If $\Sigma$ is a possibilistic Horn-NC base, then Find($\Sigma$, 0) performs
at most $n^{2}$ recursive calls, $n$ being the number of symbols of $\Sigma$.

  * Proof.

Find stops when it detects that the current base is consistent. If it is
inconsistent then $\langle(\vee):\alpha\rangle$ is deduced, and for future
calls, Find cancels the set $\\{\langle\phi:\beta\rangle\ |\
\langle\phi:\beta\rangle\in\Sigma,\beta\leq\alpha\\}$. This set trivially
contains at least one formula as $\langle(\vee):\alpha\rangle$ has been
derived. On the other hand and as discussed in the previous proof, the maximum
number of inserted formulas in the base is at most $m\times k$, where $m$ is
the number of sub-formulas in $\Sigma$ and $k$ is the number of different
weights in $\Sigma$. Thus, the number of performed recursive calls is at most
$m\times k$. Since $m,k\leq n$, then Proposition 7.7 holds. $\blacksquare$

The next proposition states that the overall complexity to determine
$\mbox{Inc}(\Sigma)$, $\Sigma\in\mathcal{\overline{H}}_{\Sigma}$, is
polynomial.

######  Proposition 7.8.

If $\Sigma$ is a possibilistic Horn-NC base, then computing
$\mbox{Inc}(\Sigma)$ takes polynomial time, i.e. the class
$\mathcal{\overline{H}}_{\Sigma}$ is polynomial.

  * Proof.

On the one hand, by Propositions 7.6 and 7.7, the number of performed: the
hypothesis holds. inferences in each recursive call and the number of
recursive calls are both polynomial. Hence the total number of inferences is
polynomially bounded. On the other hand, it is not hard to find a data
structure so that each inference in $\mathcal{UR}_{\Sigma}$ can be
polynomially performed w.r.t the size of $\Sigma$. Therefore, the complexity
to determine $\mbox{Inc}(\Sigma)$ is polynomial. $\blacksquare$

Remark. Determining a tight polynomial degree of the worst-case complexity of
computing a Horn-NC base, is planed for future work (see Section 8). It should
be mentioned that the polynomial degree of computing their counterpart Horn
clausal bases has not been specified either.

##  8 Related and Future Work

In this section we briefly discuss related work in a number of possibilistic
logical contexts and succinctly propose objectives towards which our future
work can be oriented.

$\Box$ Discovering polynomial NC classes.

In propositional logic, the valuable contribution to clausal efficiency of the
conjunction of Horn formulas and Horn-SAT algorithms is reflected by the fact
that the highly efficient DPLL solvers embed a Horn-SAT-like algorithm, so-
called Unit Propagation. Hence, searching for polynomial (clausal) super-
classes of the Horn class in propositional logic has been a key issue for
several decades towards improving clausal reasoning and has led to a great
number of such classes being currently known: hidden-Horn, generalized Horn,
Q-Horn, extended-Horn, etc. (see [32, 40] for short reviews). So it is
arguable that, just as the tractable clausal fragment has helped to grow
overall clausal efficiency, likewise widening the tractable non-clausal
fragment would grow overall non-clausal efficiency. Indeed, inspired by such
polynomial non-clausal classes, efficient non-clausal algorithms can be
devised. Nevertheless, we stress that the current tractable non-clausal
fragment is almost empty, which signifies a manifest disadvantage with respect
to its clausal counterpart. We will extrapolate the previous argumentation to
possibilistic logic and determine further NC subclasses whose inconsistency-
degree computing have polynomial complexity. In our next work, we will search
for polynomial classes obtained by lifting the possibilistic renameable (or
hidden) Horn formulas to the NC level.

$\Box$ Designing low-degree polynomial algorithms.

As we have seen in Section 7, the number of inferences required by
$\mathcal{UR}_{\Sigma}$ and the number of recursive calls to Find are both
bounded by $O(n^{2})$, where $n$ is the symbol number of the input base. Also,
all inferences in $\mathcal{UR}_{\Sigma}$ can be reasonably performed in
$O(n)$. Altogether, the complexity of determining $\mbox{Inc}(\Sigma)$ of
Horn-NC bases is in $O(n^{5})$. This complexity, though polynomial, is of
course no satisfactory for real-world applications where the size of formulas
can be relatively huge. However no much care has been taken in the proofs of
Section 7 because the goal was proving tractability, and so, first of all, a
fine-grained analysis of complexity is pending. This analysis will permit
likely to precise a tighter polynomial-degree complexity. Nevertheless, the
work presented here should be resumed in order to notably decrease the
polynomial-degree. Also, advances in several directions are obliged such as
finding suitable data-structure and proposing optimized algorithms. Finally,
we mention that this optimization effort has not been done so far in the
clausal framework either. In fact, no explicit upper bounds of complexity
algorithms that compute possibilistic clausal bases are available.

$\Box$ Combining necessity and possibility measures.

Our approach considers only necessity-valued formulas but it can be extended
to bases where formulas can have associated a possibility or a necessity
measure. Such logical context has already been studied by previous authors
[37, 41, 20]. If a formula is possibility-valued
$\langle\varphi:\Pi(\varphi)\geq\alpha\rangle$ it can be converted into the
follow: the hypothesis holds.ing necessity-valued formula
$\langle\neg\varphi:N(\neg\varphi)\leq 1-\alpha\rangle$. Hence, bases having
both possibility and necessity formulas can be converted into equivalent
necessity formulas, where the threshold $\alpha$ is additionally accompanied
by the indication of whether the formula must have a necessity level either
greater or smaller than $\alpha$. On the other side, resolution for
possibility-necessity formulas is well-known [22, 25, 26], but its extension
to the NC level towards defining a possibility/necessity NC unit-resolution
calculus seems not to be trivial and is an open question.

$\Box$ Computing general NC bases:

Since $\mathcal{\overline{H}}_{\Sigma}$ is an NC sub-class, computing
arbitrary NC bases is a natural continuation of the presented approach. As
said in the Introduction, real-world problems are generally expressed in NC
form, and so clausal reasoning requires the usage of a previous NC-to-clausal
transformation. However, such transformation is highly inadvisable as it
increases the formula size and number of variables, and losses the logical
equivalence and the formula’s original structure. Besides the clausal form is
not unique and how to guide the nondeterministic process towards a ”good”
clausal formula is not known. So, real-world efficiency is reached if formulas
are computed in its original non-clausal form. Here we have favored non-
clausal reasoning with contributions which particularly facilitate deduction
based on resolution and DPLL. Brief discussions for future work related to how
compute possibilistic general NC formulas by means of resolution and DPLL can
be found below in the items “$\Box$ Defining NC Resolution” and “$\Box$
Defining NC DPLL”.

$\Box$ Developing NC logic programming.

Possibilistic logic programming has received notable attention, and in fact,
after the pioneer work in [21], a number of approaches e.g. [10, 3, 4, 2] are
available in the state-of-the-art. However, all of them focus exclusively on
the clausal form and so are Horn-like programs where the body of rules is a
conjunction of propositions and the head is a single proposition. Our future
work in this research direction will be oriented to show that
$\mathcal{\overline{H}}_{\Sigma}$ and $\mathcal{UR}_{\Sigma}$ allow to handle
non-clausal logic programs, and more concretely:

$(1)$ To conceive a language for Possibilistic Non-Clausal Logic Programing
denoted $\mathcal{LP}_{\Sigma}$. So, instead of Horn-like rules, in
$\mathcal{LP}_{\Sigma}$, one may handle Horn-NC-like rules wherein bodies and
heads of rules are NC formulas with slight syntactical restrictions (issued
from $\mathcal{\overline{H}}_{\Sigma}$). In fact, one can check that a rule
with syntax $\langle\Pi\rightarrow\mathrm{Hnc}:\alpha\rangle$, where $\Pi$ is
an NC formula with only positive literals and Hnc is a Horn-NC formula, is
indeed a possibilistic Horn-NC formula. Therefore a set of such kind of rules,
i.e. a program, is a Horn-NC base.

