terms that may contain monopole operators [DGG14, DGG13, DGG16].

Similarly to the ${\mathcal{N}}=2$ theories ${\sf T}_{K}[C]$ of class
$\mathcal{S}$ (see Section 3.2), these three-dimen-sional theories may be
obtained from a (twisted) compactification of the six-dimensional
$(2,0)$-theory ${\mathfrak{X}}[K]$ on a three-manifold $M$. The three-
manifolds $M$ that label the three-dimensional theories ${\sf T}_{K}[M]$ of
class ${\mathcal{R}}$ are assembled by gluing topological tetrahedra. This
decomposition into tetrahedra determines an ideal triangulation $\mathscr{T}$
of their boundaries $\partial M$. The corresponding three-dimensional
superconformal field theory depends on the triangulation $\mathscr{T}$ as well
as a polarization $\varPi$ of the boundary,505050More precisely, $\varPi$ is a
polarization for an open subset of the moduli space $\mathscr{M}_{\rm
flat}(\partial M,SL(K,\mathbb{C}))$ of framed flat $SL(K,\mathbb{C})$
connections on $\partial M$. and is therefore often denoted as ${\sf
T}_{K}[M,\mathscr{T},\varPi]$. Although the discussion which follows can be
generalized rather straightforwardly to any $K$ (with the help of [DGG16,
GMN14]), we will often restrict ourselves here to $K=2$ for simplicity.

For $K=2$, we call an edge of the triangulation $\mathscr{T}$ electric or
magnetic if the corresponding one-cycle $\gamma\in
H_{1}(\overline{\Sigma},\mathbb{Z})$ is an $A$-cycle or a $B$-cycle,
respectively. Each electric edge corresponds to a manifest $U(1)$ flavour
symmetry in the three-dimensional superconformal field theory. For every such
edge $E$ there is a chiral operator $\mathcal{O}_{E}$ which transforms with
charge $+1$ under the corresponding $U(1)$ flavour symmetry. If $E$ instead
has a non-trivial magnetic charge, then an operator $\mathcal{O}_{E}$ exists
in the presence of a magnetic monopole background.

Given a four-dimensional ${\mathcal{N}}=2$ theory ${\sf T}_{K}[C]$ in an
infrared vacuum $u$, labelled by a (possibly) punctured Riemann surface $C$,
we constructed the “empty” boundary condition
$\mathscr{L}_{\mathscr{W},\varPi}$ labelled by a distinguished spectral
network $\mathscr{W}=\mathscr{W}(u,\vartheta_{\rm c})$ on $C$ together with a
polarization $\varPi$ of the Seiberg-Witten curve $\Sigma_{u}$, which is given
by the equations $\mathcal{X}^{\mathscr{W}}_{B_{i}}=1$. It should now be clear
that we may generalize this boundary condition to a “full” boundary condition
$\mathscr{L}_{M,\mathscr{W}^{\prime},\varPi^{\prime}}$ labelled by a three-
manifold $M$ (assembled as above by gluing tetrahedra) with boundary $\partial
M=C\sqcup C^{\prime}$, whose boundary triangulation
$\mathscr{T}\sqcup\mathscr{T}^{\prime}$ is dual to the pair of spectral
networks $\mathscr{W}\sqcup\mathscr{W}^{\prime}$ on $C\sqcup C^{\prime}$, and
with boundary polarizations $\varPi$ on $C$ and $\varPi^{\prime}$ on
$C^{\prime}$. The new boundary condition
$\mathscr{L}_{M,\mathscr{W}^{\prime},\varPi^{\prime}}$ is defined by the
equations

$\displaystyle\mathcal{X}^{\mathscr{W}^{\prime}}_{B^{\prime}_{i}}=1$ (5.73)

in the system of exponentiated Darboux coordinates
$\big{\\{}\mathcal{X}^{\mathscr{W}^{\prime}}_{A^{\prime}{}^{i}},\mathcal{X}^{\mathscr{W}^{\prime}}_{B^{\prime}_{i}}\big{\\}}$
obtained by abelianization with respect to the spectral network
$\mathscr{W}^{\prime}$.

In the field theory description when $K=2$, the abelian gauge fields of the
four-dimensional theory ${\sf T}_{2}[C]$ in the vacuum $u$ can be used to
gauge the flavour symmetries of the three-dimensional theory ${\sf T}_{2}[M]$
corresponding to the edges $E$ of the triangulation $\mathscr{T}$, whereas the
electric BPS hypermultiplets of ${\sf T}_{2}[C]$ may be coupled to the
corresponding chiral operators $\mathcal{O}_{E}$ of ${\sf T}_{2}[M]$. This
eliminates the dependence of the resulting four-dimensional theory on
$\mathscr{T}$ and $\varPi$. A similar argument can be made for arbitrary $K$.

The simplest new boundary conditions are those that leave the vacuum $u$
invariant. Let us examine two types of examples. First, consider a boundary
condition $\mathscr{L}_{M,\mathscr{W},\varPi^{\prime}}$ where only the
polarization $\varPi\to\varPi^{\prime}$ of the Seiberg-Witten curve
$\Sigma_{u}$ is modified. This change corresponds to an electric-magnetic
duality transformation ${\sf g}\in Sp(2r,\mathbb{Z})$ in the infrared vacuum
$u$, which is implemented by a generalized Legendre transform
$\mathcal{L}_{\sf g}$ at the level of the effective twisted superpotential,
associated to the linear symplectomorphism
$\big{(}{}^{x}_{y}\big{)}\mapsto\big{(}{}^{x^{\prime}}_{y^{\prime}}\big{)}={\sf
g}\big{(}{}^{x}_{y}\big{)}$ (see e.g. [Tul77]). For the elementary canonical
transformation $\big{(}{}^{x^{\prime}}_{y^{\prime}}\big{)}={\sf
g}_{0}\big{(}{}^{x}_{y}\big{)}=\big{(}{}^{-y}_{\ \,x}\big{)}$, this Legendre
transform has the familiar classical expression

$\displaystyle\mathcal{L}_{{\sf
g}_{0}}\,:\,\widetilde{\mathcal{W}}^{\text{eff}}(x;\epsilon)\longmapsto\widetilde{\mathcal{W}}{}^{\prime\,\rm
eff}(x^{\prime};\epsilon)=\epsilon\,x^{\prime}\cdot
x+\widetilde{\mathcal{W}}^{\text{eff}}(x;\epsilon)\ .$ (5.74)

We may also simply change the boundary triangulation
$\mathscr{T}\to\mathscr{T}^{\prime}$ on the second copy $C^{\prime}=C$ of the
boundary $\partial M=C\sqcup C^{\prime}$. Let $K=2$. Assume that $\mathscr{T}$
and $\mathscr{T}^{\prime}$ are dual to the spectral networks
$\mathscr{W}=\mathscr{W}(u,\vartheta)$ and
$\mathscr{W}^{\prime}=\mathscr{W}(u,\vartheta^{\prime})$, respectively, and
that $\mathscr{W}(u,\vartheta)$ and $\mathscr{W}(u,\vartheta^{\prime})$ are
related by a sequence of flips. As illustrated in Figure 42, each flip in the
transition from $\mathscr{W}(u,\vartheta)$ to
$\mathscr{W}(u,\vartheta^{\prime})$ corresponds to a tetrahedron in the
decomposition of $M$ into tetrahedra.

\begin{overpic}[width=346.89731pt]{FlipTetrahedron.png} \put(48.0,18.0){$f$}
\put(20.0,16.0){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$E$}}
\put(81.0,18.0){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$E$}}
\put(33.0,65.0){$\triangle$}
\put(33.0,23.0){$\subset\mathscr{W}(u,\vartheta)$}
\put(95.0,23.0){$\subset\mathscr{W}(u,\vartheta^{\prime})$} \end{overpic}
Figure 42: Each flip $f$ in the transition from $\mathscr{W}(u,\vartheta)$ to
$\mathscr{W}(u,\vartheta^{\prime})$ corresponds to a tetrahedron $\triangle$
in $M$, where the triangles adjacent to the flipped edge $E$ form the faces of
$\triangle$.

The coordinate change corresponding to such a flip is given by

$\displaystyle\mathcal{X}_{\gamma_{E}}^{\mathscr{W}^{\prime}}=\mathcal{X}_{\gamma_{E}}^{\mathscr{W}}\qquad\mbox{and}\qquad\mathcal{X}_{\gamma}^{\mathscr{W}^{\prime}}=\mathcal{X}_{\gamma}^{\mathscr{W}}\,\big{(}1+\mathcal{X}_{\gamma_{E}}^{\mathscr{W}}\big{)}^{\langle\gamma,\gamma_{E}\rangle}\
,$ (5.75)

where $\gamma_{E}$ is the one-cycle on $\Sigma_{u}$ corresponding to the edge
$E$ of $\mathscr{T}$ in Figure 42, and $\gamma$ is any other one-cycle on
$\Sigma_{u}$ (see [GMN13b, Section 7.6]). Let $\gamma$ be any one-cycle with
$\langle\gamma,\gamma_{E}\rangle=1$, and consider the Darboux coordinates
$x=\frac{\epsilon}{\pi\,{\mathrm{i}}}\log\big{(}-\mathcal{X}_{\gamma_{E}}^{\mathscr{W}}\big{)}$,
$y=\frac{1}{2\,\epsilon}\log\mathcal{X}_{\gamma}^{\mathscr{W}}$ and
$y^{\prime}=\frac{1}{2\,\epsilon}\log\mathcal{X}_{\gamma}^{\mathscr{W}^{\prime}}$.
The effective twisted superpotential then changes by

$\displaystyle\widetilde{\mathcal{W}}{}^{\prime\,\rm
eff}(x;\epsilon)-\widetilde{\mathcal{W}}^{\text{eff}}(x;\epsilon)$
$\displaystyle=\frac{1}{2}\,\int\,\Big{(}\log\mathcal{X}_{\gamma}^{\mathscr{W}^{\prime}}-\log\mathcal{X}_{\gamma}^{\mathscr{W}}\Big{)}\,\mathrm{d}x$
$\displaystyle=\frac{1}{2}\,\int\,\log\big{(}1-{\mathrm{e}}^{\,\pi\,{\mathrm{i}}\,x/\epsilon}\big{)}\,\mathrm{d}x$
$\displaystyle=-\frac{\epsilon}{2\pi\,{\mathrm{i}}}\,{\rm
Li}_{2}({\mathrm{e}}^{\,\pi\,{\mathrm{i}}\,x/\epsilon})\ .$ (5.76)

The difference (5.7) precisely matches the effective two-dimensional twisted
superpotential $\widetilde{\mathcal{W}}^{\rm eff}_{\triangle}(x;\epsilon)$ for
the elementary tetrahedron theory ${\sf
T}_{2}[\triangle,\mathscr{T},\varPi_{E}]$ in the background
$\widetilde{S}\,^{1}\times\mathbb{R}^{2}$ (the boundary of
$D_{R,\epsilon}\times\mathbb{R}^{2}$). Indeed, the theory ${\sf
T}_{2}[\triangle,\mathscr{T},\varPi_{E}]$ consists of a single free chiral
multiplet $\phi_{E}$ with charge $+1$ under the $U(1)$ flavour symmetry
associated to the edge $E$, charge $0$ under the $U(1)_{R}$ symmetry, and a
level $-\frac{1}{2}$ background Chern-Simons term for the difference of the
corresponding gauge fields $A_{\rm f}-A_{R}$. The result (5.7) is the one-loop
contribution of the chiral field $\phi_{E}$ with effective twisted mass
$\pi\,{\mathrm{i}}\,x/\epsilon$ to the effective twisted superpotential, which
is obtained as a sum over its Kaluza-Klein modes in the compactification to
$\mathbb{R}^{2}$ [Wit93]. Geometrically it corresponds to the hyperbolic
volume of the tetrahedron $\triangle$.

