpretable).

Figure 9: SID’s results of the a simplified fluid system (3D) with various
basis set (polynomials up to order $n$). When $n=1,2,3,4$, SID discovers 3,
12, 14, 14 conserved quantities respectively. Figure 10: SID’s discovered
conserved quantities (coefficients) for the simplified fluid system (3D) with
the basis set chosen to be polynomials up to order 3.

###  C.3 Summary of conserved quantities (combining experts and SID)

We have disclaimed frequently throughout the paper that SID by no means
replaces human scientists, but rather is a helpful assistant that can
facilitate the discovery process when an expert uses it. This is because: (1)
SID needs the input of basis function set from human experts; (2) The physical
interpretation of SID’s results is still left to human experts. By combining
the knowledge from human experts and SID’s findings, we summarize the
conserved quantities of the fluid system below.

####  C.3.1 2D

For the 2D case, the important quantities are:

$\displaystyle x_{cm}=\frac{1}{3}(x_{1}+x_{2}+x_{3}),\quad
y_{cm}=\frac{1}{3}(y_{1}+y_{2}+y_{3}),$ (26)
$\displaystyle\bar{x}_{1}=x_{1}-x_{cm},\quad\bar{y}_{1}=y_{1}-y_{cm},\quad\bar{x}_{2}=x_{2}-x_{cm},\quad\bar{y}_{2}=y_{2}-y_{cm},\quad\bar{x}_{3}=x_{3}-x_{cm},\quad\bar{y}_{3}=y_{3}-y_{cm},$
$\displaystyle u_{cm}=\frac{1}{3}(u_{1}+u_{2}+u_{3}),\quad
v_{cm}=\frac{1}{3}(v_{1}+v_{2}+v_{3}),$
$\displaystyle\bar{u}_{1}=u_{1}-u_{cm},\quad\bar{v}_{1}=v_{1}-v_{cm},\quad\bar{u}_{2}=u_{2}-u_{cm},\quad\bar{v}_{2}=v_{2}-v_{cm},\quad\bar{u}_{3}=u_{3}-u_{cm},\quad\bar{v}_{3}=v_{3}-v_{cm},$
$\displaystyle L_{cm}=x_{cm}v_{cm}-y_{cm}u_{cm}$ $\displaystyle
L=\bar{v}_{1}\bar{x}_{1}+\bar{v}_{2}\bar{x}_{2}+\bar{v}_{3}\bar{x}_{3}-\bar{u}_{1}\bar{y}_{1}-\bar{u}_{2}\bar{y}_{2}-\bar{u}_{3}\bar{y}_{3},$
$\displaystyle
E=\bar{u}_{1}^{2}+\bar{v}_{1}^{2}+\bar{u}_{2}^{2}+\bar{v}_{2}^{2}+\bar{u}_{3}^{2}+\bar{v}_{3}^{2},$
$\displaystyle
I=\bar{x}_{1}^{2}+\bar{y}_{1}^{2}+\bar{x}_{2}^{2}+\bar{y}_{2}^{2}+\bar{x}_{3}^{2}+\bar{y}_{3}^{2},$
$\displaystyle
J=(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}+(x_{2}-x_{3})^{2}+(y_{2}-y_{3})^{2}+(x_{3}-x_{1})^{2}+(y_{3}-y_{1})^{2},$
$\displaystyle
K=u_{1}v_{2}-u_{1}v_{3}-u_{2}v_{1}+u_{2}v_{3}+u_{3}v_{1}-u_{3}v_{2},$
$\displaystyle
A=x_{1}y_{2}-x_{1}y_{3}-x_{2}y_{1}+x_{2}y_{3}+x_{3}y_{1}-x_{3}y_{2},$
$\displaystyle
D=u_{1}(y_{2}-y_{3})+v_{1}(x_{3}-x_{2})+u_{2}(y_{3}-y_{1})+v_{2}(x_{1}-x_{3})+u_{3}(y_{1}-y_{2})+v_{3}(x_{2}-x_{1}),$
$\displaystyle\omega=u_{1}(x_{2}-x_{3})+u_{2}(x_{3}-x_{1})+u_{3}(x_{1}-x_{2})+v_{1}(y_{2}-y_{3})+v_{2}(y_{3}-y_{1})+v_{3}(y_{1}-y_{2}),$
$\displaystyle
G=\bar{u}_{1}\bar{x}_{1}+\bar{u}_{2}\bar{x}_{2}+\bar{u}_{3}\bar{x}_{3}+\bar{v}_{1}\bar{y}_{1}+\bar{v}_{2}\bar{y}_{2}+\bar{v}_{3}\bar{y}_{3}$

The conserved quantities for the 2D case are

$u_{cm},v_{cm},L_{cm},L,E,A,D,\omega,IK$

However, there is a relationship between these conserved quantities

$IK=EA-L\omega-DG$

which means there are only 8 independent conserved quantities.