$(2)$ To answer queries with an efficiency qualitatively comparable to the
clausal one, i.e. polynomial, which is possible thanks to the next two
features: (i) an $\mathcal{LP}_{\Sigma}$ program belongs to
$\mathcal{\overline{H}}_{\Sigma}$ and $\mathcal{\overline{H}}_{\Sigma}$ is a
polynomial class; and (ii) the bases in $\mathcal{\overline{H}}_{\Sigma}$ have
only one minimal model, since, as aforementioned, they are equivalent to a
Horn clausal formula.

$\Box$ Developing NC answer set programming.

After the works in possibilistic logic programming, a succession of works on
possibilistic answer set programming has been carried out, started by [48] and
continued with e.g. [51, 52, 18, 19, 6, 7]. Although, like in possibilistic
logic programming, they also focus on the clausal form, there exists an
exception as indicated in the Introduction. Indeed, NC possibilistic logic has
been formerly dealt with by the auhors in [49, 50]. However, in this work no
effectiveness issues are addressed. Instead, the authors extend, to
possibilistic logic, proper concepts of non-clausal answer set programming
within classical logic as originally defined in [44]; so their aim is distinct
from ours. Our future work in this research direction will be oriented to show
the scope of the expressiveness of the class $\mathcal{\overline{H}}_{\Sigma}$
for possibilistic NC answer set programming and to analyze the efficiency
allowed by $\mathcal{\overline{H}}_{\Sigma}$. Although answering queries in
possibilistic answer set programming is an intractable problem [48], we do not
rule out the possibility of finding tractable sub-classes.

$\Box$ Partially ordered possibilistic logic.

In this work we have assumed that possibility and necessity measures, which
rank interpretations and formulas, were in the real unit-interval $[0,1]$,
i.e. the available information is supposed to be totally ordered. Thus for two
different necessity values $\alpha,\beta$, we have always that either
$\alpha>\beta$ or $\alpha<\beta$. However, considering a more real-world
context with only partial orders has already been studied [11, 12, 16]. For
instance, partial orders avoid comparing unrelated pieces of information,
which happens when we merge multiple sources information and the merged pieces
do not have a shared reference for uncertainty. So, considering a partial pre-
order on interpretations by means of a partial pre-order on formulas is a more
real-world scenario to which the presented method is planned to be
generalized.

$\Box$ Casting richer logics into possibilistic logic.

In our logical framework, propositional logic is casted into possibilistic
logic. Casting richer logics into possibilistic logic has already been carried
out by several authors, for instance, Alsinet et al. [3] cast Gödel logic
many-valued logic in a possibilistic framework and several researchers [37,
20, 57] do similarly with description logics. See reference [25] to know other
embeddings. Since the Horn-NC class has already been defined in [39] for
regular many-valued logic, our next aim in this research line will be to embed
regular-many-valued into possibilistic logic. Specifically, we will attempt to
define the class of regular-many-valued possibilistic Horn-NC bases as well as
to generalize other notions presented in this article such as defining
regular-many-valued possibilistic NC Unit-resolution. An open question is
whether polinomiallity is preserved when two uncertainty logics are combined
within the non-clausal level.

$\Box$ Defining NC resolution.

The formalization of the existing NC resolution [46] (see also [5]) that dates
back to the 1980s is somewhat confusing. Its functional-like definition has
important weaknesses such as not precisely identifying the available
resolvents or requiring complex formal proofs [36] of its logical properties.
Another symptom of its barriers is that, contrary to our approach, it has not
led so far, to define either NC unit-resolution or NC hyper-resolution. Our
definition of NC unit-resolution is clausal-like because, as stated in Section
5, in presence of clausal formulas, $\mathcal{UR}_{\Sigma}$ coincides with
clausal resolution [23, 24]. We believe that our kind of definition is fairly
well oriented to define (full) NC Resolution and to generalize it to some
uncertainty logics, and similarly, to be analyzed and to formally prove its
logical properties.

$\Box$ Defining NC DPLL.

DPLL for possibilistic clausal formulas have already been studied [22, 41] but
DPLL for possibilistic non-clausal formulas has received no attention. We
argue below that the present article supposes an important step forward to
specify the scheme NC DPLL for possibilistic logic. The principle of DPLL
relies on: (1) the procedure Unit-Propagation; (2) a suitable heuristic to
choice the literal $\ell$ on which performs branching; and (3) the
determination of the formulas $\Sigma\wedge\ell$ and $\Sigma\wedge\neg\ell$,
namely the formulas on which DPLL should split the search. One can do the next
observations: (1) NC Unit-Propagation is based on NC unit-resolution, and (2)
formulas $\Sigma\wedge\ell$ and $\Sigma\wedge\neg\ell$ are obtained with NC
unit-resolution. Hence, the unique aspect not studied here is that of the (2)
heuristic to choice the branching literal. Thus, for future work, we will
propose the DPLL schema for possibilistic non-clausal reasoning after studying
heuristics and taking a closer look to other aspects presented in this
article.

$\Box$ Generalized Possibilistic Logic

A restriction in standard possibilistic logic is that only conjunctions of
weighted formulas are allowed. Dubois et al. [28, 29, 27] have conceived the
Generalized Possibilistic Logic (GPL) in which the disjunction and the
negation of standard possibilistic formulas are handled. In this logic, for
instance the formula $(\vee\ \ \langle\varphi:\alpha\rangle\ \
\langle\psi:\beta\rangle\,)$ belongs to the GPL’s language, and since
possibility-valued formulas are also smoothly embedded in GPL, formulas like
the next one belong to GPL:

$\\{\wedge\ \ \langle(\vee\ P\ Q):\mbox{N}\ {\bf 1}\rangle\ \langle\neg
Q:\mbox{N}\ {\bf.75}\rangle\ \neg\langle(\vee\ \neg P\ R):\mbox{N}\ {\bf
1}\rangle\ \ \ (\vee\ \langle\neg R:\Pi\ {\bf.75}\rangle\ \langle\varphi:\Pi\
{\bf.25}\rangle\ )\ \\}$

One can see that in GPL, connectives can be internal or external (GPL is a
two-tired logic) with their corresponding and different semantics, and also,
GPL formulas can be expressed in NC form. In fact, observe that the previous
formula is non-clausal. In [29] the authors prove that the satisfiability
problem associated to GPL formulas is $\mathcal{NP}$-complete. It is clear
that the standard possibilistic Horn formulas are encapsulated in GPL and so
are the Horn-NC formulas defined here. We think that in GPL, sub-classes of
external Horn formulas can also be defined, as well as sub-classes that are
both internally and externally Horn. Thus GPL turns out to be an interesting
logic in the sense that it embeds a variety of classes of Horn-like formulas
which potentially could be lifted to the NC level. So for future work, we will
study the different classes of Horn-NC GPL formulas that are definable and
then attempt to prove their complexity. It will be challenging to determine
when polymiality is preserved in the NC level. Of course, once we have
mastered the solution of GPL Horn-NC-like formulas, the solving of
unrestricted non-clausal GPL formulas can be envisaged by proposing inference
and solving mechanisms.

$\Box$ Finding Models or Inconsistency Subsets.

Our method focuses on exclusively determining $\mbox{Inc}(\Sigma)$. However,
one important issue for increasing theoretical and practical interest is the
obtaining of models or contradictory subsets of the knowledge. In some
frameworks as for instance, when the knowledge base is not definitive and is
in an experimentation phase, the only data of $\mbox{Inc}(\Sigma)$ may be of
not much help. For example, if one expects the knowledge base to be consistent
and the consistency checker finds it is inconsistent, knowing the knowledge
subset causing contradiction, called ”witness” in the literature, can be
necessary. Thus for future work, we will envisage deductive calculi oriented
to providing witnesses as a return data. In this context, a more complicated
problem is the determination of whether a knowledge base has exactly one model
or one inconsistent subset, since determining if a problem has a unique
solution is computationally more expensive than testing if has at least one
solution (see [56], Chapter 17).

##  9 Conclusions

As the encoding of practical problems is usually expressed in non-clausal
form, restricting deductive systems to handle clausal formulas obliges them to
use non-clausal-to-clausal transformations, which are very expensive in terms
of: increase of formula size and number of variables, and loss of logical
equivalence and original formula’s structure. Further, the clausal form is not
unique and however no insights are available to guide towards a ”good” clausal
form the non-clausal-to-clausal transformation. These drawbacks deprive
clausal reasoning systems to efficiently perform in real-world applications.