Implementing the modifications to the four-dimensional effective twisted
superpotential for a sequence of flips might require us to adapt the
polarization on $\Sigma_{u}$ along the way. This has the effect of an affine
symplectic transformation on the three-dimensional ${\mathcal{N}}=2$ theory
${\sf T}_{2}[M]$ (see [Wit05] and [DGG16, Appendix A]): “T-type”
transformations add background Chern-Simons couplings for flavour symmetries,
while “S-type” transformations gauge a flavour symmetry, replacing it with a
new topological $U(1)_{J}$ symmetry.

This type of generalized boundary condition does not depend on the complex
structure of $C$ and hence can only change the one-loop part of the effective
twisted superpotential $\widetilde{\mathcal{W}}^{\text{eff}}(x;\epsilon)$.
Considering the effective twisted superpotential with respect to the spectral
coordinates $\\{\mathcal{X}_{A},\mathcal{X}_{B}^{\pm}\\}$ corresponding to a
resolution of a Fenchel-Nielsen type network, as we did at the end of Section
5.4, is another example of such a generalized boundary condition. Recall that
for these choices, we found that the resulting one-loop contribution agrees
with the one-loop contribution to
$\widetilde{\mathcal{W}}^{\text{eff}}(x;\epsilon)$ in the gauge theory or
topological string scheme.

So far we have studied two types of more general infrared boundary conditions
for the four-dimensional theory ${\sf T}_{K}[C]$ in the vacuum $u$. By
choosing more complicated three-manifolds $M$, we may however also engineer
three-dimensional boundary conditions for the theory ${\sf T}_{K}[C]$ in a
different vacuum $u^{\prime}$ (because of the $Sp(2r,\mathbb{Z})$ monodromies
on the Coulomb branch $\mathscr{B}$, the three-manifold $M$ as well as the
resulting effective twisted superpotential will depend on a choice of path
from $u$ to $u^{\prime}$ on $\mathscr{B}$). In the language of [DGV15] this
means that we should consider three-manifolds with not only non-trivial “big”
boundary components, but also with non-trivial “small” boundary components,
corresponding to annuli or tori that stem from truncating the tetrahedra in
the decomposition of $M$.515151More accurately, the empty boundary condition
$\mathscr{L}_{\mathscr{W},\varPi}$ can be constructed, according to [DGV15],
as a combination of a UV-IR domain wall and an infrared boundary condition.
This combination corresponds geometrically to a three-manifold $M$ with two
boundary components $C$ and $\overline{C}$. Here $C$ is a “big” boundary
component together with a triangulation $\mathscr{T}$ dual to $\mathscr{W}$
and polarization $\varPi$, while $\overline{C}$ is only homeomorphic to $C$
and consists of “big” three-punctured spheres glued together with “small”
annuli. The more general boundary conditions discussed here may then be
constructed as a combination of an S-duality wall (which connects $C$ with
complex structure $q$ to $C^{\prime}$ with complex structure $q^{\prime}$), a
UV-IR domain wall (which flows the theory to the vacuum $u^{\prime}$) as well
as an infrared boundary condition (determined by the spectral network
$\mathscr{W}^{\prime}$ and polarization $\varPi^{\prime}$ on $C^{\prime}$).

While we do not describe such boundary conditions in detail here, these
considerations do motivate the definition of a new object: the (generalized)
effective twisted superpotential
$\widetilde{\mathcal{W}}^{\text{eff}}_{u,\vartheta,\varPi}(x;\epsilon)$. We
define this twisted superpotential as the generating function of the brane of
$\epsilon$-opers in the spectral coordinates
$\big{\\{}\mathcal{X}^{\mathscr{W}(u,\vartheta)}_{A^{i}},\mathcal{X}^{\mathscr{W}(u,\vartheta)}_{B_{i}}\big{\\}}$
obtained from abelianization with respect to _any_ spectral network
$\mathscr{W}(u,\vartheta)$, determined by $u$ and $\vartheta$, as well as the
polarization $\varPi$ on the Seiberg-Witten curve $\Sigma_{u}$. That is, the
effective twisted superpotential is extracted from the relations

$\displaystyle\log\Big{(}-\mathcal{X}_{A^{i}}^{\mathscr{W}(u,\vartheta)}\big{(}\nabla_{\epsilon}^{\rm
oper}\big{)}\Big{)}$ $\displaystyle=\frac{\pi\,{\mathrm{i}}\,x_{i}}{\epsilon}\
,$
$\displaystyle\log\mathcal{X}_{B_{i}}^{\mathscr{W}(u,\vartheta)}\big{(}\nabla_{\epsilon}^{\rm
oper}\big{)}$
$\displaystyle=2\,\frac{\partial\widetilde{\mathcal{W}}^{\text{eff}}_{u,\vartheta,\varPi}}{\partial
x_{i}}\ .$ (5.77)

Physically it corresponds to the effective twisted superpotential of the
theory ${\sf T}_{K}[C]$ in the vacuum $u$ with respect to the boundary
condition $\mathscr{L}_{\mathscr{W}(u,\vartheta),\varPi}$. This superpotential
is locally constant on $\mathscr{B}\times[0,2\pi)$, and it is analytic in $x$
and $\epsilon$ for fixed $(u,\vartheta)$. It is defined uniquely on
$\mathscr{B}\times[0,2\pi)$ up to the monodromies of the $Sp(2r,\mathbb{Z})$
bundle over the Coulomb branch $\mathscr{B}$.