Table 3: Physical interpretations of important quantities (2D) Notation | Meaning
---|---
$u_{cm}$ | velocity of center of mass in x direction
$v_{cm}$ | velocity of center of mass in y direction
$L_{cm}$ | Angular momentum of center of mass
$L$ | Angular momentum with respect to center of mass
$E$ | Energy with respect to center of mass
$A$ | Area
$D$ | Divergence
$\omega$ | Vorticity
$I$ | Moment of inertia with respect to center of mass
$G$ | Derivative of moment of inertia

Global conserved quantities can be expressed in terms of integrals, where
subscript $g$ stands for global and $\rho$ is density (suppose the support of
the fluid is $\Omega\in\mathbb{R}^{2}$, the boundary is $\partial\Omega$):

$\displaystyle\mathbf{p}_{g}=\iint_{\Omega}\rho\textbf{v}\,dA,\quad\textbf{v}\equiv(u,v),\quad{\rm(momentum,2D\
vector)}$ (27) $\displaystyle A_{g}=\iint_{\Omega}\,dA\quad{\rm(area,scalar)}$
$\displaystyle
L_{cm,g}=\mathbf{r}_{cm}\times\mathbf{p}_{g},\quad\mathbf{r}_{cm}\equiv\frac{1}{A_{g}}\iint_{\Omega}\mathbf{r}dA\quad{\rm(momentum\
of\ COM,scalar)}$ $\displaystyle
L_{g}=\left(\iint_{\Omega}\rho\textbf{r}\times\textbf{v}\,dA\right)-L_{cm,g}\quad{\rm(momentum\
in\ COM,scalar)}$ $\displaystyle
E_{g}=\left(\iint_{\Omega}\rho|\textbf{v}|^{2}\,dA\right)-\frac{|\mathbf{p}_{g}|^{2}}{2\rho
A_{g}}\quad{\rm(energy\ in\ COM,scalar)}$ $\displaystyle
D_{g}=\oint_{\partial\Omega}\,\mathbf{v}\cdot d\mathbf{n}\equiv
0\quad{\rm(outflow,scalar)}$ $\displaystyle C_{g}=\iint_{\Omega}\omega
dA=\oint_{\partial\Omega}\textbf{v}\cdot\,d\textbf{l}\quad{\rm(circulation,scalar)}$

####  C.3.2 3D

For the 3D case, the important quantities are:

$\displaystyle x_{cm}=\frac{1}{4}(x_{1}+x_{2}+x_{3}+x_{4}),\quad
y_{cm}=\frac{1}{4}(y_{1}+y_{2}+y_{3}+y_{4}),\quad
z_{cm}=\frac{1}{4}(z_{1}+z_{2}+z_{3}+z_{4})$ (28)
$\displaystyle\bar{x}_{1}=x_{1}-x_{cm},\quad\bar{y}_{1}=y_{1}-y_{cm},\quad\bar{z}_{1}=z_{1}-z_{cm},$
$\displaystyle\bar{x}_{2}=x_{2}-x_{cm},\quad\bar{y}_{2}=y_{2}-y_{cm},\quad\bar{z}_{2}=z_{2}-z_{cm},$
$\displaystyle\bar{x}_{3}=x_{3}-x_{cm},\quad\bar{y}_{3}=y_{3}-y_{cm},\quad\bar{z}_{3}=z_{3}-z_{cm},$
$\displaystyle\bar{x}_{4}=x_{4}-x_{cm},\quad\bar{y}_{4}=y_{4}-y_{cm},\quad\bar{z}_{4}=z_{4}-z_{cm},$
$\displaystyle u_{cm}=\frac{1}{4}(u_{1}+u_{2}+u_{3}+u_{4}),\quad
v_{cm}=\frac{1}{4}(v_{1}+v_{2}+v_{3}+v_{4}),\quad
w_{cm}=\frac{1}{4}(w_{1}+w_{2}+w_{3}+w_{4}),$
$\displaystyle\bar{u}_{1}=u_{1}-u_{cm},\quad\bar{v}_{1}=v_{1}-v_{cm},\quad\bar{w}_{1}=w_{1}-w_{cm}$
$\displaystyle\bar{u}_{2}=u_{2}-u_{cm},\quad\bar{v}_{2}=v_{2}-v_{cm},\quad\bar{w}_{2}=w_{2}-w_{cm},$
$\displaystyle\bar{u}_{3}=u_{3}-u_{cm},\quad\bar{v}_{3}=v_{3}-v_{cm},\quad\bar{w}_{3}=w_{3}-w_{cm},$
$\displaystyle\bar{u}_{4}=u_{4}-u_{cm},\quad\bar{v}_{4}=v_{4}-v_{cm},\quad\bar{w}_{4}=w_{4}-w_{cm},$