To overcome such limitations and avoid the costs induced by the normal form
transformation, we process formulas in non-clausal form, concretely in
negation normal form (NNF). This form allows an arbitrary nesting of
conjunctions and disjunctions and only limits the scope of the negation
connective. NNF can be obtained deterministically and solely causes a
negligible, easily assumable increase of the formula size.

Thus, along the lines of previous works in propositional and regular many-
valued logics [40, 39], we have extrapolated the previous argumentation to
possibilistic logic, the most extended approach to deal with knowledge
impregnated of uncertainty and presenting partial inconsistencies. Thus our
first contribution has been lifting the possibilistic Horn class to the non-
clausal level obtaining a new possibilistic class, which has been called Horn
Non-Clausal, denoted $\mathcal{\overline{H}}_{\Sigma}$ and shown that it is a
sort of non-clausal analogous of the standard Horn class. Indeed, we have
proven that $\mathcal{\overline{H}}_{\Sigma}$ subsumes syntactically the Horn
class and that both classes are semantically equivalent. We have also proven
that all possibilistic NC bases whose clausal form is Horn belong to
$\mathcal{\overline{H}}_{\Sigma}$.

In order to compute the inconsistency degree of
$\mathcal{\overline{H}}_{\Sigma}$ members, we have established the calculus
Possibilistic Non-Clausal Unit-Resolution, denoted $\mathcal{UR}_{\Sigma}$. We
formally proved that $\mathcal{UR}_{\Sigma}$ correctly computes the
inconsistency degree of any $\mathcal{\overline{H}}_{\Sigma}$ base.
$\mathcal{\overline{H}}_{\Sigma}$ was nonexistent in the literature and
extends the propositional logic calculus given in [40] to possibilistic logic.

After having specified $\mathcal{\overline{H}}_{\Sigma}$ and
$\mathcal{UR}_{\Sigma}$, we have studied the computational problem of
computing the inconsistency degree of $\mathcal{\overline{H}}_{\Sigma}$ via
$\mathcal{UR}_{\Sigma}$ and determined that it is polynomial, and hence,
$\mathcal{\overline{H}}_{\Sigma}$ is the first found class to be
possibilistic, non-clausal and polynomial.

Our formulation of $\mathcal{UR}_{\Sigma}$ is unambiguously clausal-like
since, when applied to clausal formulas, $\mathcal{UR}_{\Sigma}$ indeed
coincides with clausal unit-resolution. This aspect is relevant in the sense
that it lays the foundations towards redefining NC resolution in a clausal-
like manner which could avoid the barriers caused by the existing functional-
like definition (see related work). We believe that this clausal-like
definition of NC resolution will allow to generalize it to some other
uncertainty logics.

Finally, in this work we also attempted to show that effective NC reasoning
for possibilistic and for some other uncertainty logics is an open research
field and, in view of our outcomes, we consider the presented research line is
rather promising. A symptom of such consideration is the possibility of our
method to be extended to different possibilistic logic contexts giving rise to
a number of future research directions that were briefly discussed:

Computing possibilistic arbitrary NC bases; discovering additional tractable
NC subclasses; conceiving low-degree-polynomial algorithms; extending
generalist possibilistic logic to NC; combining necessity and possibility
measures; considering partially ordered possibility measures; developing
possibilistic NC logic programming; developing possibilistic NC answer set
programming; casting richer logics in possibilistic logic; defining
possibilistic NC resolution; defining possibilistic NC DPLL; and finding
models.

Some of the above listed future objectives can also be searched in the context
of other non-classical logics such as Łukasiewicz logics, Gödel logic, product
logic, etc.

Funding Source. Spanish project ISINC (PID2019-111544GB-C21).

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##  10 Proofs of Section 4

Lemma 4.8. A NC disjunction $\varphi=(\vee\
\varphi_{1}\ldots\varphi_{i}\ldots\varphi_{k})$ with $k\geq 1$ disjuncts
pertains to $\mathcal{\overline{H}}$ iff $\varphi$ has $k-1$ negative
disjuncts and one Horn-NC disjunct, formally

$\varphi=(\vee\
\varphi_{1}\ldots\varphi_{i}\ldots\varphi_{k})\in\mathcal{\overline{H}}\mbox{\em\
\ iff}\quad\exists i\ \mbox{\em s.t.}\ \varphi_{i}\in\mathcal{\overline{H}}\ \
\mbox{\em and}\ \ \forall j\neq i,\varphi_{j}\in\mathcal{N}^{-}.$

  * Proof.

If: As the formulas $\forall j,j\neq i,\varphi_{j}$ have no positive literals,
the non-negative disjunctions of $\varphi=(\vee\
\varphi_{1}\ldots\varphi_{i}\ldots\varphi_{k})$ are those of $\varphi_{i}$
plus $\varphi_{i}$ and $\varphi$ themselves. Given that by hypothesis
$\varphi_{i}\in\mathcal{\overline{H}}$ and that $\forall j,j\neq
i,\varphi_{j}$ has no positive literals then all of them pertain to
$\mathcal{\overline{H}}$. Iff: It can be easily proved by contradiction that
if any of the two conditions of the lemma are unsatisfied, i.e. (i) $\exists
i,\varphi_{i}\notin\mathcal{\overline{H}}$ or (ii) $\exists i,j,i\neq
j,\varphi_{i},\varphi_{j}\notin\mathcal{N}^{-}$, then
$\varphi\notin\mathcal{\overline{H}}$. $\blacksquare$

Theorem 4.12. We have that $\mathcal{\widehat{H}}=\mathcal{\overline{H}}$.

  * Proof.

We prove first $\mathcal{\widehat{H}}\subseteq\mathcal{\overline{H}}$ and then
$\mathcal{\widehat{H}}\supseteq\mathcal{\overline{H}}$.

$\bullet$ $\mathcal{\widehat{H}}\subseteq\mathcal{\overline{H}}$ is easily
proven by structural induction as outlined below:

(1) $\mathcal{L}\subset\mathcal{\overline{H}}$ trivially holds.

(2) The non-recursive $\mathcal{\widehat{H}}$ conjunctions are literal
conjunctions, which trivially verify Definition 4.3 and so are in
$\mathcal{\overline{H}}$. Assume that
$\mathcal{\widehat{H}}\subseteq\mathcal{\overline{H}}$ holds until a given
inductive step and that $\varphi_{i}\in\mathcal{\widehat{H}},\
\varphi_{i}\in\mathcal{\overline{H}},\ 1\leq i\leq k$. In the next recursion,
any $\varphi=\\{\wedge\ \varphi_{1}\ldots\varphi_{i}\ldots\varphi_{k}\\}$ may
be added to $\mathcal{\widehat{H}}$. On the other hand, by induction
hypothesis, we have $\varphi_{i}\in\mathcal{\overline{H}},1\leq i\leq k$, and
so by Lemma 4.6, $\varphi\in\mathcal{\overline{H}}$. Therefore
$\mathcal{\widehat{H}}\subseteq\mathcal{\overline{H}}$ holds.

(3) The non-recursive disjunctions in $\mathcal{\widehat{H}}$ are obviously
Horn clauses, which trivially fulfill Definition 4.3 and so are in
$\mathcal{\overline{H}}$. Then assuming that for a given recursive level
$\mathcal{\widehat{H}}\subseteq\mathcal{\overline{H}}$ holds, in the next
recursion, only disjunctions $\varphi$ in (3) are added to
$\mathcal{\widehat{H}}$. But the condition of (3) and that of Lemma 4.8 are
equal; so by Lemma 4.8, $\varphi$ is in $\mathcal{\overline{H}}$ also.
Therefore $\mathcal{\widehat{H}}\subseteq\mathcal{\overline{H}}$ holds.