## References

  * [AS65] M. Abramowitz and I. A. Stegun “Handbook of Mathematical Functions” Dover Publications, Inc., New York, 1965
  * [Aga+13] M. Aganagic, N. Haouzi, C. Kozcaz and S. Shakirov “Gauge/Liouville triality”, 2013 arXiv:1309.1687 [hep-th]
  * [AHS14] M. Aganagic, N. Haouzi and S. Shakirov “$A_{n}$-triality”, 2014 arXiv:1403.3657 [hep-th]
  * [Ald+10] L. F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde “Loop and surface operators in $\mathcal{N}=2$ gauge theory and Liouville modular geometry” In _J. High Energy Phys._ 01, 2010, pp. 113 arXiv:0909.0945 [hep-th]
  * [AGT10] L. F. Alday, D. Gaiotto and Y. Tachikawa “Liouville correlation functions from four-dimensional gauge theories” In _Lett.Math.Phys._ 91, 2010, pp. 167–197 arXiv:0906.3219 [hep-th]
  * [Ali20] M. Alim “Difference equation for the Gromov-Witten potential of the resolved conifold”, 2020 arXiv:2011.12759 [math.AG]
  * [Ali21] M. Alim “Intrinsic non-perturbative topological strings”, 2021 arXiv:2102.07776 [hep-th]
  * [Ali+13] M. Alim, S. Cecotti, C. Córdova, S. Espahbodi, A. Rastogi and C. Vafa “BPS quivers and spectra of complete ${\mathcal{N}}=2$ quantum field theories” In _Comm. Math. Phys._ 323, 2013, pp. 1185–1227 arXiv:1109.4941 [hep-th]
  * [Ali+21] M. Alim, A. Saha, J. Teschner and I. Tulli “Mathematical structures of non-perturbative topological string theory: from GW to DT invariants”, 2021 arXiv:2109.06878 [hep-th]
  * [All19] D. G. L. Allegretti “Voros symbols as cluster coordinates” In _J. Topol._ 12, 2019, pp. 1031–1068 arXiv:1802.05479 [math.CA]
  * [All21] D. G. L. Allegretti “Stability conditions, cluster varieties, and Riemann-Hilb-ert problems from surfaces” In _Adv. Math._ 380, 2021, pp. 107610 arXiv:1912.05938 [math.AG]
  * [AD95] P. C. Argyres and M. R. Douglas “New phenomena in $SU(3)$ supersymmetric gauge theory” In _Nucl. Phys. B_ 448, 1995, pp. 93–126 arXiv:hep-th/9505062
  * [AS07] P. C. Argyres and N. Seiberg “S-duality in ${\mathcal{N}}=2$ supersymmetric gauge theories” In _J. High Energy Phys._ 12, 2007, pp. 088 arXiv:0711.0054 [hep-th]
  * [Ash+15] S. K. Ashok, M. Billó, E. Dell’Aquila, M. Frau, R. R. John and A. Lerda “Non-perturbative studies of ${\mathcal{N}}=2$ conformal quiver gauge theories” In _Fortsch. Phys._ 63, 2015, pp. 259–293 arXiv:1502.05581 [hep-th]
  * [AB82] M. F. Atiyah and R. Bott “The Yang-Mills equations over Riemann surfaces” In _Phil. Trans. Roy. Soc. London A_ 308, 1982, pp. 523–615
  * [BDS96] T. Banks, M. R. Douglas and N. Seiberg “Probing F-theory with branes” In _Phys. Lett. B_ 387, 1996, pp. 278–281 arXiv:hep-th/9605199
  * [Bao+14] L. Bao, V. Mitev, E. Pomoni, M. Taki and F. Yagi “Non-Lagrangian theories from brane junctions” In _J. High Energy Phys._ 01, 2014, pp. 175 arXiv:1310.3841 [hep-th]
  * [BBS20] A. Barbieri, T. Bridgeland and J. Stoppa “A quantized Riemann-Hilbert problem in Donaldson-Thomas theory” In _Int. Math. Res. Not._ 2020, 2020, pp. rnaa294 arXiv:1905.00748 [math.AG]
  * [BD05] A. Beilinson and V. Drinfeld “Opers”, 2005 arXiv:math.AG/0501398
  * [Ber+95] M. Bershadsky, A. Johansen, V. Sadov and C. Vafa “Topological reduction of $4d$ SYM to $2d$ sigma models” In _Nucl. Phys. B_ 448, 1995, pp. 166–186 arXiv:hep-th/9501096
  * [BB04] O. Biquard and P. Boalch “Wild non-abelian Hodge theory on curves” In _Compos. Math._ 140, 2004, pp. 179–204 arXiv:math.DG/0111098
  * [Boa14] P. Boalch “Geometry and braiding of Stokes data; Fission and wild character varieties” In _Ann. Math._ 179.1 Princeton Universitythe Institute for Advanced Study, 2014, pp. 301–365
  * [Bon+17] G. Bonelli, O. Lisovyy, K. Maruyoshi, A. Sciarappa and A. Tanzini “On Painlevé/gauge theory correspondence” In _Lett. Math. Phys_ 107.2, 2017, pp. 2359–2413 arXiv:1612.06235 [hep-th]
  * [BM19] T. D. Brennan and G. W. Moore “Index-like theorems from line defect vevs” In _J. High Energy Phys._ 09, 2019, pp. 073 arXiv:1903.08172 [hep-th]
  * [Bri19] T. Bridgeland “Riemann-Hilbert problems from Donaldson–Thomas theory” In _Invent. Math._ 216.1 Springer, 2019, pp. 69–124 arXiv:1611.03697 [math.AG]
  * [Bri20] T. Bridgeland “Riemann-Hilbert problems for the resolved conifold and non-perturbative partition functions” In _J. Diff. Geom._ 115, 2020, pp. 395–435 arXiv:1703.02776 [math.AG]
  * [BM20] T. Bridgeland and D. Masoero “On the monodromy of the deformed cubic oscillator”, 2020 arXiv:2006.10648 [math.CA]
  * [Bul+16] M. Bullimore, T. Dimofte, D. Gaiotto and J. Hilburn “Boundaries, mirror symmetry, and symplectic duality in $3d$ $\mathcal{N}=4$ gauge theory” In _J. High Energy Phys._ 10, 2016, pp. 108 arXiv:1603.08382 [hep-th]
  * [CCV11] S. Cecotti, C. Cordova and C. Vafa “Braids, walls, and mirrors”, 2011 arXiv:1110.2115 [hep-th]
  * [CV13] S. Cecotti and C. Vafa “Classification of complete ${\mathcal{N}}=2$ supersymmetric theories in 4 dimensions” In _Surv. Diff. Geom._ 18, 2013, pp. 19–101 arXiv:1103.5832 [hep-th]
  * [CD10] O. Chacaltana and J. Distler “Tinkertoys for Gaiotto duality” In _J. High Energy Phys._ 11, 2010, pp. 099 arXiv:1008.5203 [hep-th]
  * [CK01] S. A. Cherkis and A. Kapustin “Nahm transform for periodic monopoles and ${\mathcal{N}}=2$ super Yang-Mills theory” In _Comm. Math. Phys._ 218, 2001, pp. 333–371 arXiv:hep-th/0006050
  * [CKVP98] P. Claus, R. Kallosh and A. Van Proeyen “M5-brane and superconformal $(0,2)$ tensor multiplet in six dimensions” In _Nucl. Phys. B_ 518, 1998, pp. 117–150 arXiv:hep-th/9711161
  * [CGM17] S. Codesido, A Grassi and M. Mariño “Spectral theory and mirror curves of higher genus” In _Ann. Henri Poincaré_ 18, 2017, pp. 559–622 arXiv:1507.02096 [hep-th]
  * [CLT20] I. Coman, P. Longhi and J. Teschner “From quantum curves to topological string partition functions II”, 2020 arXiv:2004.04585 [hep-th]
  * [CPT18] I. Coman, E. Pomoni and J. Teschner “From quantum curves to topological string partition functions”, 2018 arXiv:1811.01978 [hep-th]
  * [CPT19] I. Coman, E. Pomoni and J. Teschner “Toda conformal blocks, quantum groups, and flat connections” In _Comm. Math. Phys._ 375, 2019, pp. 1117–1158 arXiv:1712.10225 [hep-th]
  * [CPT20] I. Coman, E. Pomoni and J. Teschner “Trinion conformal blocks from topological strings” In _J. High Energy Phys._ 09, 2020, pp. 078 arXiv:1906.06351 [hep-th]
  * [Dij97] R. Dijkgraaf “Les Houches lectures on fields, strings and duality” In _NATO Advanced Study Institute: Les Houches Summer School on Theoretical Physics, Session 64: Quantum Symmetries_ , 1997, pp. 3–147 arXiv:hep-th/9703136
  * [DGG16] T. Dimofte, M. Gabella and A. B. Goncharov “K-decompositions and $3d$ gauge theories” In _J. High Energy Phys._ 11, 2016, pp. 151 arXiv:1301.0192 [hep-th]
  * [DGG13] T. Dimofte, D. Gaiotto and S. Gukov “3-manifolds and $3d$ indices” In _Adv. Theor. Math. Phys._ 17.5, 2013, pp. 975–1076 arXiv:1112.5179 [hep-th]
  * [DGG14] T. Dimofte, D. Gaiotto and S. Gukov “Gauge theories labelled by three-manifolds” In _Comm. Math. Phys._ 325, 2014, pp. 367–419 arXiv:1108.4389 [hep-th]
  * [DGV15] T. Dimofte, D. Gaiotto and R. Veen “RG domain walls and hybrid triangulations” In _Adv. Theor. Math. Phys._ 19, 2015, pp. 137–276 arXiv:1304.6721 [hep-th]
  * [DW96] R. Donagi and E. Witten “Supersymmetric Yang-Mills theory and integrable systems” In _Nucl. Phys. B_ 460, 1996, pp. 299–334 arXiv:hep-th/9510101
  * [Don83] S. K. Donaldson “A new proof of a theorem of Narasimhan and Seshadri” In _J. Diff. Geom._ 18, 1983, pp. 269–277
  * [Dru+10] N. Drukker, J. Gomis, T. Okuda and J. Teschner “Gauge theory loop operators and Liouville theory” In _J. High Energy Phys._ 02, 2010, pp. 057 arXiv:0909.1105 [hep-th]
  * [DMO09] N. Drukker, D. R. Morrison and T. Okuda “Loop operators and S-duality from curves on Riemann surfaces” In _J. High Energy Phys._ 09, 2009, pp. 031 arXiv:0907.2593 [hep-th]
  * [Dum18] O. Dumitrescu “A journey from the Hitchin section to the oper moduli” In _Proc. Symp. Pure Math._ 98, 2018, pp. 107–138 arXiv:1701.00155 [math.AG]
  * [Dum+21] O. Dumitrescu, L. Fredrickson, G. Kydonakis, R. Mazzeo, M. Mulase and A. Neitzke “From the Hitchin section to opers through nonabelian Hodge” In _J. Diff. Geom._ 117, 2021, pp. 223–253 arXiv:1607.02172 [math.DG]
  * [FF11] B. Feigin and E. Frenkel “Quantization of soliton systems and Langlands duality” In _Adv. Stud. Pure Math._ 61, 2011, pp. 185–274 arXiv:0705.2486 [math.QA]
  * [Fre07] E. Frenkel “Lectures on the Langlands program and conformal field theory” In _Les Houches School of Physics: Frontiers in Number Theory, Physics and Geometry_ , 2007, pp. 387–533 arXiv:hep-th/0512172
  * [Gab+17] M. Gabella, P. Longhi, C. Y. Park and M. Yamazaki “BPS graphs: From spectral networks to BPS quivers” In _J. High Energy Phys._ 07, 2017, pp. 032 arXiv:1704.04204 [hep-th]
  * [GRW15] A. Gadde, S. S. Razamat and B. Willett “‘Lagrangian’ for a non-Lagrangian field theory with ${\mathcal{N}}=2$ supersymmetry” In _Phys. Rev. Lett._ 115.17, 2015, pp. 171604 arXiv:1505.05834 [hep-th]
  * [Gai12] D. Gaiotto “${\mathcal{N}}=2$ dualities” In _J. High Energy Phys_ 08, 2012, pp. 034 arXiv:0904.2715 [hep-th]
  * [Gai14] D. Gaiotto “Opers and TBA”, 2014 arXiv:1403.6137 [hep-th]
  * [GMN10] D. Gaiotto, G. W. Moore and A. Neitzke “Four-dimensional wall-crossing via three-dimensional field theory” In _Comm.Math.Phys._ 299, 2010, pp. 163–224 arXiv:0807.4723 [hep-th]
  * [GMN13] D. Gaiotto, G. W. Moore and A. Neitzke “Framed BPS states” In _Adv. Theor. Math. Phys._ 17.2, 2013, pp. 241–397 arXiv:1006.0146 [hep-th]
  * [GMN13a] D. Gaiotto, G. W. Moore and A. Neitzke “Spectral networks” In _Ann. Henri Poincaré_ 14, 2013, pp. 1643–1731 arXiv:1204.4824 [hep-th]
  * [GMN13b] D. Gaiotto, G. W. Moore and A. Neitzke “Wall-crossing, Hitchin systems, and the WKB approximation” In _Adv. Math._ 234 Elsevier, 2013, pp. 239–403 arXiv:0907.3987 [hep-th]
  * [GMN14] D. Gaiotto, G. W. Moore and A. Neitzke “Spectral networks and snakes” In _Ann. Henri Poincaré_ 15, 2014, pp. 61–141 arXiv:1209.0866 [hep-th]
  * [Gal+13] D. Galakhov, P. Longhi, T. Mainiero, G. W. Moore and A. Neitzke “Wild wall crossing and BPS giants” In _J. High Energy Phys._ 11 Springer, 2013, pp. 046 arXiv:1305.5454 [hep-th]
  * [GLM15] D. Galakhov, P. Longhi and G. W. Moore “Spectral networks with spin” In _Comm. Math. Phys._ 340.1, 2015, pp. 171–232 arXiv:1408.0207 [hep-th]
  * [Gol84] W. M. Goldman “The symplectic nature of fundamental groups of surfaces” In _Adv. Math._ 54, 1984, pp. 200–225
  * [Gor+95] A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov “Integrability and Seiberg-Witten exact solution” In _Phys. Lett. B_ 355, 1995, pp. 466–474 arXiv:hep-th/9505035
  * [GGM20] A. Grassi, J. Gu and M. Mariño “Non-perturbative approaches to the quantum Seiberg-Witten curve” In _J. High Energy Phys._ 07, 2020, pp. 106 arXiv:1908.07065 [hep-th]
  * [GHN21] A. Grassi, Q. Hao and A. Neitzke “Exact WKB methods in $SU(2)$ $N_{\rm f}=1$”, 2021 arXiv:2105.03777 [hep-th]
  * [GHM16] A. Grassi, Y. Hatsuda and M. Mariño “Topological strings from quantum mechanics” In _Ann. Henri Poincaré_ 17, 2016, pp. 3177–3235 arXiv:1410.3382 [hep-th]
  * [Gun67] R. C. Gunning “Special coordinate coverings of Riemann surfaces” In _Math. Ann._ 170, 1967, pp. 67–86
  * [HHN20] Q. Hao, L. Hollands and A. Neitzke “BPS states in the Minahan-Nemesch-ansky $E_{7}$ theory” In _J. High Energy Phys._ 04, 2020, pp. 039 arXiv:1905.09879 [hep-th]
  * [HMS95] J. A. Harvey, G. W. Moore and A. Strominger “Reducing S-duality to T-duality” In _Phys. Rev. D_ 52, 1995, pp. 7161–7167 arXiv:hep-th/9501022
  * [Hit87] N. J. Hitchin “The self-duality equations on a Riemann surface” In _Proc. London Math. Soc._ 55, 1987, pp. 59–131
  * [HKS11] L. Hollands, C. A. Keller and J. Song “From $SO/Sp$ instantons to $W$-algebra blocks” In _J. High Energy Phys._ 03, 2011, pp. 053 arXiv:1012.4468 [hep-th]
  * [HKS11a] L. Hollands, C. A. Keller and J. Song “Towards a $4d/2d$ correspondence for Sicilian quivers” In _J. High Energy Phys._ 10, 2011, pp. 100 arXiv:1107.0973 [hep-th]
  * [HK18] L. Hollands and O. Kidwai “Higher length-twist coordinates, generalized Heun’s opers, and twisted superpotentials” In _Adv. Theor. Math. Phys._ 22 International Press of Boston, 2018, pp. 1713–1822 arXiv:1710.04438 [hep-th]
  * [HN16] L. Hollands and A. Neitzke “Spectral networks and Fenchel–Nielsen coordinates” In _Lett. Math. Phys._ 106.6 Springer, 2016, pp. 811–877 arXiv:1312.2979 [math.GT]
  * [HN17] L. Hollands and A. Neitzke “BPS states in the Minahan-Nemeschansky ${E_{6}}$ theory” In _Comm. Math. Phys._ 353, 2017, pp. 317–351 arXiv:1607.01743 [hep-th]
  * [HN20] L. Hollands and A. Neitzke “Exact WKB and abelianization for the $T_{3}$ equation” In _Comm. Math. Phys._ 380, 2020, pp. 131–186 arXiv:1906.04271 [hep-th]
  * [HKPK13] M.-x. Huang, A.-K. Kashani-Poor and A. Klemm “The $\Omega$-deformed B-model for rigid $\mathcal{N}=2$ theories” In _Ann. Henri Poincaré_ 14, 2013, pp. 425–497 arXiv:1109.5728 [hep-th]
  * [IM21] M. Ionita and B. Morrissey “Spectral networks and nonabelianization”, 2021 arXiv:2103.12285 [math.AG]
  * [IMS19] K. Ito, M. Mariño and H. Shu “TBA equations and resurgent quantum mechanics” In _J. High Energy Phys._ 01, 2019, pp. 228 arXiv:1811.04812 [hep-th]
  * [IOT12] Y. Ito, T. Okuda and M. Taki “Line operators on $S^{1}\times\mathbb{R}^{3}$ and quantization of the Hitchin moduli space” In _J. High Energy Phys._ 04, 2012, pp. 010 arXiv:1111.4221 [hep-th]
  * [IK20] K. Iwaki and O. Kidwai “Topological recursion and uncoupled BPS
structures I: BPS spectrum and free energies”, 2020 arXiv:2010.05596 [math-ph]