$\displaystyle
A^{x}_{1}=-y_{2}z_{3}+y_{2}z_{4}+y_{3}z_{2}-y_{3}z_{4}-y_{4}z_{2}+y_{4}z_{3},\quad
A^{y}_{1}=x_{2}z_{3}-x_{2}z_{4}-x_{3}z_{2}+x_{3}z_{4}+x_{4}z_{2}-x_{4}z_{3},$
(29) $\displaystyle
A^{z}_{1}=-x_{2}y_{3}+x_{2}y_{4}+x_{3}y_{2}-x_{3}y_{4}-x_{4}y_{2}+x_{4}y_{3},\quad
A^{x}_{2}=y_{1}z_{3}-y_{1}z_{4}-y_{3}z_{1}+y_{3}z_{4}+y_{4}z_{1}-y_{4}z_{3},$
$\displaystyle
A^{y}_{2}=-x_{1}z_{3}+x_{1}z_{4}+x_{3}z_{1}-x_{3}z_{4}-x_{4}z_{1}+x_{4}z_{3},\quad
A^{z}_{2}=x_{1}y_{3}-x_{1}y_{4}-x_{3}y_{1}+x_{3}y_{4}+x_{4}y_{1}-x_{4}y_{3},$
$\displaystyle
A^{x}_{3}=-y_{1}z_{2}+y_{1}z_{4}+y_{2}z_{1}-y_{2}z_{4}-y_{4}z_{1}+y_{4}z_{2},\quad
A^{y}_{3}=x_{1}z_{2}-x_{1}z_{4}-x_{2}z_{1}+x_{2}z_{4}+x_{4}z_{1}-x_{4}z_{2}$
$\displaystyle
A^{z}_{3}=-x_{1}y_{2}+x_{1}y_{4}+x_{2}y_{1}-x_{2}y_{4}-x_{4}y_{1}+x_{4}y_{2},\quad
A^{x}_{4}=y_{1}z_{2}-y_{1}z_{3}-y_{2}z_{1}+y_{2}z_{3}+y_{3}z_{1}-y_{3}z_{2}$
$\displaystyle
A^{y}_{4}=-x_{1}z_{2}+x_{1}z_{3}+x_{2}z_{1}-x_{2}z_{3}-x_{3}z_{1}+x_{3}z_{2},\quad
A^{z}_{4}=x_{1}y_{2}-x_{1}y_{3}-x_{2}y_{1}+x_{2}y_{3}+x_{3}y_{1}-x_{3}y_{2},$
$\displaystyle
B^{x}_{1}=-v_{2}w_{3}+v_{2}w_{4}+v_{3}w_{2}-v_{3}w_{4}-v_{4}w_{2}+v_{4}w_{3},\quad
B^{y}_{1}=u_{2}w_{3}-u_{2}w_{4}-u_{3}w_{2}+u_{3}w_{4}+u_{4}w_{2}-u_{4}w_{3}$
$\displaystyle
B^{z}_{1}=-u_{2}v_{3}+u_{2}v_{4}+u_{3}v_{2}-u_{3}v_{4}-u_{4}v_{2}+u_{4}v_{3},\quad
B^{x}_{2}=v_{1}w_{3}-v_{1}w_{4}-v_{3}w_{1}+v_{3}w_{4}+v_{4}w_{1}-v_{4}w_{3},$
$\displaystyle
B^{y}_{2}=-u_{1}w_{3}+u_{1}w_{4}+u_{3}w_{1}-u_{3}w_{4}-u_{4}w_{1}+u_{4}w_{3},\quad
B^{z}_{2}=u_{1}v_{3}-u_{1}v_{4}-u_{3}v_{1}+u_{3}v_{4}+u_{4}v_{1}-u_{4}v_{3},$
$\displaystyle
B^{x}_{3}=-v_{1}w_{2}+v_{1}w_{4}+v_{2}w_{1}-v_{2}w_{4}-v_{4}w_{1}+v_{4}w_{2},\quad
B^{y}_{3}=u_{1}w_{2}-u_{1}w_{4}-u_{2}w_{1}+u_{2}w_{4}+u_{4}w_{1}-u_{4}w_{2}$
$\displaystyle
B^{z}_{3}=-u_{1}v_{2}+u_{1}v_{4}+u_{2}v_{1}-u_{2}v_{4}-u_{4}v_{1}+u_{4}v_{2},\quad
B^{x}_{4}=v_{1}w_{2}-v_{1}w_{3}-v_{2}w_{1}+v_{2}w_{3}+v_{3}w_{1}-v_{3}w_{2}$
$\displaystyle
B^{y}_{4}=-u_{1}w_{2}+u_{1}w_{3}+u_{2}w_{1}-u_{2}w_{3}-u_{3}w_{1}+u_{3}w_{2},\quad
B^{z}_{4}=u_{1}v_{2}-u_{1}v_{3}-u_{2}v_{1}+u_{2}v_{3}+u_{3}v_{1}-u_{3}v_{2},$