$\bullet$ $\mathcal{\widehat{H}}\supseteq\mathcal{\overline{H}}$. Given that
the structures to define $\mathcal{{NC}}$ and $\mathcal{\widehat{H}}$ in
Definition 2.3 and Definition 4.11, respectively, are equal, the potential
inclusion of each NC formula $\varphi$ in $\mathcal{\widehat{H}}$ is
systematically considered. Further, the statement: if
$\varphi\in\mathcal{\overline{H}}$ then $\varphi\in\mathcal{\widehat{H}}$, is
proven by structural induction on the depth of formulas, by applying a
reasoning similar to that of the previous
$\mathcal{\widehat{H}}\subseteq\mathcal{\overline{H}}$ case and by also using
Lemmas 4.6 and 4.8. $\blacksquare$

Proof of Theorems 4.16 and 4.18. Before proving both theorems, the preliminary
Theorem 10.2 is required.

######  Definition 10.1.

For every $\varphi\in\mathcal{NC}$, we define $cl(\varphi)$ as the unique
clausal formula that results from applying $\vee/\wedge$ distributivity to
$\varphi$ until a clausal formula, viz. $cl(\varphi)$, is obtained.

######  Theorem 10.2.

Let $\varphi=(\vee\
\varphi_{1}\ldots\varphi_{i}\ldots\varphi_{k}\,)\in\mathcal{\overline{H}}$.
$cl(\varphi)\in\mathcal{H}$ iff $\varphi$ has $k-1$ negative disjuncts and one
disjunct s.t. $cl(\varphi_{i})\in\mathcal{H}$, formally:

$cl(\,(\vee\
\varphi_{1}\ldots\varphi_{i}\ldots\varphi_{k}\,)\,)\in\mathcal{H}$
$\mathrm{iff}$ $(1)\ \exists i,\mbox{\,s.t.}\ cl(\varphi_{i})\in\mathcal{{H}}\
\mbox{\,and}\ \,(2)\ \forall j,j\neq i,\varphi_{j}\in\mathcal{N}^{-}.$

  * Proof.

If-then. By refutation: let $cl(\,(\vee\
\varphi_{1}\ldots\varphi_{i}\ldots\varphi_{k})\,)\in\mathcal{H}$ and prove
that if (1) or (2) are violated, then $cl(\varphi)\notin\mathcal{H}$.

    1. $\bullet$

$(1)\ \nexists i,\mbox{\,s.t.}\ cl(\varphi_{i})\in\mathcal{{H}}$

$-$ If we take the case $k=1$, then $\varphi=\varphi_{1}$.

$-$ But $cl(\varphi_{1})\notin\mathcal{H}$ implies
$cl(\varphi)\notin\mathcal{H}$.

    2. $\bullet$

(2) $\exists j,j\neq i,\varphi_{j}\notin\mathcal{N}^{-}.$

$-$ Suppose that, besides $\varphi_{i}$, $\varphi_{j}\notin\mathcal{N}^{-}$
with $\,j\neq i$.

$-$ We take $k=2,\,\varphi_{1}=P$ and $\varphi_{2}=Q$.

$-$ So, $\varphi=(\vee\ \varphi_{1}\ \varphi_{2})=(\vee\ P\ Q)$, and hence
$cl(\varphi)\notin\mathcal{H}$.

Only-If. Without loss of generality, we take $(\vee\
\varphi_{1}\ldots\varphi_{i}\ldots\varphi_{k-1})=\varphi^{-}\in\mathcal{N}^{-}\mbox{
and }\varphi_{k}\in\mathcal{\overline{H}}$, and prove:

$cl(\varphi)=cl(\,(\vee\ \
\varphi_{1}\ldots\varphi_{i}\ldots\varphi_{k-1}\,\varphi_{k})\,)=cl(\,(\vee\
\varphi^{-}\ \varphi_{k})\,)\in\mathcal{H}.$

$-$ To obtain $cl(\varphi)$, one must obtain first $cl(\varphi^{-})$ and
$cl(\varphi_{k})$, and so

$(i)\ \ cl(\varphi)=cl(\,(\vee\ \ \varphi^{-}\ \varphi_{k})\,)=cl(\,(\vee\ \
cl(\varphi^{-})\ \ cl(\varphi_{k})\,)\,).$

$-$ By definition of $\varphi^{-}\in\mathcal{N}^{-}$,

$(ii)\ \ cl(\varphi^{-})=\\{\wedge\ D^{-}_{1}\ldots
D^{-}_{m-1}\,D^{-}_{m}\\}$; the $D^{-}_{i}$’s are negative clauses.

$-$ Since $\varphi_{k}\in\mathcal{\overline{H}}$,

$(iii)\ \ cl(\varphi_{k})=\mathrm{H}=\\{\wedge\ h_{1}\ \ldots
h_{n-1}\,h_{n}\\}$; the $h_{i}$’s are Horn clauses.

$-$ By ($i$), ($ii$) and ($iii$),

$cl(\varphi)=cl(\,(\vee\ \ \\{\wedge\ D^{-}_{1}\
\,\ldots\,D^{-}_{m-1}\,D^{-}_{m}\\}\ \ \\{\wedge\ h_{1}\ \ldots
h_{n-1}\,h_{n}\,\\}\,)\,).$

$-$ Applying $\vee\//\wedge$ distributivity to $cl(\varphi)$ and noting
$C_{i}=(\vee\ D^{-}_{1}\ h_{i}\,)$,

$cl(\varphi)=cl(\,\\{\wedge\ \ \\{\wedge\ C_{1}\ldots C_{i}\ldots C_{n}\\}\ \
(\vee\ \,\\{\wedge\ D^{-}_{2}\ldots D^{-}_{m-1}\,D^{-}_{m}\,\\}\ \
\mathrm{H}\,)\,\\}\ ).$

$-$ Since the $C_{i}=(\vee\ D^{-}_{1}\ h_{i}\,)$’s are Horn clauses,

$\\{\wedge\ C_{1}\ldots C_{i}\ldots C_{n}\\}=\mathrm{H}_{1}\in\mathcal{H}$.

$cl(\varphi)=cl(\,\\{\wedge\ \ \mathrm{H}_{1}\ \ (\vee\ \ \\{\wedge\
D^{-}_{2}\ldots D^{-}_{m-1}\,D^{-}_{m}\,\\}\ \ \mathrm{H}\,)\,\\}\ ).$

$-$ For $j<m$ we have,

$cl(\varphi)=cl(\ \\{\wedge\ \mathrm{H}_{1}\
\ldots\mathrm{H}_{j-1}\mathrm{H}_{j}\ \ (\vee\ \ \\{\wedge\ D^{-}_{j+1}\ldots
D^{-}_{m-1}\,D^{-}_{m}\\}\ \ \mathrm{H}\,)\,\\}\ ).$

$-$ For $j=m$, $cl(\varphi)=\\{\wedge\ \mathrm{H}_{1}\
\ldots\mathrm{H}_{m-1}\,\mathrm{H}_{m}\
\mathrm{H}\,\\}=\mathrm{H}^{\prime}\in\mathcal{H}.$

$-$ Hence $cl(\varphi)\in\mathcal{H}$. $\blacksquare$

Theorem 4.16. $\forall\varphi\in\mathcal{\overline{H}}$ we have
$cl(\varphi)\in\mathcal{H}$.

  * Proof.

We consider Definition 4.11 of $\mathcal{\overline{H}}$. The proof is done by
structural induction on the depth $r(\varphi)$ of any
$\varphi\in\mathcal{\overline{H}}$ and defined below, where $\ell$ is a
literal:

$r(\varphi)=\left\\{\begin{array}[]{l l l}\vspace{.15cm}\par 0&\varphi=[\odot\
\ell_{1}\ \ldots\ell_{k-1}\ \ell_{k}]\ \mbox{or}\ \varphi=\ell.\\\
1+max\,\\{r(\varphi_{1}),\ldots,r(\varphi_{k-1}),\,r(\varphi_{k})\\}&\varphi=[\odot\
\varphi_{1}\ \ldots\varphi_{k-1}\ \varphi_{k}].\\\ \end{array}\right.$

$\bullet$ Base Case: $r(\varphi)=0.$

– Clearly, $r(\varphi)=0$ entails $\varphi=[\odot\ \ell_{1}\ \ldots\ell_{k-1}\
\ell_{k}]\in\mathcal{H}$ and $\varphi=\ell\in\mathcal{H}$.