  * [IK21] K. Iwaki and O. Kidwai “Topological recursion and uncoupled BPS structures II: Voros symbols and the $\tau$-function”, 2021 arXiv:2108.06995 [math-ph]
  * [IN14] K. Iwaki and T. Nakanishi “Exact WKB analysis and cluster algebras” In _J. Phys. A_ 47 IOP Publishing, 2014, pp. 474009
  * [JN18] S. Jeong and N. A. Nekrasov “Opers, surface defects, and Yang-Yang functional”, 2018 arXiv:1806.08270 [hep-th]
  * [JN20] S. Jeong and N. A. Nekrasov “Riemann-Hilbert correspondence and blown up surface defects” In _J. High Energy Phys._ 12, 2020, pp. 006 arXiv:2007.03660 [hep-th]
  * [KRS09] A. Kapustin, L. Rozansky and N. Saulina “Three-dimensional topological field theory and symplectic algebraic geometry I” In _Nucl. Phys. B_ 816, 2009, pp. 295–355 arXiv:0810.5415 [hep-th]
  * [KW07] A. Kapustin and E. Witten “Electric-magnetic duality and the geometric Langlands program” In _Comm. Num. Theor. Phys._ 1, 2007, pp. 1–236 arXiv:hep-th/0604151 [hep-th]
  * [KT05] T. Kawai and Y. Takei “Algebraic Analysis of Singular Perturbation Theory” American Mathematical Society, 2005
  * [Kle+96] A. Klemm, W. Lerche, P. Mayr, C. Vafa and N. P. Warner “Selfdual strings and ${\mathcal{N}}=2$ supersymmetric field theory” In _Nucl.Phys.B_ 477, 1996, pp. 746–766 arXiv:hep-th/9604034
  * [KS21] T. Koike and R. Schäfke “On the Borel summability of WKB solutions of Schrödinger equations with polynomial potentials and its applications” In _To appear_ , 2021
  * [Kon93] H. Konno “Construction of the moduli space of stable parabolic Higgs bundles on a Riemann surface” In _J. Math. Soc. Japan_ 45, 1993, pp. 253–276
  * [Kon92] M. Kontsevich “Intersection theory on the moduli space of curves and the matrix Airy function” In _Comm. Math. Phys._ 147, 1992, pp. 1–23
  * [KS08] M. Kontsevich and Y. Soibelman “Stability structures, motivic Donaldson-Thomas invariants and cluster transformations”, 2008 arXiv:0811.2435 [math.AG]
  * [Ler97] W. Lerche “Introduction to Seiberg-Witten theory and its stringy origin” In _Nucl. Phys. Proc. Suppl. B_ 55, 1997, pp. 83–117 arXiv:hep-th/9611190
  * [Lon18] P. Longhi “Wall-crossing invariants from spectral networks” In _Ann. Henri Poincaré_ 19.3, 2018, pp. 775–842 arXiv:1611.00150 [hep-th]
  * [LP16] P. Longhi and C. Y. Park “ADE spectral networks” In _J. High Energy Phys._ 08, 2016, pp. 087 arXiv:1601.02633 [hep-th]
  * [Mar20] M. Martone “The constraining power of Coulomb branch geometry: Lectures on Seiberg-Witten theory” In _Young Researchers Integrability School and Workshop 2020: A modern primer for superconformal field theories_ , 2020 arXiv:2006.14038 [hep-th]
  * [MS17] K. Maruyoshi and J. Song “Enhancement of supersymmetry via renormalization group flow and the superconformal index” In _Phys. Rev. Lett._ 118.15, 2017, pp. 151602 arXiv:1606.05632 [hep-th]
  * [Mat95] M. Matone “Instantons and recursion relations in ${\mathcal{N}}=2$ SUSY gauge theory” In _Phys. Lett. B_ 357, 1995, pp. 342–348 arXiv:hep-th/9506102
  * [MN96] J. A. Minahan and D. Nemeschansky “An ${\mathcal{N}}=2$ superconformal fixed point with $E_{6}$ global symmetry” In _Nucl. Phys. B_ 482.1-2 Elsevier, 1996, pp. 142–152 arXiv:hep-th/9608047
  * [MM10] A. Mironov and A. Morozov “Nekrasov functions and exact Bohr-Sommer-feld integrals” In _J. High Energy Phys._ 04, 2010, pp. 040 arXiv:0910.5670 [hep-th]
  * [MP15] V. Mitev and E. Pomoni “Toda 3-point functions from topological strings” In _J. High Energy Phys._ 06, 2015, pp. 049 arXiv:1409.6313 [hep-th]
  * [Mit+15] V. Mitev, E. Pomoni, M. Taki and F. Yagi “Fiber-base duality and global symmetry enhancement” In _J. High Energy Phys._ 04, 2015, pp. 052 arXiv:1411.2450 [hep-th]
  * [Moo10] G. W. Moore “PiTP lectures on BPS states and wall-crossing in $d=4$, ${\mathcal{N}}=2$ theories” Lecture notes available at
http://www.physics.rutgers.edu/~gmoore/PiTP-LectureNotes.pdf, 2010

  * [Moo12] G. W. Moore “Applications of the six-dimensional $(2,0)$-theory to physical mathematics” Lecture notes for Felix Klein lectures available at http://www.physics.rutgers.edu/~gmoore/FelixKleinLectureNotes.pdf, 2012
  * [Nah78] W. Nahm “Supersymmetries and their representations” In _Nucl. Phys. B_ 135, 1978, pp. 149–166
  * [Nak96] H. Nakajima “Hyperkähler structures on moduli spaces of parabolic Higgs bundles on Riemann surfaces” In _Lect. Notes Pure Appl. Math._ 179, 1996, pp. 199–208
  * [Nei16] A. Neitzke “Hitchin systems in ${\mathcal{N}}=2$ field theory” In _New Dualities of Supersymmetric Gauge Theories_ Springer, 2016, pp. 53–77 arXiv:1412.7120 [hep-th]
  * [Nei20] A. Neitzke “swn-plotter” Mathematica program available at
https://gauss.math.yale.edu/~an592/, 2020