$\displaystyle
E=\bar{u}_{1}^{2}+\bar{v}_{1}^{2}+\bar{w}_{1}^{2}+\bar{u}_{2}^{2}+\bar{v}_{2}^{2}+\bar{w}_{2}^{2}+\bar{u}_{3}^{2}+\bar{v}_{3}^{2}+\bar{w}_{3}^{2}+\bar{u}_{4}^{2}+\bar{v}_{4}^{2}+\bar{w}_{4}^{2},$
(30) $\displaystyle
L_{x}=\bar{w}_{1}\bar{y}_{1}+\bar{w}_{2}\bar{y}_{2}+\bar{w}_{3}\bar{y}_{3}+\bar{w}_{4}\bar{y}_{4}-\bar{v}_{1}\bar{z}_{1}-\bar{v}_{2}\bar{z}_{2}-\bar{v}_{3}\bar{z}_{3}-\bar{v}_{4}\bar{z}_{4},$
$\displaystyle
L_{y}=\bar{u}_{1}\bar{z}_{1}+\bar{u}_{2}\bar{z}_{2}+\bar{u}_{3}\bar{z}_{3}+\bar{u}_{4}\bar{z}_{4}-\bar{w}_{1}\bar{x}_{1}-\bar{w}_{2}\bar{x}_{2}-\bar{w}_{3}\bar{x}_{3}-\bar{w}_{4}\bar{x}_{4},$
$\displaystyle
L_{z}=\bar{v}_{1}\bar{x}_{1}+\bar{v}_{2}\bar{x}_{2}+\bar{v}_{3}\bar{x}_{3}+\bar{v}_{4}\bar{x}_{4}-\bar{u}_{1}\bar{y}_{1}-\bar{u}_{2}\bar{y}_{2}-\bar{u}_{3}\bar{y}_{3}-\bar{u}_{4}\bar{y}_{4},$
$\displaystyle L^{cm}_{x}=w_{cm}y_{cm}-v_{cm}z_{cm},\quad
L^{cm}_{y}=u_{cm}z_{cm}-w_{cm}x_{cm},\quad
L^{cm}_{z}=v_{cm}x_{cm}-u_{cm}y_{cm},$ $\displaystyle
V=\frac{1}{3}(x_{1}A^{x}_{1}+y_{1}A^{y}_{1}+z_{1}A^{z}_{1}+x_{2}A^{x}_{2}+y_{2}A^{y}_{2}+z_{2}A^{z}_{2}+x_{3}A^{x}_{3}+y_{3}A^{y}_{3}+z_{3}A^{z}_{3}+x_{4}A^{x}_{4}+y_{4}A^{y}_{4}+z_{4}A^{z}_{4}),$
$\displaystyle\omega_{x}=v_{1}A^{z}_{1}-w_{1}A^{y}_{1}+v_{2}A^{z}_{2}-w_{2}A^{y}_{1}+v_{3}A^{z}_{3}-w_{3}A^{y}_{3}+v_{4}A^{z}_{4}-w_{4}A^{y}_{4},$
$\displaystyle\omega_{y}=w_{1}A^{x}_{1}-u_{1}A^{z}_{1}+w_{2}A^{x}_{2}-u_{2}A^{z}_{1}+w_{3}A^{x}_{3}-u_{3}A^{z}_{3}+w_{4}A^{x}_{4}-u_{4}A^{z}_{4},$
$\displaystyle\omega_{z}=u_{1}A^{y}_{1}-v_{1}A^{x}_{1}+u_{2}A^{y}_{2}-v_{2}A^{x}_{2}+u_{3}A^{y}_{3}-v_{3}A^{x}_{3}+u_{4}A^{y}_{4}-v_{4}A^{x}_{4},$
$\displaystyle
D=u_{1}A^{x}_{1}+v_{1}A^{y}_{1}+w_{1}A^{z}_{1}+u_{2}A^{x}_{2}+v_{2}A^{y}_{2}+w_{2}A^{z}_{2}+u_{3}A^{x}_{3}+v_{3}A^{y}_{3}+w_{3}A^{z}_{3}+u_{4}A^{x}_{4}+v_{4}A^{y}_{4}+w_{4}A^{z}_{4},$
$\displaystyle
I=\bar{x}_{1}^{2}+\bar{y}_{1}^{2}+\bar{z}_{1}^{2}+\bar{x}_{2}^{2}+\bar{y}_{2}^{2}+\bar{z}_{2}^{2}+\bar{x}_{3}^{2}+\bar{y}_{3}^{2}+\bar{z}_{3}^{2}+\bar{x}_{4}^{2}+\bar{y}_{4}^{2}+\bar{z}_{4}^{2},$
$\displaystyle
I_{xx}=\bar{x}_{1}^{2}+\bar{x}_{2}^{2}+\bar{x}_{3}^{2}+\bar{x}_{4}^{2},\quad
I_{yy}=\bar{y}_{1}^{2}+\bar{y}_{2}^{2}+\bar{y}_{3}^{2}+\bar{y}_{4}^{2},\quad
I_{zz}=\bar{z}_{1}^{2}+\bar{z}_{2}^{2}+\bar{z}_{3}^{2}+\bar{z}_{4}^{2},$
$\displaystyle
I_{xy}=\bar{x}_{1}\bar{y}_{1}+\bar{x}_{2}\bar{y}_{2}+\bar{x}_{3}\bar{y}_{3}+\bar{x}_{4}\bar{y}_{4},\quad
I_{yz}=\bar{y}_{1}\bar{z}_{1}+\bar{y}_{2}\bar{z}_{2}+\bar{y}_{3}\bar{z}_{3}+\bar{y}_{4}\bar{z}_{4},\quad
I_{zx}=\bar{z}_{1}\bar{x}_{1}+\bar{z}_{2}\bar{x}_{2}+\bar{z}_{3}\bar{x}_{3}+\bar{z}_{4}\bar{x}_{4},$
$\displaystyle
G=\bar{u}_{1}\bar{x}_{1}+\bar{v}_{1}\bar{y}_{1}+\bar{w}_{1}\bar{z}_{1}+\bar{u}_{2}\bar{x}_{2}+\bar{v}_{2}\bar{y}_{2}+\bar{w}_{2}\bar{z}_{2}+\bar{u}_{3}\bar{x}_{3}+\bar{v}_{3}\bar{y}_{3}+\bar{w}_{3}\bar{z}_{3}+\bar{u}_{4}\bar{x}_{4}+\bar{v}_{4}\bar{y}_{4}+\bar{w}_{4}\bar{z}_{4},$