– So $cl(\varphi)=\varphi\in\mathcal{H}$.

$\bullet$ Induction hypothesis: $\forall\varphi,\ r(\varphi)\leq n,\ \
\varphi\in\mathcal{\overline{H}}\mbox{ \ entails \
}cl(\varphi)\in\mathcal{H}.$

$\bullet$ Induction proof: $r(\varphi)=n+1$.

By Definition 4.11, lines (2) ad (3) below arise:

(2) $\varphi=\\{\wedge\ \varphi_{1}\ldots\varphi_{i}\ldots\varphi_{k}\\}$,
where $k\geq 1$.

$-$ By definition of $r(\varphi)$,

$r(\varphi)=n+1$ entails $1\leq i\leq k,\ r(\varphi_{i})\leq n.$

$-$ By induction hypothesis,

$\varphi_{i}\in\mathcal{\overline{H}}\mbox{\, and \,}r(\varphi_{i})\leq
n\mbox{ \ entail \ }cl(\varphi_{i})\in\mathcal{H}$.

$-$ It is obvious that,

$cl(\varphi)=\\{\wedge\ \ cl(\varphi_{1})\ \ldots\ cl(\varphi_{i})\ldots
cl(\varphi_{k})\,\\}.$

$-$ Therefore,

$cl(\varphi)=\\{\wedge\
\mathrm{H}_{1}\ldots\mathrm{H}_{i}\ldots\mathrm{H}_{k}\\}=\mathrm{H}\in\mathcal{H}$.

(3) $\varphi=(\vee\ \varphi_{1}\ldots\varphi_{i}\ldots\varphi_{k-1}\
\varphi_{k})\in\mathcal{\overline{H}}$, where:

$k\geq 1$, $0\leq i\leq k-1,\ \varphi_{i}\in\mathcal{N}^{-}$ and
$\varphi_{k}\in\mathcal{\overline{H}}.$

$-$ By definition of $r(\varphi)$,

$r(\varphi)=n+1$ entails $r(\varphi_{k})\leq n.$

$-$ By induction hypothesis,

$d(\varphi_{k})\,\leq n$ and $\varphi_{k}\,\in\mathcal{\overline{H}}$ entail
$cl(\varphi_{k})\,\in\mathcal{H}.$

$-$ By Theorem 10.2, only-if,

$0\leq i\leq k-1,\ \varphi_{i}\in\mathcal{N}^{-}$ and
$cl(\varphi_{k})\in\mathcal{H}$ entail:

$cl(\,(\vee\ \varphi_{1}\ldots\varphi_{i}\ldots\varphi_{k-1}\
\varphi_{k})\,)\in\mathcal{H}$. $\blacksquare$

Theorem 4.18. $\forall\varphi\in\mathcal{NC}$: if $cl(\varphi)\in\mathcal{H}$
then $\varphi\in\mathcal{\overline{H}}$.

  * Proof.

It is done by structural induction on the depth $d(\varphi)$ of $\varphi$
defined as

$d(\varphi)=\left\\{\begin{array}[]{l l l}\vspace{.1cm}\par
0&\varphi\in\mathcal{C}.\\\
1+max\,\\{d(\varphi_{1}),\ldots,d(\varphi_{i}),\ldots\,,d(\varphi_{k})\\}&\varphi=[\odot\
\varphi_{1}\ldots\varphi_{i}\ldots\varphi_{k}].\\\ \end{array}\right.$

$\bullet$ Base case: $d(\varphi)=0$ and $cl(\varphi)\in\mathcal{H}$.

$-$ $d(\varphi)=0$ entails $\varphi\in\mathcal{C}$.

$-$ If $\varphi\notin\mathcal{H}$, then $cl(\varphi)\notin\mathcal{H}$,
contradicting the assumption.

$-$ Hence $\varphi\in\mathcal{H}$ and so by Definition 4.11,
$\varphi\in\mathcal{\overline{H}}$.

$\bullet$ Inductive hypothesis: $\forall\varphi\in\mathcal{NC},\
d(\varphi)\leq n,\ cl(\varphi)\in\mathcal{H}\mbox{ \ entail \
}\varphi\in\mathcal{\overline{H}}.$

$\bullet$ Induction proof: $d(\varphi)=n+1$.

By Definition 2.3 of $\mathcal{NC}$, cases $(i)$ and $(ii)$ below arise.

$(i)$ $cl(\varphi)=cl(\,\\{\wedge\
\varphi_{1}\ldots\varphi_{i}\ldots\,\varphi_{k}\\}\,)\in\mathcal{H}$ and
$k\geq 1$.

$-$ Since $\varphi$ is a conjunction, $1\leq i\leq
k,\,cl(\varphi_{i})\in\mathcal{H}.$

$-$ By definition of $d(\varphi)$,

$d(\varphi)=n+1$ entails $1\leq i\leq k,\,d(\varphi_{i})\leq n.$

$-$ By induction hypothesis,

$1\leq i\leq k,\ \,d(\varphi_{i})\leq n,\ cl(\varphi_{i})\in\mathcal{H}$
$\mbox{\ entail\ }\varphi_{i}\in\mathcal{\overline{H}}.$

$-$ By Definition 4.11, line (2),

$1\leq i\leq k,\ \varphi_{i}\in\mathcal{\overline{H}}$ entails
$\varphi\in\mathcal{\overline{H}}$.

$(ii)$ $cl(\varphi)=cl(\,(\vee\
\varphi_{1}\ldots\varphi_{i}\ldots\varphi_{k-1}\
\varphi_{k})\,)\in\mathcal{H}$ and $k\geq 1$.

$-$ By Theorem 10.2, if-then,

$0\leq i\leq k-1,\ \varphi_{i}\in\mathcal{N}^{-}$ and
$cl(\varphi_{k})\in\mathcal{H}.$

$-$ By definition of $d(\varphi)$,

$d(\varphi)=n+1$ entails $d(\varphi_{k})\leq n.$

$-$ By induction hypothesis,

$d(\varphi_{k})\,\leq n$ and $cl(\varphi_{k})\in\mathcal{H}$ entail
$\varphi_{k}\,\in\mathcal{\overline{H}}.$

$-$ By Definition 4.11, line (3),

$0\leq i\leq k-1,\ \varphi_{i}\in\mathcal{N}^{-}$ and
$\varphi_{k}\in\mathcal{\overline{H}}$ entail:

$(\vee\ \varphi_{1}\ldots\varphi_{i}\ldots\varphi_{k-1}\
\varphi_{k})=\varphi\in\mathcal{\overline{H}}$.

$\blacksquare$


# Data-driven prediction of complex flow field over an axisymmetric body of
revolution using Machine Learning

J. P. Panda Address all correspondence to this author.  Department of
Mechanical Engineering,
DIT University, Dehradun, UK, India
e-mail<EMAIL_ADDRESS>H. V. Warrior  Department of Ocean
Engineering and Naval Architecture
IIT Kharagpur, WB, India
e-mail<EMAIL_ADDRESS>
###### Abstract

Computationally efficient and accurate simulations of the flow over
axisymmetric bodies of revolution (ABR) has been an important desideratum for
engineering design. In this article the flow field over an ABR is predicted
using machine learning (ML) algorithms, using trained ML models as surrogates
for classical computational fluid dynamics (CFD) approaches. The flow field is
approximated as functions of x and y coordinates of locations in the flow
field and the velocity at the inlet of the computational domain. The data
required for the development of the ML models were obtained from high fidelity
Reynolds stress transport model (RSTM) based simulations. The optimal hyper-
parameters of the trained ML models are determined using validation. The
trained ML models can predict the flow field rapidly and exhibits orders of
magnitude speed up over conventional CFD approaches. The predicted results of
pressure, velocity and turbulence kinetic energy are compared with the
baseline CFD data, it is found that the ML based surrogate model predictions
are as accurate as CFD results. This investigation offers a framework for fast
and accurate predictions for a flow scenario that is critically important in
engineering design.