  * [Nek03] N. A. Nekrasov “Seiberg-Witten prepotential from instanton counting” In _Adv. Theor. Math. Phys._ 7.5 International Press of Boston, 2003, pp. 831–864 arXiv:hep-th/0206161
  * [Nek20] N. A. Nekrasov “Blowups in BPS/CFT correspondence and Painlevé VI”, 2020 arXiv:2007.03646 [hep-th]
  * [NP12] N. A. Nekrasov and V. Pestun “Seiberg-Witten geometry of four-dimension-al ${\mathcal{N}}=2$ quiver gauge theories”, 2012 arXiv:1211.2240 [hep-th]
  * [NRS11] N. A. Nekrasov, A. Rosly and S. L. Shatashvili “Darboux coordinates, Yang-Yang functional, and gauge theory” In _Nucl. Phys. Proc. Suppl._ 216.1 Elsevier, 2011, pp. 69–93 arXiv:1103.3919 [hep-th]
  * [NS10] N. A. Nekrasov and S. L. Shatashvili “Quantization of integrable systems and four-dimensional gauge theories” In _XVIth International Congress On Mathematical Physics_ World Scientific, 2010, pp. 265–289 arXiv:0908.4052 [hep-th]
  * [NW10] N. A. Nekrasov and E. Witten “The Omega deformation, branes, integrability, and Liouville theory” In _J. High Energy Phys._ 09, 2010, pp. 092 arXiv:1002.0888 [hep-th]
  * [Nik20] N. Nikolaev “Exact solutions for the singularly perturbed Riccati equation and exact WKB analysis”, 2020 arXiv:2008.06492 [math.CA]
  * [Nik21] N. Nikolaev “Abelianisation of logarithmic $\mathfrak{sl}_{2}$-connections” In _Selecta Math._ 27.5, 2021, pp. 78 arXiv:1902.03384 [math.AG]
  * [Oku16] T. Okuda “Line operators in supersymmetric gauge theories and the $2d$-$4d$ relation” In _New Dualities of Supersymmetric Gauge Theories_ Springer, 2016, pp. 195–222 arXiv:1412.7126 [hep-th]
  * [Olv+10] “NIST Handbook of Mathematical Functions” Cambridge University Press, 2010
  * [Pes12] V. Pestun “Localization of gauge theory on a four-sphere and supersymmetric Wilson loops” In _Comm. Math. Phys._ 313, 2012, pp. 71–129 arXiv:0712.2824 [hep-th]
  * [Pes+17] V. Pestun “Localization techniques in quantum field theories” In _J. Phys. A_ 50.44, 2017, pp. 440301 arXiv:1608.02952 [hep-th]
  * [SW94] N. Seiberg and E. Witten “Electric-magnetic duality, monopole condensation, and confinement in ${\mathcal{N}}=2$ supersymmetric Yang-Mills theory” In _Nucl. Phys. B_ 426, 1994, pp. 19–52 arXiv:hep-th/9407087
  * [SW94a] N. Seiberg and E. Witten “Monopoles, duality and chiral symmetry breaking in ${\mathcal{N}}=2$ supersymmetric QCD” In _Nucl. Phys. B_ 431.3 Elsevier, 1994, pp. 484–550 arXiv:hep-th/9408099
  * [SW97] N. Seiberg and E. Witten “Gauge dynamics and compactification to three-dimensions” In _The Mathematical Beauty of Physics_ World Scientific, 1997, pp. 333–366 arXiv:hep-th/9607163
  * [Sim92] C. T. Simpson “Higgs bundles and local systems” In _Publ. Math. IHÉS_ 75, 1992, pp. 5–95
  * [Sim97] C. T. Simpson “The Hodge filtration on non-abelian cohomology” In _Proc. Symp. Pure Math._ 62, 1997, pp. 217–281
  * [Son12] J. Song “$4d/2d$ correspondence: Instantons and W-algebras”, 2012
  * [Str96] A. Strominger “Open $p$-branes” In _Phys. Lett. B_ 383, 1996, pp. 44–47 arXiv:hep-th/9512059
  * [Sza16] R. J. Szabo “${\mathcal{N}}=2$ gauge theories, instanton moduli spaces and geometric representation theory” In _J. Geom. Phys._ 109, 2016, pp. 83–121 arXiv:1507.00685 [hep-th]
  * [Tac14] Y. Tachikawa “${\mathcal{N}}=2$ supersymmetric dynamics for pedestrians” In _Lect. Notes Phys._ 890, 2014, pp. 1–205 arXiv:1312.2684 [hep-th]
  * [Tac15] Y. Tachikawa “A review of the $T_{N}$ theory and its cousins” In _Progr. Theor. Exp. Phys._ 2015.11, 2015, pp. 11B102 arXiv:1504.01481 [hep-th]
  * [Tac16] Y. Tachikawa “A review on instanton counting and W-algebras” In _New Dualities of Supersymmetric Gauge Theories_ Springer, 2016, pp. 79–120 arXiv:1412.7121 [hep-th]
  * [Tak17] Y. Takei “WKB analysis and Stokes geometry of differential equations” In _Analytic, Algebraic and Geometric Aspects of Differential Equations_ , 2017, pp. 263–304
  * [Tes16] “New Dualities of Supersymmetric Gauge Theories” In _Math. Phys. Stud._ , Mathematical Physics Studies Springer, 2016
  * [Tes16a] J. Teschner “Quantisations of some spectral curves” In _Oberwolfach Rept._ 9, 2016, pp. 433–435
  * [TV15] J. Teschner and G. S. Vartanov “Supersymmetric gauge theories, quantization of $\mathscr{M}_{\mathrm{flat}}$, and conformal field theory” In _Adv. Theor. Math. Phys._ 19, 2015, pp. 1–135 arXiv:1302.3778 [hep-th]
  * [Tul77] W. M. Tulczyjew “The Legendre transformation” In _Ann.Inst.Henri Poincaré A_ 27, 1977, pp. 101–114
  * [Wit91] E. Witten “Two-dimensional gravity and intersection theory on moduli space” In _Surv. Diff. Geom._ 1, 1991, pp. 243–310
  * [Wit93] E. Witten “Phases of ${\mathcal{N}}=2$ theories in two dimensions” In _Nucl. Phys. B_ 403, 1993, pp. 159–222 arXiv:hep-th/9301042
  * [Wit96] E. Witten “Some comments on string dynamics” In _Future Perspectives in String Theory_ World Scientific, 1996, pp. 501–523 arXiv:hep-th/9507121
  * [Wit97] E. Witten “Solutions of four-dimensional field theories via M-theory” In _Nucl. Phys. B_ 500, 1997, pp. 3–42 arXiv:hep-th/9703166
  * [Wit05] E. Witten “$SL(2,\mathbb{Z})$ action on three-dimensional conformal field theories with abelian symmetry” In _From Fields to Strings: Circumnavigating Theoretical Physics_ World Scientific, 2005, pp. 1173–1200 arXiv:hep-th/0307041
  * [Yan20] F. Yan “Exact WKB and the quantum Seiberg-Witten curve for $4d$ ${\mathcal{N}}=2$ pure $SU(3)$ Yang-Mills, Part I: Abelianization”, 2020 arXiv:2012.15658 [hep-th]


# DeepOKAN: Deep Operator Network Based on Kolmogorov Arnold Networks for
Mechanics Problems

Diab W. Abueidda
Civil and Urban Engineering Department
New York University Abu Dhabi
National Center for Supercomputing Applications
University of Illinois at Urbana-Champaign
& Panos Pantidis
Civil and Urban Engineering Department
New York University Abu Dhabi
& Mostafa E. Mobasher
Civil and Urban Engineering Department
New York University Abu Dhabi
da3205<EMAIL_ADDRESS>
###### Abstract

The modern digital engineering design often requires costly repeated
simulations for different scenarios. The prediction capability of neural
networks (NNs) makes them suitable surrogates for providing design insights.
However, only a few NNs can efficiently handle complex engineering scenario
predictions. We introduce a new version of the neural operators called
DeepOKAN, which utilizes Kolmogorov Arnold networks (KANs) rather than the
conventional neural network architectures. Our DeepOKAN uses Gaussian radial
basis functions (RBFs) rather than the B-splines. The DeepOKAN is used to
develop surrogates for different mechanics problems. This approach should pave
the way for further improving the performance of neural operators. Based on
the current investigations, we observe that DeepOKANs require a smaller number
of learnable parameters than current MLP-based DeepONets to achieve comparable
accuracy.

_Keywords_ Computational solid mechanics $\cdot$ Deep operator networks
$\cdot$ Gaussian radial basis functions $\cdot$ Neural networks $\cdot$
Orthotropic elasticity $\cdot$ Transient analysis

##  1 Introduction

Contemporary science and engineering increasingly depend on advanced physics-
based computational models. Numerical simulations offer critical insights for
understanding and predicting complex physical phenomena and engineering
systems. Real-world problems often involve varying loads, boundary and initial
conditions, material properties, and domain geometries. Due to their inherent
complexity, multi-physics nature, dimensionality, time dependency, and
fidelity requirements, finite element analysis (FEA) models can be
computationally intensive, even with high-performance computing platforms [1,
2, 3]. Consequently, relying solely on traditional high-fidelity simulation
models for tasks such as computer-aided design, material discovery, and
digital twins is often impractical, especially when exploring numerous design
scenarios or geometries [4]. In this context, surrogate neural network (NN)
models emerge as a promising machine learning approach, capable of rapidly
inferring solutions to physical problems without requiring expensive numerical
simulations once they are trained [5, 6]. These models hold significant
potential across various application domains, including real-time simulations
for predictions and controls, design, topology and shape optimizations,
sensitivity analysis, and uncertainty quantification, which require extensive
forward evaluations with varying parameters [7, 8, 9]. For a comprehensive
overview of the application of neural networks in computational mechanics, see
[10].

However, most of these approaches necessitate retraining or transfer learning
when input parameters such as loads, boundary conditions, material properties,
or geometry are altered. This requirement is similarly observed in
conventional numerical methods like finite elements (FE), where each new input
parameter value mandates a distinct simulation. Recently, researchers devised
NNs trained to approximate the underlying physics or mathematical operator for
a class of problems, a process known as operator learning [11]. Researchers
have proposed various architectures for operator learning, with two notable
examples being the Fourier neural operator (FNO) and the deep operator network
(DeepONet). The Fourier Neural Operator (FNO) was first introduced by Li et
al. [12] to solve partial differential equations with parametric inputs.
Drawing inspiration from the Fourier transform used in differential equation
solutions, the input function is processed through multiple Fourier layers.
The encoded information is then mapped onto the output function space. Each
Fourier layer applies the fast Fourier transform (FFT) to its input and
filters out high-frequency modes. FNO and its enhanced versions have been
successfully used for Burger’s equation, Darcy flow, and the Navier-Stokes
equation [12], as well as for elasticity and plasticity problems in solid
mechanics [13]. Nonetheless, since the FNO employs FFT, it encounters
challenges when dealing with complex geometries or intricate non-periodic
boundary conditions. Additionally, Fourier-based operations are
computationally expensive when addressing high-dimensional problems.