$\displaystyle
K=x_{1}B^{x}_{1}+y_{1}B^{y}_{1}+z_{1}B^{z}_{1}+x_{2}B^{x}_{2}+y_{2}B^{y}_{2}+z_{2}B^{z}_{2}+x_{3}B^{x}_{3}+y_{3}B^{y}_{3}+z_{3}B^{z}_{3}+x_{4}B^{x}_{4}+y_{4}B^{y}_{4}+z_{4}B^{z}_{4},$
(31) $\displaystyle
K_{xx}=x_{1}B^{x}_{1}+x_{2}B^{x}_{2}+x_{3}B^{x}_{3}+x_{4}B^{x}_{4},\quad
K_{yy}=y_{1}B^{y}_{1}+y_{2}B^{y}_{2}+y_{3}B^{y}_{3}+y_{4}B^{y}_{4},\quad
K_{zz}=z_{1}B^{z}_{1}+z_{2}B^{z}_{2}+z_{3}B^{z}_{3}+z_{4}B^{z}_{4},$
$\displaystyle
K_{xy}=x_{1}B^{y}_{1}+x_{2}B^{y}_{2}+x_{3}B^{y}_{3}+x_{4}B^{y}_{4},\quad
K_{yx}=y_{1}B^{x}_{1}+y_{2}B^{x}_{2}+y_{3}B^{x}_{3}+y_{4}B^{x}_{4},$
$\displaystyle
K_{yz}=y_{1}B^{z}_{1}+y_{2}B^{z}_{2}+y_{3}B^{z}_{3}+y_{4}B^{z}_{4},\quad
K_{zy}=y_{1}B^{z}_{1}+y_{2}B^{z}_{2}+y_{3}B^{z}_{3}+y_{4}B^{z}_{4},$
$\displaystyle
K_{zx}=z_{1}B^{x}_{1}+z_{2}B^{x}_{2}+z_{3}B^{x}_{3}+z_{4}B^{x}_{4},\quad
K_{xz}=x_{1}B^{z}_{1}+x_{2}B^{z}_{2}+x_{3}B^{z}_{3}+x_{4}B^{z}_{4},$
$\displaystyle
J=(A^{x}_{1})^{2}+(A^{y}_{1})^{2}+(A^{z}_{1})^{2}+(A^{x}_{2})^{2}+(A^{y}_{2})^{2}+(A^{z}_{2})^{2}+(A^{x}_{3})^{2}+(A^{y}_{3})^{2}+(A^{z}_{3})^{2}+(A^{x}_{4})^{2}+(A^{y}_{4})^{2}+(A^{z}_{4})^{2},$
$\displaystyle
C_{1}=u_{2}x_{3}-u_{2}x_{4}-u_{3}x_{2}+u_{3}x_{4}+u_{4}x_{2}-u_{4}x_{3}+v_{2}y_{3}-v_{2}y_{4}-v_{3}y_{2}+v_{3}y_{4}+v_{4}y_{2}-v_{4}y_{3}+w_{2}z_{3}-w_{2}z_{4}-w_{3}z_{2}+w_{3}z_{4}+w_{4}z_{2}-w_{4}z_{3},$
$\displaystyle
C_{2}=-u_{1}x_{3}+u_{1}x_{4}+u_{3}x_{1}-u_{3}x_{4}-u_{4}x_{1}+u_{4}x_{3}-v_{1}y_{3}+v_{1}y_{4}+v_{3}y_{1}-v_{3}y_{4}-v_{4}y_{1}+v_{4}y_{3}-w_{1}z_{3}+w_{1}z_{4}+w_{3}z_{1}-w_{3}z_{4}-w_{4}z_{1}+w_{4}z_{3},$
$\displaystyle
C_{3}=u_{1}x_{2}-u_{1}x_{4}-u_{2}x_{1}+u_{2}x_{4}+u_{4}x_{1}-u_{4}x_{2}+v_{1}y_{2}-v_{1}y_{4}-v_{2}y_{1}+v_{2}y_{4}+v_{4}y_{1}-v_{4}y_{2}+w_{1}z_{2}-w_{1}z_{4}-w_{2}z_{1}+w_{2}z_{4}+w_{4}z_{1}-w_{4}z_{2},$
$\displaystyle
C_{4}=-u_{1}x_{2}+u_{1}x_{3}+u_{2}x_{1}-u_{2}x_{3}-u_{3}x_{1}+u_{3}x_{2}-v_{1}y_{2}+v_{1}y_{3}+v_{2}y_{1}-v_{2}y_{3}-v_{3}y_{1}+v_{3}y_{2}-w_{1}z_{2}+w_{1}z_{3}+w_{2}z_{1}-w_{2}z_{3}-w_{3}z_{1}+w_{3}z_{2},$

The conserved quantities of the 3D case are

$u_{cm},v_{cm},w_{cm},L^{cm}_{x},L^{cm}_{y},L^{cm}_{z},L_{x},L_{y},L_{z},E,V,D,C_{1},C_{2},C_{3},C_{4}$

The relationships between the conserved quantities are

$C_{1}+C_{2}+C_{3}+C_{4}=0,\quad
u_{cm}L^{cm}_{x}+v_{cm}L^{cm}_{y}+w_{cm}L^{cm}_{z}=0,$

which means there are only 14 independent conserved quantities.