$U$Mean velocity $u$Velocity in x- direction $v$Velocity in y- direction
$w$Velocity in z- direction $w$Inlet velocity $Re_{v}$Volumetric Reynolds
number $R_{ij}$Reynolds stress tensor $b_{ij}$Reynolds stress anisotropy
$\delta_{ij}$Kronecker delta $k$turbulent kinetic energy $\rho$density
$\nu$Kinematic viscosity

##  1 Introduction

The flow over axisymmetric bodies of revolution has been an important topic of
study for decades due to their central importance in fields such as aerospace
engineering, ocean and naval engineering, etc. A primary parameter in the
design of LTA systems (Lighter-Than-Air) such as Airships and Aerostats
involves determining the drag at various speeds over their aerodynamic
envelopes that are very accurately represented as axisymmetric bodies of
revolution in CFD simulations [1]. Novel space launch vehicle often use the
hammerhead design that are approximated by ABRs in CFD simulations[2].
Similarly, ABR simulations are essential in the design of new autonomous
underwater vehicle (AUV) hulls [3]. While such ABR configurations have been
the central focus of theoretical and empirical studies, such historical data
is limited in spatial and temporal resolution. All these factors necessitate
repeated CFD simulations for different ABR configurations across different
fields of engineering design. This is a significant computational burden over
numerous iterations involved in the engineering design process. As a practical
step to reduce this burden many investigators utilize simpler eddy-viscosity
based turbulence models. But these simpler models have severe limitations and
can make flawed predictions for ABR simulations due to the flow separation and
streamline curvature[4, 2]. In this scenario a framework that can offer
computationally inexpensive, robust and accurate predictions would be
advantageous to these applications.

While numerical simulations based on computational fluid dynamics methods are
the most popular alternative for flow field prediction especially in complex
engineering cases, the rapid evolving of machine learning and data storage
capacities, researchers working in the field of fluid dynamics, have started
applying ML principles for flow field modeling and prediction for two
dimensional and low Reynolds number flows [5]. More recently, researchers
working in the field of turbulence modeling have applied ML concepts for
developing turbulence models at different levels, starting from eddy viscosity
models (EVM) to Reynolds stress models and large eddy simulations (LES). In
EVM, data-driven models were learned for components of Reynolds stresses from
high fidelity data sets of direct numerical simulations (DNS) or experiments.
In LES, models were proposed for the sub-grid scale stresses. [6] proposed
data-driven models for the Reynolds stress anisotropy tensor by using deep
neural networks (DNN). It was noticed that the data-driven model prediction of
the flow field was better than the predictions of baseline eddy viscosity and
nonlinear eddy viscosity models. [7] used DNN to develop models for the
pressure strain correlation of turbulence [8]. The model was trained using
high fidelity DNS data sets of turbulent channel flow at different Reynolds
numbers. [9] have proposed a model for the subgrid-scale stress in LES. It was
observed that the ML-based surrogate model was more accurate than the
Smagorinsky models for isotropic turbulent flow predictions. Few other popular
nominations of application of ML in turbulence modeling are [10, 11, 12].

(a)

(b)

Figure 1: a) Geometry of the ABR. The length and diameter of ABR are 1.4 and
0.14 meters respectively, b) mesh for CFD simulations

In addition to the application of ML in turbulence modeling, several
researchers have applied ML algorithms in the direct modeling of flow fields
by correlating velocity and pressure with the input parameters or the boundary
conditions. [13] used DNN for prediction of flow field over airfoils. They
used x, y, Reynolds number, angle of attack, and shape parameters as input
data to the DNN model and outputs are pressure and velocity. x, y are the
coordinates of different locations in the fluid domain. Numerical simulations
data at different Reynolds numbers mainly in the laminar regime were performed
to prepare a data-set for training and testing of the DNN flow prediction
model. [14] used CNN for the prediction of the pressure distribution across an
airfoil. The data set for the CNN model development was prepared using
numerical simulations of deformed airfoils. [15] used DNN to develop surrogate
models for transonic flow prediction past an airfoil. Using the DNN based
model they predicted the flow field past a RAE2822 airfoil under varying
freestream Mach numbers and angles of attack. Their main aim is to reduce the
computational cost without compromising the accuracy of prediction. [15]
utilized CNN to predict flow field in a scramjet isolator. Numerical
simulation data at different Mach numbers and back pressures were utilized to
develop the surrogate model. Using CNN they mainly mapped a relationship
between the wall pressure and velocity field on the isolator. [16] used CNN
and LSTM for flow field prediction along bluff bodies. They used flow field
data of 100 different bluff bodies for training and validation of the ML
model. Out of 100 bluff body data, 80 data were used for training, and the
rest 20 data were used to validate the ML model. [17] predicted the unsteady
volume wake flow field by using CNN. Past information of velocity and pressure
were used to train and validate the model. Important work has also been done
to estimate errors and uncertainties in the predictions of turbulence models
using machine learning approaches. For example, [18] used an ensemble learning
approach constrained by physics-based requirements to generate uncertainty
bounds RANS model predictions. This investigation also utilized physics-based
constraints to improve the ML model by imparting physics-based domain
knowledge to it. [19] predicted the flow field past bluff bodies using ANN.
The data set for ANN model development was generated using CFD simulations.
The ANN model was used to predict the unknown flow fields. [20] utilized a
novel deep learning architecture (point-net) for a one-to-one mapping of the
flow field for flow past irregular geometries. The spatial position of the
mesh nodes was taken as inputs to the DL model and corresponding velocity and
pressure were taken as output. The non-linear relationship between the input
and output was learned by the DL model. Numerical simulation data for laminar
incomprehensible flow past a cylinder was used to train the model.

In this work, deep neural networks (DNN) and random forests (RF) were used to
develop surrogate models for the data-driven prediction of the flow field of
an ABR. Its shape is very much similar to the DARPA sub-off sub-marine model
[21, 22]. The data required to train and test the ML models were obtained from
RSTM based simulations over the ABR. The x,y coordinates, and velocity at the
inlet of the domain were considered as inputs of the ML model and the outputs
to the ML model were velocity, pressure, and turbulence kinetic energy. The ML
models have the capability to learn the non-linear relationship between the
inputs and outputs. Most of the researchers have developed ML-based surrogate
models for flow past airfoils at low-Reynolds numbers, but in this article,
the turbulent flow field (U, P, and k) both in the boundary layer and wake
region of the ABR is predicted using ML-based surrogate models, that is the
novelty of this research.

##  2 ML Algorithms

ML algorithms such as polynomial regression (PLR), support vector machines
(SVR), random forests(RF), gradient boosted trees(GBT) and deep neural
networks (DNN) can be used for flow field prediction problems since a function
can be approximated by use of such techniques. In this article, we only have
considered RF and DNN for the prediction of flow fields both in the boundary
layer and wake region of the ABR.

Figure 2: The flow field prediction DNN. The optimized DNN has 5 layers with
120 neurons in each layers. In the diagram, x, y correspond to x and
y-coordinates respectively and V is the velocity at the inlet of the
computational domain. The arrow marks signifies the information flow from
features to levels. q is the quantity of interest (u,p and k) that needs to be
predicted.

###  2.1 Artificial Neural Networks (ANN)

Artificial Neural Networks (ANN) (fig.2) are inspired by biological neural
networks. These are machine learning systems and are also known as multi-layer
perception (MLP). ANN with more than two hidden layers is known as deep neural
networks (DNN). The function of biological neural networks (BNN) is to carry
out a specific work when activated. The BNNs are the combination of neurons
interconnected by synapses. In contrast to BNNs, ANN/MLP consists of
artificial neurons. A perceptron (fig.2a) is a basic unit of ANN. A non-linear
activation function $\eta$ activates the linear combination of input data from
the (l-1)th layer multiplied by a weight:

$\displaystyle q_{i}^{l}=\eta(\sum_{j}W_{ij}^{l}q_{j}^{l-1}).$ (1)

A number of perceptrons can be combined to form an ANN. A typical ANN is
presented in fig.2b. The back-propagation algorithm can be used to calculate
gradients for the weights of an ANN. The loss of the ML model can be minimized
by using the gradient descent method. The activation functions, which can be
used in MLP are sigmoid $\eta(\beta)=1/(1+e^{-\beta})$, hyperbolic
tangent(tanh) $\eta(\beta)=(e^{-\beta}-e^{-\beta}).(e^{-\beta}+e^{-\beta})$
and RELU $\eta(\beta)=max[0,\beta]$.