Lately, Lu et al. [14, 15] introduced the DeepONet, which emerged as another
capable architecture for operator learning. DeepONet comprises two sub-
networks: a branch network for encoding input functions and a trunk network
for encoding input domain geometry. Initially, both networks were designed as
multi-layer perceptron (MLP) networks. In the seminal study, DeepONet
effectively mapped between unknown parametric functions and solution spaces
for several linear and nonlinear partial differential equations (PDEs) while
also learning explicit operators such as integrals. DeepONets have been
increasingly used to tackle scientific and engineering challenges, such as the
inverse design of nanoscale heat transfer systems [16], brittle fracture [17],
digital twins [18], and uncertainty quantification [19]. Lu et al. [20]
thoroughly compared FNO and DeepONet. Several researchers have proposed
different architectures for the branch and/or trunk networks to enhance
DeepONet’s versatility and capability to capture complex scenarios [21, 22,
23, 24, 25, 26].

The significance of MLPs is immense, as they are the standard models in
machine learning for approximating nonlinear functions, owing to their
expressive power as assured by the universal approximation theorem [27, 28].
Recently, Liu et al. [29] proposed a promising alternative to MLPs called
Kolmogorov-Arnold networks (KANs). While MLPs are based on the universal
approximation theorem [28], KANs are based on the Kolmogorov-Arnold
representation theorem [30, 31, 32]. Similar to MLPs, KANs have fully
connected architectures. Nevertheless, unlike MLPs, which use fixed activation
functions on nodes (neurons), KANs employ activation functions with learnable
weights on edges. This difference can make KANs more accurate and
interpretable than MLPs for several modeling scenarios. The potential of using
the Kolmogorov-Arnold representation theorem to construct neural networks has
been explored in various studies [33, 34, 35, 36, 37] before the work of Liu
et al. [29]. However, most research has adhered to the original depth-$2$,
width-$(2n+1)$ representation and has not utilized modern techniques such as
backpropagation for training. Liu et al. [29] generalized the original
Kolmogorov-Arnold representation to arbitrary widths and depths, thereby
revitalizing and contextualizing it within the contemporary deep learning
framework. They also conducted extensive empirical experiments to demonstrate
its potential as a foundational artificial intelligence and science model,
highlighting its accuracy and interoperability. Although the work of Liu et
al. [29] was published recently, it inspired other researchers to investigate
the topic further [38, 39].

This study introduces the DeepOKAN, where the branch and trunk of DeepONet are
KANs rather than the traditional MLPs. While the original implementation
utilized splines for function approximation, we have replaced them with radial
basis functions (RBFs). This modification has resulted in a significant
computational speedup, enhancing the model’s efficiency without compromising
accuracy. Adopting RBFs leverages their scalability and flexibility, making
the approach more suitable for high-dimensional applications. The DeepOKAN is
used to solve several mechanics problems. The paper’s structure is outlined as
follows: Section 2 provides an overview of KANs and Gaussian RBFs. It also
scrutinizes the DeepOKAN. In Section 3, the DeepOKAN is developed as a neural
operator for a few numerical examples. The current version of the paper has
one example only. The other examples will be added soon. The paper concludes
in Section 4 by summarizing the key results and indicating potential
directions for future research.

##  2 Methods

###  2.1 Kolmogorov-Arnold representation theorem

The Kolmogorov-Arnold representation theorem, also known as the superposition
theorem, is a fundamental result in approximation theory. It asserts that any
continuous multivariate function on a bounded domain can be represented as a
composition of a finite number of univariate functions and a set of linear
operations (additions). Specifically, for any smooth function
$f:[0,1]^{n_{\text{in}}}\rightarrow\mathbb{R}^{n_{\text{out}}}$, the
Kolmogorov-Arnold theorem asserts that there exist continuous univariate
functions $\psi_{q}$ and $\phi_{p,q}$ such that:

$\displaystyle f(\boldsymbol{x})$
$\displaystyle=\sum_{q=1}^{2n_{\text{in}}+1}\psi_{q}\left(\sum_{p=1}^{n_{\text{in}}}\phi_{p,q}(x_{p})\right),$
(1)

where $\boldsymbol{x}=(x_{1},\ldots,x_{d_{\text{in}}})$,
$\phi_{q,p}:[0,1]\rightarrow\mathbb{R}$, and
$\psi_{q}:\mathbb{R}\rightarrow\mathbb{R}$.

###  2.2 Kolmogorov-Arnold network

The Kolmogorov-Arnold representation theorem states that any continuous
multivariable function on a bounded domain can be represented as a
superposition of continuous functions of one variable and addition. While no
strict restrictions exist on choosing these univariate functions beyond their
continuity, practical considerations may influence their specific forms based
on the problem context and desired properties. In the KAN implementation by
Liu et al. [29], each KAN layer is composed of the sum of the spline function
and the SiLU activation function, with learnable coefficients. Splines face a
significant curse of dimensionality (COD) due to their inability to leverage
compositional structures. On the other hand, MLPs are less affected by COD
because of their feature learning capabilities. Still, they are less accurate
than splines in low-dimensional settings due to their inefficiency in
optimizing univariate functions. A model must capture the compositional
structure (external degrees of freedom) and effectively approximate univariate
functions (internal degrees of freedom) to learn a function accurately. A more
detailed description of KANs can be found in the work of [29].

Here, we provide a brief description. A KAN comprises several KAN layers.
Figure 1 shows a KAN with two KAN layers. A KAN layer with
$n_{in}$-dimensional inputs and $n_{out}$-dimensional outputs can be
represented as a matrix of 1D functions, denoted as:

$\displaystyle\boldsymbol{\Psi}=\\{\phi_{q,p}\\},\quad
p=1,2,\ldots,n_{\text{in}},\quad q=1,2,\ldots,n_{\text{out}},$ (2)

where the functions $\phi_{q,p}$ have learnable parameters.

Figure 1: Illustration of the activation functions flowing through the
network.

The shape of KAN can be expressed as an integer array
$\left[n_{0},n_{1},\ldots,n_{L}\right]$, where $n_{l}$ is the number of nodes
(neurons) in the $l^{\text{th}}$ layer (see Figure 1). The $i^{\text{th}}$
neuron in the $l^{\text{th}}$ layer is denoted by $(l,i)$, while the
activation value of the $(l,i)$-neuron is represented by $x_{l,i}$. There are
$n_{l}n_{l+1}$ activation functions between the consecutive layers $l$ and
$l+1$. The activation function connecting the neurons $(l,i)$ and $(l+1,j)$ is
denoted by:

$\displaystyle\phi_{l,i,j},\quad l=0,\ldots,L-1,\quad i=1,\ldots,n_{l},\quad
j=1,\ldots,n_{l+1}.$ (3)

As shown in Figure 1, the activation functions appear on the edges rather than
the nodes, unlike MLPs. Specifically, the pre-activation of $\phi_{l,i,j}$ is
$x_{l,i}$, while the post-activation of $\phi_{l,i,j}$ is $\tilde{x}_{l,i,j}$.
Then, the $\tilde{x}_{l,i,j}$’s are summed to determine $x_{l+1,j}$, which is
the activation value at the $(l+1,j)$-neuron:

$\displaystyle
x_{l+1,j}=\sum_{i=1}^{n_{l}}\tilde{x}_{l,i,j}=\sum_{i=1}^{n_{l}}\phi_{l,i,j}(x_{l,i}).$
(4)

This can be presented in a matrix form as follows:

$\displaystyle\boldsymbol{x}_{l+1}=\underbrace{\begin{pmatrix}\phi_{l,1,1}(\cdot)&\phi_{l,1,2}(\cdot)&\cdots&\phi_{l,1,n_{l}}(\cdot)\\\
\phi_{l,2,1}(\cdot)&\phi_{l,2,2}(\cdot)&\cdots&\phi_{l,2,n_{l}}(\cdot)\\\
\vdots&\vdots&\ddots&\vdots\\\
\phi_{l,n_{l+1},1}(\cdot)&\phi_{l,n_{l+1},2}(\cdot)&\cdots&\phi_{l,n_{l+1},n_{l}}(\cdot)\end{pmatrix}}_{\boldsymbol{\Psi}_{l}}\boldsymbol{x}_{l},$
(5)

where $\boldsymbol{\Psi}_{l}$ is the function matrix corresponding to the
$l^{\text{th}}$ KAN layer. Then, a general KAN is expressed as a composition
of $L$ layers:

$\displaystyle\text{KAN}(\boldsymbol{x})=(\boldsymbol{\Psi}_{L-1}\circ\boldsymbol{\Psi}_{L-2}\circ\cdots\circ\boldsymbol{\Psi}_{1}\circ\boldsymbol{\Psi}_{0})\boldsymbol{x}.$
(6)

The original Kolmogorov–Arnold representation (see Equation 1) is deemed a
special case of the KAB, with two layers and a shape
$\left[n_{0},2n_{0}+1,1\right]$. On the other hand, the well-known MLPs
alternate between linear transformation $\boldsymbol{W}$ and nonlinear
activation functions $\sigma$:

$\displaystyle\text{MLP}(\boldsymbol{x})=(\boldsymbol{W}_{L-1}\circ\sigma\circ\boldsymbol{W}_{L-2}\circ\sigma\circ\cdots\circ\boldsymbol{W}_{1}\circ\sigma\circ\boldsymbol{W}_{0})\boldsymbol{x}.$
(7)

MLPs distinctly handle linear transformations $\boldsymbol{W}$ and
nonlinearities $\sigma$, whereas KANs combine both within $\boldsymbol{\Psi}$.