Global conserved quantities can be expressed in terms of integrals, where
subscript $g$ stands for global and $\rho$ is density (suppose the support of
the fluid is $\Omega\in\mathbb{R}^{3}$, the boundary is $\partial\Omega$, $C$
stands for any closed loop on $\partial\Omega$):

$\displaystyle V_{g}=\iiint_{\Omega}\,dV\quad{\rm(volume,scalar)}$ (32)
$\displaystyle\mathbf{p}_{g}=\iiint_{\Omega}\rho\textbf{v}\,dV,\quad\textbf{v}=(u,v,w),\quad{\rm(momentum,3D\
vector)}$ $\displaystyle
E_{g}=\left(\iiint_{\Omega}\rho|\textbf{v}|^{2}\,dV\right)-\frac{|\mathbf{p}_{g}|^{2}}{2\rho
V_{g}}\quad{\rm(energy\ in\ COM,scalar)}$ $\displaystyle
L_{cm,g}=\mathbf{r}_{cm}\times\mathbf{p}_{g}\quad\mathbf{r}_{cm}\equiv\frac{1}{V_{g}}\iiint_{\Omega}\mathbf{r}dV\quad{\rm(momentum\
of\ COM,3D\ vector)}$ $\displaystyle
L_{g}=\left(\iiint_{\Omega}\rho\textbf{r}\times\textbf{v}\,dV\right)-L_{cm,g}\quad{\rm(momentum\
in\ COM,3D\ vector)}$ $\displaystyle
D_{g}=\oiint_{\partial\Omega}\,\mathbf{v}\cdot d\mathbf{n}\equiv
0\quad{\rm(outflow,scalar)}$ $\displaystyle
C_{g}=\oint_{C\in\partial\Omega}\textbf{v}\cdot\,d\textbf{l}\quad{\rm(circulation,scalar)}$

Table 4: Physical interpretations of important quantities (3D) Notation | Meaning
---|---
$u_{cm}$ | velocity of center of mass in x direction
$v_{cm}$ | velocity of center of mass in y direction
$w_{cm}$ | velocity of center of mass in z direction
$L^{cm}_{x}$ | Angular momentum of center of mass in x direction
$L^{cm}_{y}$ | Angular momentum of center of mass in y direction
$L^{cm}_{z}$ | Angular momentum of center of mass in z direction
$L_{x}$ | Angular momentum with respect to center of mass in x direction
$L_{y}$ | Angular momentum with respect to center of mass in y direction
$L_{z}$ | Angular momentum with respect to center of mass in z direction
$E$ | Energy with respect to center of mass
$V$ | Volume
$D$ | Divergence
$C_{i}$ | Circulations $i=1,2,3,4$
$\mathbf{I}$ | Moment of inertia with respect to center of mass
$\mathbf{\omega}$ | Vorticity
$G$ | Derivative of moment of inertia


# Exploratory Data Analysis on Code-mixed Misogynistic Comments

1st Sargam Yadav  School of Informatics and Creative Arts
Dundalk Institute of Technology
Dundalk, Ireland
<EMAIL_ADDRESS>2nd Abhishek Kaushik  School of Informatics and Creative
Arts
Dundalk Institute of Technology
Dundalk, Ireland
<EMAIL_ADDRESS>3rd Kevin Mc Daid  School of Informatics and
Creative Arts
Dundalk Institute of Technology
Dundalk, Ireland
<EMAIL_ADDRESS>
###### Abstract

The problems of online hate speech and cyberbullying have significantly
worsened since the increase in popularity of social media platforms such as
YouTube and Twitter (X). Natural Language Processing (NLP) techniques have
proven to provide a great advantage in automatic filtering such toxic content.
Women are disproportionately more likely to be victims of online abuse.
However, there appears to be a lack of studies that tackle misogyny detection
in under-resourced languages. In this short paper 111This paper is accepted in
the 16th ISDSI-Global Conference 2023 https://isdsi2023.iimranchi.ac.in/, we
present a novel dataset of YouTube comments in mix-code Hinglish collected
from YouTube videos which have been weak labelled as ‘Misogynistic’ and ‘Non-
misogynistic’. Pre-processing and Exploratory Data Analysis (EDA) techniques
have been applied on the dataset to gain insights on its characteristics. The
process has provided a better understanding of the dataset through sentiment
scores, word clouds, etc.

###### Index Terms:

Code-mixed Languages, Machine learning, Hinglish, Natural language processing,
Misogyny

##  I Introduction

In the era of the internet and digital communication, social media platforms
have been integrated as a viable component in many companies, businesses, and
even the daily lives of people. They serve as the hub of connectivity and
communication. However, this increase in interconnectedness has brought about
several new issues such as cyberbullying, online hate speech [1], and
misinformation [2]. Platforms employ techniques such as keyword filtering and
manual content moderation to remove hateful and offensive content. However,
this approach generally proves to be insufficient, especially for under-
resourced and mix-code languages [3]. This is because while detecting hate
speech, it is important to keep the context of the conversation in mind [4].
Users from multicultural and multilingual countries generally combine their
local languages with English in online posts. For example, ‘Hinglish’ is a
combination of Hindi and English, and consists of Romanized Hindi text mixed
with English words [5].

In this paper, we present a novel dataset for misogyny detection in code-
switched Hinglish. The dataset was collected from YouTube videos and has been
weakly annotated by a single annotator into the following two categories:
Misogynistic (‘MGY’) and Not-misogynistic (NOT). EDA has been performed on the
dataset to understand its nature and characteristics, find patterns, detect
anomalies, gain insight, and receive feedback. Modelling decisions will be
made based on the findings of the study. The rest of the paper is structured
as follows: Section II discusses the motivation behind the conducting the
study, the hypothesis, and research questions. Section III discusses the
relevant literature in the field of misogyny and hate speech detection in
code-mixed and other languages. Section IV provides details of the dataset. In
Section V, EDA is performed on the dataset to find patterns in the data. In
Section VI, we discuss the findings and insights from the study and how they
answer the research questions. Section VII concludes the study.