###  2.2 Random forests (RF)

Figure 3: Architecture of of a random forest, a) a single decision tree b) a
random forest. The figure is reproduced from [23]

[24] proposed the random forest algorithm. A decision tree (fig.3a) is the
building block of an RF. The group of features and data are represented in
boxes in the decision tree. The desired output can be obtained in the decision
tree by applying. The output variable in an RF can be estimated by combining
the performance of multiple regression trees (fig.3b). The RF is an assembled
learning technique, which works on the bagging method concept. The regression
trees in RF are prepared by utilizing a subset of random samples drawn with
replacement from the data used in training. The trees are not allowed to prun
in the RF and random vectors are generated to depict the growth of the trees.
At each and every node of the RF, for the splitting of the data-set, a random
combination of features is selected.

##  3 Details of the Learning Dataset

In data-driven machine learning applications, the learning dataset is
centrally important as it determines the amount, quality, and nature of
information available to the machine learning model framework. The learning
dataset is traditionally split into a training dataset, used for optimization
to tune the model coefficients; the validation dataset used to select
hyperparameters of the model, and the testing dataset used to generate an
unbiased estimate of the model’s generalization error. In this investigation,
we utilize a learning dataset generated using Reynolds Stress Transport Model
(RSTM) based simulations. These are often used as surrogates for data from
direct numerical studies or experimental investigations, especially in the
generation of ML models. As opposed to eddy viscosity-based models where a
constitutive relation is used to relate the turbulent quantities to the mean
stresses, RSTM simulations utilize separate evolution equations for each
component of the Reynolds stress tensor[25]. RSTM simulations account for the
directional effects of turbulence evolution, the intercomponent energy
transfer effected by the non-local action of pressure, and are able to deal
with high degrees of anisotropy[26, 27]. RSTM simulations are also accurate
for cases with high degrees of streamline curvature and flow separation as is
observed in ABR simulations[28, 29]. The Reynolds stress transport equations
can be written as follows:

$\begin{split}&\partial_{t}\overline{u_{i}u_{j}}+U_{k}\frac{\partial\overline{u_{i}u_{j}}}{\partial
x_{k}}=P_{ij}-\frac{\partial T_{ijk}}{\partial x_{k}}-\eta_{ij}+\phi_{ij},\\\
&\mbox{where},\\\ &P_{ij}=-\overline{u_{k}u_{j}}\frac{\partial U_{i}}{\partial
x_{k}}-\overline{u_{i}u_{k}}\frac{\partial U_{j}}{\partial x_{k}},\\\ &\
T_{ijk}=\overline{u_{i}u_{j}u_{k}}-\nu\frac{\partial\overline{u_{i}u_{j}}}{\partial{x_{k}}}+\delta_{jk}\overline{u_{i}\frac{p}{\rho}}+\delta_{ik}\overline{u_{j}\frac{p}{\rho}},\\\
&\epsilon_{ij}=-2\nu\overline{\frac{\partial u_{i}}{\partial
x_{k}}\frac{\partial u_{j}}{\partial x_{k}}}\\\
&\phi_{ij}=\overline{\frac{p}{\rho}(\frac{\partial u_{i}}{\partial
x_{j}}+\frac{\partial u_{j}}{\partial x_{i}})}\\\ \end{split}$ (2)

$P_{ij}$ denotes the production of turbulence, $T_{ijk}$ is the diffusive
transport, $\epsilon_{ij}$ is the dissipation rate tensor and $\phi_{ij}$ is
the pressure strain correlation. The pressure fluctuations are governed by a
Poisson equation:

$\frac{1}{\rho}{\nabla}^{2}(p)=-2\frac{\partial{U}_{j}}{\partial{x}_{i}}\frac{\partial{u}_{i}}{\partial{x}_{j}}-\frac{\partial^{2}u_{i}u_{j}}{\partial
x_{i}\partial x_{j}}$ (3)

The fluctuating pressure term can be separated as rapid and slow pressure term
$p=p^{S}+p^{R}$. The fluctuations in rapid and slow pressure satisfies the
following equations.

$\frac{1}{\rho}{\nabla}^{2}(p^{S})=-\frac{\partial^{2}}{\partial x_{i}\partial
x_{j}}{(u_{i}u_{j}-\overline{u_{i}u_{j}})}$ (4)
$\frac{1}{\rho}{\nabla}^{2}(p^{R})=-2\frac{\partial{U}_{j}}{\partial{x}_{i}}\frac{\partial{u}_{i}}{\partial{x}_{j}}$
(5)

The slow and rapid pressure terms account for the turbulence-turbulence and
turbulence-mean strain interactions respectively. The rational mechanics
approach was used to model the pressure strain correlation [30]

$\phi_{ij}^{R}=4k\frac{\partial{U}_{l}}{\partial{x_{k}}}(M_{kjil}+M_{ikjl})$
(6)

where,

$M_{ijpq}=\frac{-1}{8\pi
k}\int\frac{1}{r}\frac{\partial^{2}R_{ij}(r)}{\partial r_{p}\partial r_{p}}dr$
(7)

where, $R_{ij}(r)=\langle u_{i}(x)u_{j}(x+r)\rangle$ For homogeneous
turbulence the complete pressure strain correlation can be written as

$\phi_{ij}=\epsilon
A_{ij}(b)+kM_{ijkl}(b)\frac{\partial\overline{v}_{k}}{\partial{x_{l}}}$ (8)

In this work, we have used the quadratic Reynolds stress model (SSG pressure
strain correlation model [31]):

$\begin{split}&\phi_{ij}=-(C_{1}\rho\epsilon+C_{1}^{*}P)b_{ij}+C_{2}\rho\epsilon(b_{ik}b_{kj}-\frac{1}{3}b_{mn}b_{mn}\delta_{ij})+\\\
&(C_{3}-C_{3}^{*}\sqrt{b_{ij}b_{ij}})\rho KS_{ij}+C_{4}\rho
K(b_{ik}S_{jk}+b_{jk}S_{ik}-2/3b_{mn}S_{mn}\delta_{ij})\\\ &+C_{5}\rho
K(b_{ik}W_{jk}+b_{jk}W_{ik})\end{split}$ (9)

Here $b_{ij}=\frac{\overline{u_{i}u_{j}}}{2k}-\frac{\delta_{ij}}{3}$ is the
Reynolds stress anisotropy tensor, $S_{ij}$ is the mean rate of strain and
$W_{ij}$ is the mean rate of rotation.

The numerical simulations were performed using the commercial CFD solver ANSYS
Fluent [32] at 11 different Reynolds numbers for the flow past ABR, where the
in-compressible form of Navier-Stokes equations are solved. We have used
SIMPLE(Semi-Implicit Method for Pressure Linked Equations) scheme for the
pressure velocity coupling. The mesh was generated in ANSYS meshing utility.
Adequate y+ values were used for the RSTM model and inflation with a ratio of
1.2 was considered up to 10 layers in the boundary layer. The simulations were
performed for 2-dimensional turbulent flow cases. The free stream velocities
of water are 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2, 2.2 and 2.4 m/s
respectively.

Figure 4: Point cloud across the ABR.

##  4 Data preparation for ML model development

Numerical results were collected at 10000 different locations across the AUV
to form the train-test data-set. We have considered a sub-domain (x=-1 2.4 and
y=-0.5 0.5) across the ABR for data collection. The actual size of the CFD
domain is, x=-1 5 and y=-1 1. At the nose tip of the ABR, x=0, and y=0. A
point cloud (fig. 4) was generated across the AUV in CFD post. For 11 Reynolds
numbers, the total dataset size is 110000. We have made six different columns
in the excel sheet, which are x, y, V, U, P, and k. x,y, and V are the input
features, and U, P, k are the output levels respectively. V stands for free
stream velocity, the rest of the symbols has their usual meaning. We have
prepared two different training and testing data sets for the development of
ML-based predictive models. Dataset1.0 has all other data except 1.0 m/s data
as a training set and the data at 1.0 m/s is kept for testing the ML models.
Similarly, dataset2.0 has all other data except 2.0 m/s data as a training
set, and the data at 2.0 m/s is kept for testing the ML models. All the
features in the datasets were normalized to a common scale using the min-max
scaler of the SK-Learn [33]. The range of the features was taken as [-1,1].
Data normalization helps in making a common scale for the features and
maintains the differences in a range of values, that preserves general
distribution and ratios in the features of the dataset.