###  2.3 RBF-KAN

As mentioned earlier, Liu et al. [29] proposed using B-splines as activation
functions within the KAN framework. Here, we propose using RBFs for KANs. An
RBF is a real-valued function whose value depends only on the distance from a
central point. The Gaussian RBF is a specific type from the radial basis
function family. The transformation of each input $x_{i}$ using a Gaussian RBF
$\boldsymbol{R}$ is given by:

$\displaystyle R^{l}(x_{i}^{l},g_{j}^{l})$
$\displaystyle=\exp\left(-\left(\frac{x_{i}^{l}-\text{g}_{j}^{l}}{\beta}\right)^{2}\right),\quad
i=1,2,\ldots,n,\quad j=1,2,\ldots,m$ (8)

where $n$ is the input dimension (layer $l$ size), and $m$ represents the
number of grid points (RBF centers). The superscript $l$ denotes the layer
number: $l=1,\ldots,L$, where $L$ is the number of layers.. $g_{j}$ represents
the $j^{\text{th}}$ point in the RBF grid; $g_{j}\in\boldsymbol{G}$. The RBF
centers $\boldsymbol{G}$ can be learnable parameters or fixed points (non-
learnable) defined during the initialization of the RBF transformation within
the RBF-KAN framework. $\beta$ is the scaling factor for the RBF, defined as:

$\displaystyle\beta$
$\displaystyle=\frac{g_{\text{max}}-g_{\text{min}}}{m-1}.$ (9)

$g_{\text{max}}$ and $g_{\text{min}}$ denote the maximum and minimum values of
the grid, respectively. After transforming the inputs, the transformed
features are combined linearly using some learnable weight matrix
$\boldsymbol{W}$. The output $\boldsymbol{x}^{l+1}$ is computed as:

$\displaystyle\boldsymbol{x}^{l+1}$
$\displaystyle=\boldsymbol{W}^{l}\boldsymbol{R}^{l}\left(\boldsymbol{x}^{l},\boldsymbol{G}^{l}\right)$
(10)

where $\boldsymbol{R}$ is a vector of size $(n\times m,)$, $\boldsymbol{W}$ is
a matrix with a size of $(o,n\times m)$, where $o$ is the length of the vector
$\boldsymbol{x}^{l+1}$. While constructing an RBF-KAN, one can stack several
RBF-KAN layers. During the training of an RBF-KAN, $\boldsymbol{W}$ and
$\boldsymbol{G}$ are obtained by minimizing a loss function $\mathcal{L}$,
which requires finding the gradients of $\mathcal{L}$ with respect to
$\boldsymbol{W}$ and $\boldsymbol{G}$ using backpropagation:

$\displaystyle\frac{\partial\mathcal{L}}{\partial\boldsymbol{W}^{l}}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\boldsymbol{x}^{L}}\cdot\boldsymbol{W}^{L-1}\cdot\frac{\partial\boldsymbol{R}^{L-1}}{\partial\boldsymbol{x}^{L-1}}\cdot\boldsymbol{W}^{L-2}\cdot\cdots\cdot\boldsymbol{W}^{l+1}\cdot\frac{\partial\boldsymbol{R}^{l+1}}{\partial\boldsymbol{x}^{l+1}}\cdot\boldsymbol{R}^{l}$
(11) $\displaystyle\frac{\partial\mathcal{L}}{\partial\boldsymbol{G}^{l}}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\boldsymbol{x}^{L}}\cdot\boldsymbol{W}^{L-1}\cdot\frac{\partial\boldsymbol{R}^{L-1}}{\partial\boldsymbol{x}^{L-1}}\cdot\boldsymbol{W}^{L-2}\cdot\cdots\cdot\boldsymbol{W}^{l+1}\cdot\frac{\partial\boldsymbol{R}^{l+1}}{\partial\boldsymbol{x}^{l+1}}\cdot\boldsymbol{W}^{l}\cdot\frac{\partial\boldsymbol{R}^{l}}{\partial\boldsymbol{G}^{l}}$

where $\boldsymbol{x}^{L}$ is the final output of the RBF-KAN, and
$\frac{\partial\mathcal{L}}{\partial\boldsymbol{x}^{L}}$ depends on the choice
of the loss function.
$\frac{\partial\boldsymbol{R}^{l}}{\partial\boldsymbol{x}^{l}}$ and
$\frac{\partial\boldsymbol{R}^{l}}{\partial\boldsymbol{G}^{l}}$ are found by
differentiating Equation 8:

$\displaystyle\frac{\partial R^{l}}{\partial x_{i}^{l}}$
$\displaystyle=-\frac{2(x_{i}^{l}-g_{j}^{l})}{\beta^{2}}\exp\left(-\left(\frac{x_{i}^{l}-g_{j}^{l}}{\beta}\right)^{2}\right)$
(12) $\displaystyle\frac{\partial R^{l}}{\partial g_{j}^{l}}$
$\displaystyle=\frac{2(x_{i}^{l}-g_{j}^{l})}{\beta^{2}}\exp\left(-\left(\frac{x_{i}^{l}-g_{j}^{l}}{\beta}\right)^{2}\right).$

###  2.4 Neural operators

Researchers developed neural networks trained to approximate the underlying
physics or mathematical operator for a class of problems, a process known as
operator learning. Specifically, neural operators were proposed to learn
nonlinear operators, mapping input functions into corresponding output
functions. For neural operators, in the infinite functional space $Q$, the
parameter $q\in Q$ represents the input functions and $s\in S$ refers to the
unknown solutions of the PDE in the functional space $S$. It is assumed that
for each $q$ in $Q$, a unique solution exists $s=s(q)$ in $S$ corresponding to
the governing PDE, which also satisfies the boundary conditions (BCs).
Consequently, the mapping solution operator $F:Q\rightarrow S$ can be defined
as:

$\displaystyle F(q)=s(q).$ (13)

Specifically, for a collection of $N$ points $\boldsymbol{X}$ on a domain,
each denoted by its coordinates $(x_{i},y_{i})$, the neural operator considers
both the functions $q_{j}$ in its branch and positions $\boldsymbol{X}$ in its
trunk and predicts the solution operator $\hat{F}(q)(\boldsymbol{X})$ by
fusing intermediate encoded outputs $b_{i}$ (from branch) and $t_{i}$ (from
trunk) in a dot product enhanced by bias $B$, as shown in Figure 2. In a
larger sense, it is possible to think of $\hat{F}(q)(\boldsymbol{X})$ as a
function of $\boldsymbol{X}$ conditioning on input $q$.

Figure 2: Schematic for neural operators.

The parameter $q$ can encompass a variety of functional inputs that influence
the underlying PDE solution. For instance, $q$ may represent material
properties such as thermal conductivity or elastic modulus. Additionally, $q$
could denote load constants or time-dependent input load function, discretized
at $T$ time steps to form an input load vector $\boldsymbol{q}$. This allows
the modeling of transient processes where the applied loads vary over time, as
shown in one of the examples later. This study leverages neural operators as
regressors. Thus, one needs to devise an appropriate loss function. Typical
loss functions for regression problems are mean square error and mean absolute
error. This paper uses the root mean square deviation (RMSD) as the loss
function to be minimized and determine the optimized network parameter. The
RMSD is defined as:

$\displaystyle\mathcal{L}=\text{RMSD}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(s_{i}-\hat{s}_{i})^{2}}$
(14)

where $\hat{s}$ denotes the solution obtained from the neural operator. This
paper considers two types of neural operators: DeepONet and DeepOKAN.

###  2.5 DeepOKAN vs. DeepONet

A DeepONet (see Figure 3) model, which uses MLP networks in both its branch
and trunk, serves as a performance baseline. This paper introduces the
DeepOKAN (see Figure 4), which replaces the MLP networks with KANs.
Specifically, it employs Gaussian RBF KANs, though other KAN types, such as
multi-quadratic RBF KANs and B-splines KANs, could also be explored in the
context of neural operators. KAN models require significantly fewer learnable
parameters than MLP models while achieving equivalent or better accuracy [29].
RBFs have been used before in the context of machine learning [40], but their
success was limited. However, the recent development of KANs promises more
effective integration. By leveraging the theoretical foundations laid by
Kolmogorov and Arnold, KANs offer a compact and efficient representation of
functions. This efficiency, combined with higher accuracies, positions KANs as
a promising alternative to traditional MLPs across various applications.

Figure 3: Illustration of the DeepONet. Figure 4: Illustration of the
DeepOKAN. $\phi^{t}$ and $\phi^{b}$ represent the activation functions
corresponding to the trunk and branch networks, respectively. $p$ and $v$ are
hyperparameters defining the width of layer $l=1$ in the branch and trunk,
respectively. Different layers can possess different widths.

##  3 Numerical examples

###  3.1 2D orthotropic elasticity

####  3.1.1 FEA and data generation

Consider a homogeneous, orthotropic, elastic body undergoing small
deformations. In the absence of body forces and inertial forces, the
equilibrium equation can be expressed as follows:

$\displaystyle\boldsymbol{\nabla}\cdot\boldsymbol{\sigma}$
$\displaystyle=\boldsymbol{0}{,}\quad\boldsymbol{x}\in\Omega,$ (15)
$\displaystyle\boldsymbol{\hat{u}}$
$\displaystyle=\overline{\boldsymbol{u}},\quad\\!\boldsymbol{x}\in\Gamma_{u},$
$\displaystyle\boldsymbol{\sigma}\cdot\boldsymbol{n}$
$\displaystyle=\overline{\boldsymbol{t}},\quad\boldsymbol{x}\in\Gamma_{t}.$

Here, $\boldsymbol{\sigma}$ represents the Cauchy stress tensor,
$\boldsymbol{n}$ denotes the normal unit vector, $\boldsymbol{\nabla}\cdot$
signifies the divergence operator, and $\boldsymbol{\nabla}$ indicates the
gradient operator. Given the assumption of small deformations, the strain
$\boldsymbol{\varepsilon}$ can be described by:

$\displaystyle\boldsymbol{\varepsilon}$
$\displaystyle=\frac{1}{2}(\boldsymbol{\nabla}\boldsymbol{u}+\boldsymbol{\nabla}\boldsymbol{u}^{T})$
(16)

The relationship between the strain and stress is written as:

$\displaystyle\boldsymbol{\sigma}$
$\displaystyle=\mathbb{C}:\boldsymbol{\varepsilon},$ (17)

where $\mathbb{C}$ is the fourth-order elastic tensor defining linear elastic
behavior. In the case of plane-stress orthotropy, it is assumed that the two-
dimensional material has two planes of symmetry aligned with the global $x$
and $y$ axes. For the orthotropic plane-stress case, the compliance matrix
$\mathbf{S}$ is expressed as:

$\displaystyle\left[\mathbf{S}\right]=\begin{bmatrix}\frac{1}{E_{x}}&-\frac{\nu_{xy}}{E_{x}}&0\\\
-\frac{\nu_{yx}}{E_{y}}&\frac{1}{E_{y}}&0\\\
0&0&\frac{1}{G_{xy}}\end{bmatrix},$ (18)

with $E_{x}$ and $E_{y}$ representing Young’s moduli in the orthotropic
directions, $G_{xy}$ being the shear modulus, and $\nu_{xy}$ being the in-
plane Poisson’s ratio. The constitutive relation symmetry is ensured by:

$\displaystyle\nu_{xy}$ $\displaystyle=\frac{E_{x}}{E_{y}}\nu_{yx}.$ (19)

$\mathbf{S}$ has to be inverted to obtain the stiffness matrix.