##  II Motivation

The motivation behind conducting this study is to provide a new dataset of
2229 YouTube comments for misogyny detection in Hinglish, perform exploratory
data analysis on the dataset, and gain insight from the findings to guide
future modelling decisions. There is a lack of robust datasets for misogyny
detection in mix-code languages [6]. Data was gathered with established
ethical approval. The dataset has only one annotator, and another one is
currently working on annotation, hence it is only sparsely annotated. The main
reason for submitting the paper was to solicit expert feedback on the study’s
workflow.

Hypothesis: EDA techniques can derive valuable insights from a code-mixed
Hinglish dataset for misogyny detection and find patterns and relationships in
the comments and labels.

  1. 1.

Research Question 1: Can EDA techniques help uncover valuable insights from a
dataset of Hinglish comments for the misogyny detection?

  2. 2.

Research Question 2: What valuable insights and patterns are uncovered? Which
EDA techniques provide valuable insights?

##  III Literature Review

In this section, we will discuss the relevant literature in the fields of
misogyny detection and code-mixed languages.

###  III-A Misogyny Detection

There has been some effort in misogyny detection in languages other than
English. Fersini et al. [7] conducted the Automatic misogyny identification
(AMI) in Italian and English. Anzovino et al. [8] used a similar taxonomy for
misogyny detection and achieved the highest score using a Linear Support
Vector Machine. Kirk et al. [9] conducted the Explainable Detection of Online
Sexism at SemEval-2023 that consisted of 3 subtasks. Subtask 3 consisted of 11
fine-grained vectors. The highest scores were achieved by Transformer based
models such as DeBERTa-v3.

###  III-B Code-mixed Languages

The efforts in hate speech detection for languages such as Marathi, Hindi, and
more, have seen a dramatic rise [3] [10]. The Identification of Conversational
Hate-Speech in Code-Mixed Languages (ICHCL) shared task focused mainly on
Hindi and Hinglish. Similarly, the TRAC subtask [11] was also conducted in
mix-code Hinglish. Other works that conduct sentiment analysis on mix-code
data include the study done by Kaur et al. [12], where they implemented
machine learning models and different feature engineering techniques on a
YouTube cookery dataset in mix-code Hinglish. Donthula et al. [13] continued
the experimentation by using deep learning models, contextual embedding
techniques, and statistical methods such as TF-IDF and CountVec. In the study
done by Yadav et al. [14], BERT-based models were used to perform sentiment
analysis on the cookery dataset. This approach was expanded on by Yadav et al.
[15], who utilised contextual embeddings from BERT-based models such as
RoBERTa. This approach outperformed the approach taken by Kaur et al. [12] and
Yadav et al. [14].

##  IV Dataset

In this section, we will discuss the details of the dataset. The dataset was
collected from YouTube videos discussing current social issues in India. Table
I provides a list of the videos, the channel names, and other information
regarding the videos. All comments that are not in English or Romanized Hindi
have been removed from the dataset. Any replies to comment that did not form a
logical sentence in itself have also been removed. Table II depicts the number
of comments belonging to each of the classes. The classes are highly
imbalanced, as is usually the case in hate speech and misogyny detection [7],
since there are very few positive samples in the real world. The ‘MGY’ class
consists of only 181 comments.

Name |  Channel |  No. of Subscribers |  Date Uploaded |  No. of views |  Extracted comments |  Final comments
---|---|---|---|---|---|---
Know why Delhi is called the Rape Capital |  ThePrint |  2.32 million |  Dec 15, 2019 |  987,073 |  11,344 |  1149
Use vs Abuse - Debate over declaring Marital Rape a crime |  The Deshbhakt |  3.43 million |  Jan 26, 2022 |  316,794 |  4362 |  621
What people think about bringing the Marital rape law in India? |  InUth |  44.8K |  Jul 4, 2022 |  26,724 |  384 |  92
Misogyny in Indian Politics: Time to end this disease! |  The Deshbhakt |  3.43 million |  Jan 19, 2022 |  441,144 |  1702 |  362
TABLE I: Video Titles Label name | Count
---|---
NOT | 2048
MGY | 181
Total | 2,229
TABLE II: Label counts

##  V Exploratory Data Analysis

In this section, we will discuss the details of the research methodology and
analyse the results.

###  V-A Pre-processing

Data cleaning and pre-processing was performed on the dataset before using EDA
techniques. All non-ASCII characters, punctuation, stopwords, non-English
characters, hyperlinks, extra spaces and lines, were removed from the dataset.
Stemming was performed using Snowball Stemmer.

###  V-B Analysis

We will now discuss the various EDA techniques and their results on the
dataset. The most common tokens in the dataset are: ‘i’ (888), ‘rape’ (682),
‘women’ (661), ‘men’ (499), and ‘hai’ (492). The average number of characters
in each comment are 115.22 and the average number of words are 20.96.

Fig 1 displays a word cloud for all the comments in the dataset. The most
frequent words in the dataset are ‘women’, ‘men’, ‘law’, ‘rape’, etc. Table
III shows the average word count for the sequences in each class. Comments
belonging to the ‘MGY’ class are 1.6 times longer on an average than comments
belonging to the ‘NOT’ class.