(a)

(b)

(c)

Figure 5: Loss history of the DNN training, the dotted and solid lines
correspond to training and validation losses respectively. a) u, b) p and c) k
losses.

##  5 ML model Training and validation

For training and testing of ANN Tensorflow with Keras API [34] and for the
random forest, SKLearn [33] was used. The ML models are utilized here to find
a mapping function between the inputs and output quantities of interest. So
the task accomplished is basically a regression.

The mapping function for both the ML algorithms can be written as:

$\begin{split}(U,P,k)=f_{ML}(x,y,V)\end{split}$ (10)

where, U, P, k are the outputs and x, y and V are the inputs to the ML models.
$f_{ML}$ is the approximation function. The neural networks were trained in
such a manner that the final error of prediction is minimized. The loss vs
epoch figures is shown in figure LABEL:fig:_loss. Here mean squared error
(MSE) was considered as loss for the training of the ANN. The MSE is mainly
used to determine the performance of an algorithm, it can be defined as:

$MSE=\frac{1}{N}\sum_{i=1}^{N}(\phi_{i}-\hat{\phi}_{i})^{2}$ (11)

The training and test $R^{2}$ scores are presented in table 1 and 2. The
$R^{2}$ can be defined as:

$R^{2}=1-\frac{\sum_{i}(\phi_{i}-\hat{\phi}_{i})^{2}}{\sum_{i}(\phi_{i}-\bar{\phi_{i}})^{2}}$
(12)

The neural networks are named as NNx, where x is the digit, that corresponds
to number of layers in that neural network. Each layer in the NNx has 120
neurons. From the table, it is clear that there is a sharp increase in both
train and test values of $R^{2}$, which signifies a better correlation between
the input and output data. We did not increase the number of layers further to
avoid over-fitting (over-fitting is a case that arises when with an increase
in the size of neural network train $R^{2}$ increases, but there is the
reduction of $R^{2}$ for the validation). The train and test $R^{2}$ for
different random forests are shown in table 2. The important hyper-parameters
of random forests are the number of decision trees and their depth. From the
table, it is noticed that the Random forest with 50 decision trees and a depth
of 50 has the best value of $R^{2}$ for both training and testing.

Models | NN1 | NN2 | NN3 | NN4 | NN5
---|---|---|---|---|---
Train $R^{2}$ | 0.948 | 0.991 | 0.988 | 0.991 | 0.988
Test $R^{2}$ | 0.949 | 0.849 | 0.936 | 0.946 | 0.966
Table 1: $R^{2}$(coefficient of determination) of $v$ predictions by training-test cases by different NN models. The number beside NN signifies the number of layers in the NN. There are 120 neurons in each layer of the NN. Models | RF1 | RF2 | RF3 | RF4 | RF5
---|---|---|---|---|---
Train $R^{2}$ | 0.982 | 0.996 | 0.995 | 0.995 | 0.995
Test $R^{2}$ | 0.481 | 0.930 | 0.914 | 0.921 | 0.935
Table 2: $R^{2}$(coefficient of determination) of $v$ predictions by training-
test cases by different RF models. RF1, RF2, RF3, RF4 and RF5 has hyper-
parameters (10,10), (20,20), (30,30), (40,40), (50,50) respectively. In the
brackets the first and second number correspond to number of decision trees
and their maximum depth respectively.

(a)

(b)

Figure 6: Effect of data size on the predictive capability of ML models

###  5.1 Effect of data-set size on predictive capability of the ML models

In supervised learning, the dataset size can have a significant effect on the
model training. Small training dataset results in a poor approximation
function. In order to study the effect of dataset size on the predictive
capability of the ML models, we have prepared three separate training datasets
each having 30000, 50000, and 100000 data. The data was collected from CFD
posts using appropriate point clouds. The dataset with 30000 samples has 3000
data for each of the Reynolds numbers. In a similar fashion, we have created
the other two datasets. All the three datasets were used in the training of
DNN and RF separately, with the optimized hyperparameters. This study may be
named as Data Sensitivity Study (DSS), similar to the mesh sensitivity study
performed in CFD simulations. The DSS is proposed in this paper for the first
time. The DSS results are presented in the figure. 6. The first and second
figures correspond to the DNN and RF model perditions respectively. The
predictions are for the test data of 1 m/s. In both, the figure’s solid lines
correspond to the baseline RSM results of the velocity field in the wake
region of ABR at an inlet velocity of 1 m/s. The dashed, dotted and dashed-dot
lines correspond to velocity predictions using ML models trained with 100000,
50000, and 30000 data respectively. Figure. 6a and b correspond to DNN and RF
model predictions respectively. From both the figures it is clear that with an
increase in dataset size the predictive capability of both DNN and RF
increases. We did not increase data size further to save computational time.
We have performed DSS for the other two variables e.g pressure and turbulence
kinetic energy, but for brevity, we have presented only the prediction results
of the velocity field in the wake region of the ABR.

(a)

(b)

(c)

Figure 7: Profiles of velocity, pressure and turbulence kinetic energy in the
wake region of the ABR for inlet velocity 1 m/s, Solid line RSM, dashed line
neural network, dotted line random forest, y/r=0.

Figure 8: Profiles of velocity, pressure and turbulence kinetic energy in the
wake region of the ABR for inlet velocity 2 m/s, Solid line RSM, dashed line
neural network, dotted line random forest, y/r=0.

Figure 9: Profiles of velocity in the boundary layer of the ABR for inlet
velocity (a) 1 m/s and (b) 2 m/s, Solid line RSM, dashed line neural network,
dotted line random forest, x/r=10.

##  6 Prediction of unknown flow field

We have applied the optimized DNN and RF for the prediction of the unknown
flow cases, those were not used for training the models. As discussed in the
data preparation section, those are flowing past the ABR at 1 and 2 m/s
respectively. In figures. 7, the DNN and RF Predictions are presented by
dashed and dotted lines respectively. The solids lines baseline results of RSM
models. The DNN predictions of the velocity, pressure, and turbulent kinetic
energy match well with RSM results. The RF prediction of the velocity field is
as good as the RSM results but the TKE and pressure predictions poor in
contrast to the DNN predictions. Similar is the case for flow predictions at 2
m/s as shown in figure 8. The figures 7 and 8 correspond to the distribution
of velocity, pressure, and turbulence kinetic energy in the wake region of the
ABR. In addition to wakefield prediction, the flow field in the boundary layer
of the ABR was also presented in the figure 9. From both the figures, it is
clear that the DNN prediction of the flow field is better than the predictions
of the RF model. This concludes that the DNNs can act as a universal
approximator in predicting the flow field of the ABR. Hence DNN based
surrogate data-driven ML models can be applied in the prediction of the flow
field in the preliminary design stages of the ABR.

##  7 Conclusions

In this article, we introduce a framework that can offer computationally
inexpensive, robust, and accurate predictions for flow simulations over
Axisymmetric Bodies of Revolution. We have applied different ML algorithms
such as artificial neural networks and random forests for flow field
prediction around an ABR. The flow statistics around the ABR were utilized to
develop the ML models. Mainly, regression functions for the velocity, pressure
and turbulence kinetic energy were approximated using the input velocity and
x,y coordinates as inputs to the ML models. The optimal hyper-parameters of
the ML models were presented. It is noticed that the ML predicted flow fields
are as accurate as of the actual flow field as obtained by the RSM model. The
ML models presented in this article will be useful for fast flow field
predictions along the ABR in its preliminary design stage. In the future
course of work, the Reynolds numbers and shape parameters can be incorporated
in the modeling basis for improved prediction of the flow field around the
ABR.

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