Figure 1: Word Cloud for All Comments | MGY | NOT
---|---|---
Average Word Count | 32.72 | 19.91
TABLE III: Average Word Count for Each Class Figure 2: Histogram of Sentiment
Polarity vs the Comment Count

Fig 2 displays a histogram that shows the distribution of sentiment polarity
score for all the comments in the dataset. The x-axis represents the sentiment
analysis scores of the comments obtained using TextBlob, and ranges from -1
(lowest) to 1 (highest). The y-axis displays the total count of comments. Each
of the bins depicts a range of polarity scores. The histogram is slightly
skewed to the right, depicting that the number of comments with a positive
score is slightly higher. The highest number of comments occur in the bin
between 0 and 0.0625. The histogram is widely spread out indicating that
comments have a wide variety of sentiment scores. Fig 3 is a histogram that
displays the distribution of comment length with a Kernel Density Estimate
(KDE) curve. The x-axis represents the number of words in each comment. The
y-axis represents the number of comments that belong to each bin. As the
histogram is highly skewed to the left, the number of long comments is higher.

Figure 3: Histogram of Comment Length vs the Comment Count Figure 4: Boxplot
of Binary Label Distribution and Polarity

Fig 4 depicts a boxplot with a swarm plot overlay. The x-axis depicts the two
binary classes, and the y-axis represents the sentiment scores. Each box
presents the interquartile range (IQR).The small dots outside the box are
individual comments. Fig 5 displays a boxplot that depicts the binary labels
(‘NOT’ and ‘MGY’) on the x-axis and the comment length on the y-axis. The
central tendency of the box depicting class ‘MGY’ is higher than that of the
box depicting class ‘NOT’. The box for class ‘MGY’ is bigger than for ‘NOT’,
which indicates that the comment length is generally longer for class ‘NOT’.
The plot also depicts an outlier in the class ‘NOT’, with a comment of length
of over 1750 characters.

Figure 5: Boxplot of Binary Label Distribution and Comment Length

###  V-C Principal Component Analysis

PCA has been performed on the dataset to reduce dimensionality of the data and
find patterns. TF-IDF vectorizer with the maximum number of features = 3000
was used to obtain numerical features from the dataset. All the comments from
the dataset have been selected for analysis. Fig 6 displays a plot of the
Elbow method, which shows that the optimal number of clusters was 3.

Figure 6: Elbow Method for PCA Figure 7: PCA Plot for All Comments

The top keywords for cluster 1 are women, peopl, men, like, india, girl, feel,
wear, this, indian. The top keywords for cluster 2 are hai, ko, ki, ke, bhi,
se, ka, ye, aur, nahi. The top keywords for cluster 3 are rape, law, case,
marit, consent, husband, wife, sex, women, men. Thus, cluster 2 consists
mostly of mix-code tokens. An example of a comment in cluster 1 is: ‘Marriage
doesn’t mean only sex. It’s emotional physical and spiritual’. An example of a
comment in cluster 2 is: ‘Soch par fakrah hay !! Savage!!’ An example of a
comment in cluster 3 is: ‘the fact there is not any law regarding such a big
issue tells a lot about our country’.

Feature | Cluster 1 | Cluster 2 | Cluster 3
---|---|---|---
Cluster Size | 466 | 1381 | 382
Sentiment score | 0.0294 | 0.0137 | 0.0259
Average length | 16 | 26 | 30
TABLE IV: Cluster Characteristics using PCA

Fig 7 displays the PCA plot of the dataset which shows 3 distinct clusters.
Table IV displays scores for the 3 clusters obtained from the PCA plot.
Cluster 2 has the highest number of comments and the lowest polarity score.
This cluster contains mostly mix-code sentences and makes up the largest
portion of the dataset. Cluster 3 has the highest positive sentiment score and
the highest average length in words.

##  VI Discussion

  1. 1.

Research Question 1: Can EDA techniques help uncover valuable insights from a
dataset of Hinglish comments for the misogyny detection? In this paper, EDA
techniques, such as comparison of samples across classes, PCA, keyword
extraction, sentiment analysis, word cloud, and more, have been applied to the
dataset. The analysis has provided valuable insights into the comment
distribution across the dataset, polarity of comments, keywords, and more.
Keyword extraction is an efficient strategy for hate speech classification
through ‘hate’ keywords [16] [17].

  2. 2.

Research Question 2: What valuable insights and patterns are uncovered? Which
EDA techniques provide valuable insights? The findings of the study suggest
that misogynistic comments are generally longer than non-misogynistic
comments. This aligns with previous studies that discuss sequence length and
the corresponding class [18]. Most comments consist of less than 20 words, and
most of the comments show slightly positive sentiment score using TextBlob.

##  VII Conclusion

This study was performed to conduct EDA on a code-mixed Hinglish dataset for
misogyny detection. Several techniques such as word cloud, sentiment analysis,
PCA, etc., were applied to the dataset. The findings have uncovered some
interesting and useful correlations and patterns in the data, which can guide
the modelling process. PCA obtained three clusters of comments, where code-
mixed comments were clustered separately. Further analysis could reveal other
important factors, and will be conducted alongside the training and evaluation
of machine learning and deep learning models. In future studies, we will
finish the fair annotation process with the Kappa coefficient and use machine
learning algorithms on the dataset to investigate the implementation stage.

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