In this example, we consider a two-dimensional (2D) domain subjected to
boundary conditions and traction forces shown in Figure 5. The bottom and
right edges of the domain are subjected to roller boundary conditions.
Traction forces are applied at the top edge of the domain, represented by two
components, $t_{x}$ and $t_{y}$. These forces are sampled from uniform
distributions within a specified range: $t_{x}$, $t_{y}$ $\in$
$\left[-0.3,0.3\right]$. As demonstrated in Equations 18 and 19, for a 2D
orthotropic material, four constants are required: $E_{x}$, $E_{y}$,
$\nu_{xy}$, and $G_{xy}$. We sample $E_{x}$, $E_{y}$, and $\nu_{xy}$ from
specified uniform distributions: $E_{x},E_{y}\in\left[5,20\right]$, and
$\nu_{xy}\in\left[0.15,0.35\right]$. $G_{xy}$ is computed using the following
relationship:

$\displaystyle G_{xy}$
$\displaystyle=\frac{1}{2}\left(\frac{E_{x}}{2\left(1+\nu_{xy}\right)}+\frac{E_{y}}{2\left(1+\nu_{yx}\right)}\right).$
(20)

Equation 20 does not generally hold for orthotropic materials. However, we
compute $G_{xy}$ using this equation to ensure that it remains within a
physically plausible range, as for most materials, it is observed that they
are less stiff in shear than in tension or compression. Hence, we would like
to highlight that it is an assumption we made, and one might choose a
different range to generate the dataset for training. Although $G_{xy}$ is
calculated using Equation 20 in the data generation phase, it is still fed to
the neural operators to learn its impact on the material deformation. We
assume zero body force and negligible inertia. This leaves us with six
parameters to be fed to the branch network: $t_{x}$, $t_{y}$, $E_{x}$,
$E_{y}$, $\nu_{xy}$, and $G_{xy}$. The coordinates are fed to the trunk
network. A total of 5000 samples were generated using this approach. The
dataset was then split into 80% for training and 20% for testing.

Figure 5: Schematic of the geometry and boundary conditions of the 2D
orthotropic material example.

####  3.1.2 Results

In order to compare the training efficiency of the DeepOKAN and DeepONet
models, we create the following variations. For the DeepOKAN, both the KAN
trunk and branch networks have a width of 6 and 1 hidden layer. Specifically,
the resulting architectures (including input and output layers) for the branch
and trunk are $\left[6,6,r=4\right]$ and $\left[2,6,r=4\right]$, respectively.
This yields a setup with 768 learnable parameters. For the DeepONet, the
benchmark MLP has 2 hidden layers and 16 neurons per hidden layer for both the
trunk and branch. The branch and trunk architectures have the following
shapes: $\left[6,16,16,r=4\right]$ and $\left[2,16,16,r=4\right]$,
respectively. This results in 738 learnable parameters, and the purpose of
that model is to have approximately the same number of learnable parameters as
in the KAN case. To further increase the approximation capability of the
DeepONets, we also adopt the following variations for the MLP hidden layers:
$\left[16,16,16\right]$, $\left[16,16,16,16\right]$, and $\left[64,64\right]$,
which result in 1282, 1826, and 9090 learnable parameters, respectively. We
train all the networks for 200, 1000, and 5000 epochs using the Adam
optimizer. We use a learning rate $lr=0.01$ and a learning scheduler every 500
epochs with $\gamma=0.5$. Specifically, the learning rate scheduler is
designed to reduce the learning rate by a factor of 0.5 every 500 epochs,
allowing the model to make more fine-grained adjustments during training as it
approaches convergence. This helps prevent the model from overshooting the
optimal solution as learning progresses.

The evolution of the loss function for all the models is shown in Fig. 6.
Here, we remark that the learning capability of all the DeepONet variations
seems to get quickly stalled, and the loss value remains essentially constant
after a few training epochs. This observation holds true regardless of the
number of the DeepONet learnable parameters. On the contrary, the DeepOKAN
model seems to follow a more healthy and steadily descending trend in its cost
function as the number of training epochs increases. This is the first sign
that the DeepOKAN shows more numerical flexibility during its training than
the DeepONet counterparts.

Figure 6: Loss function evolution for the 2D orthotropic elasticity problem,
when the networks are trained for a. 200, b. 1000 and c. 5000 Adam epochs.
Figure 7: a. Histogram of the L2-norm for all the test examples. b. Maximum
L2-norm value for each case of training epochs.

Next, we evaluate the accuracy of the predictions, and Fig. 7a shows a
histogram of the L2-norm values for all the testing examples. The left column
corresponds to the DeepOKAN, while the other graphs correspond to the four
DeepONet variations (color description shown in the legend). Each row
corresponds to a different number of training epochs (200, 1000, and 5000).
Also, to better elucidate the difference in the training efficiency between
DeepOKAN and DeepONets, we plot in Fig. 7b the maximum L2-norm value from all
the operators. Then, we can make the following observations. First, we note
that within the context we investigate, the DeepONet performance remains
essentially unaffected by the number of learnable parameters, and increasing
the training epochs has only a mild positive impact. On the contrary, the
DeepOKAN performance shows a clear improvement with the increased training
time. This is shown by the shift of the histograms to the lower range, as the
number of epochs increases, and also by the distinct decrease of the maximum
L2-norm value as shown in Fig. 7b.

Figure 8: a. True values of $u_{x}$ for 20 randomly selected test samples. b.
Predictions and absolute error using the DeepOKAN trained for 5000 Adam
epochs. c. Predictions and absolute error using the DeepONet [16,16] trained
for 5000 Adam epochs.

Finally, while monitoring the L2-norm of all examples provides a holistic
measure of performance evaluation, comparing the actual prediction fields is
also instructive. For this reason, we randomly pick 20 examples from the test
dataset, and we plot in Fig 8 the following contours: a) true values, b)
predictions and absolute error for the DeepOKAN trained at 5000 epochs, and c)
predictions and absolute error for the DeepONet [16,16] trained at 5000
epochs. The colorbars for the predictions are shown in Fig. 8a, and they are
kept the same for Figs. 8b and 8c, while the colorbars of the absolute error
are all capped between 0 and 0.005. Clearly, we observe that the DeepOKAN
predicts with much higher accuracy the queried field, as evidenced by both the
prediction and the absolute error contours comparison. Finally, we note that
for our comparison, we selected the DeepONet, which had roughly the same
number of parameters as the DeepOKAN. However, similar conclusions can be
drawn for all the other variations we investigated.

##  4 Conclusions, limitations, and future work

Neural operators are machine learning models that approximate the solutions of
partial differential equations by learning the mapping between function
spaces. They provide an efficient and accurate tool for solving complex, high-
dimensional problems in physics and engineering. One of the most popular
neural operators is the DeepONet. The DeepONet uses MLP networks in the branch
and trunk, where the MLP is based on interleaving affine transformations and
nonlinear activation functions. MLPs treat the linear transformations and
activation functions separately. This paper proposes a new neural operator
based on KANs called DeepOKAN. The proposed operator uses KANs in the branch
and trunk components, where KANs have learnable activation functions. One may
consider different bases for these activation functions. The original work by
Liu et al. [29] used B-splines. This paper uses Gaussian RBFs as a basis for
the KANs, appearing in both the branch and trunk. The classical data-driven
DeepONet framework is effective; however, the proposed DeepOKAN outperforms
it. Specifically, within the context of 2D orthotropic elasticity, we observe
that a DeepOKAN outperforms a DeepONet with the same number of learnable
parameters. We also observe that training DeepOKAN for more epochs yielded a
computational boost that was not observed by the DeepONets. More detailed
analysis is being conducted on this problem and other solid mechanics cases to
explore the differences in training efficiency, accuracy, and expressivity
between different operator variations.

## Acknowledgment

This work was partially supported by the Sand Hazards and Opportunities for
Resilience, Energy, and Sustainability (SHORES) Center, funded by Tamkeen
under the NYUAD Research Institute Award CG013. The authors wish to thank the
NYUAD Center for Research Computing for providing resources, services, and
skilled personnel. This work was partially supported by Sandooq Al Watan
Applied Research and Development (SWARD), funded by Grant No.: SWARD-F22-018.

## Data availability

The data supporting the study’s findings will be available upon paper
acceptance.

## References

  * [1] D. Ammosov, M. Vasilyeva, Online multiscale finite element simulation of thermo-mechanical model with phase change, Computation 11 (4) (2023) 71.
  * [2] Y. Efendiev, T. Y. Hou, Multiscale finite element methods: theory and applications, Vol. 4, Springer Science & Business Media, 2009.
  * [3] M. E. Mobasher, H. Waisman, Dual length scale non-local model to represent damage and transport in porous media, Computer Methods in Applied Mechanics and Engineering 387 (2021) 114154.
  * [4] M. Torzoni, M. Tezzele, S. Mariani, A. Manzoni, K. E. Willcox, A digital twin framework for civil engineering structures, Computer Methods in Applied Mechanics and Engineering 418 (2024) 116584.
  * [5] S. Kushwaha, J. He, D. Abueidda, I. Jasiuk, Designing impact-resistant bio-inspired low-porosity structures using neural networks, Journal of Materials Research and Technology 27 (2023) 767–779.
  * [6] M. Valizadeh, S. J. Wolff, Convolutional neural network applications in additive manufacturing: A review, Advances in Industrial and Manufacturing Engineering 4 (2022) 100072.
  * [7] B. Paermentier, D. Debruyne, R. Talemi, A machine learning based sensitivity analysis of the gtn damage parameters for dynamic fracture propagation in x70 pipeline steel, International Journal of Fracture 227 (1) (2021) 111–132.
  * [8] A. Olivier, M. D. Shields, L. Graham-Brady, Bayesian neural networks for uncertainty quantification in data-driven materials modeling, Computer methods in applied mechanics and engineering 386 (2021) 114079.
  * [9] C. M. Parrott, D. W. Abueidda, K. A. James, Multidisciplinary topology optimization using generative adversarial networks for physics-based design enhancement, Journal of Mechanical Design 145 (6) (2023) 061704.
  * [10] L. Herrmann, S. Kollmannsberger, Deep learning in computational mechanics: a review, Computational Mechanics (2024) 1–51.
  * [11] Z. Li, Neural operator: Learning maps between function spaces, in: 2021 Fall Western Sectional Meeting, AMS, 2021.
  * [12] Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, A. Anandkumar, Fourier neural operator for parametric partial differential equations, arXiv preprint arXiv:2010.08895.
  * [13] Z. Li, D. Z. Huang, B. Liu, A. Anandkumar, Fourier neural operator with learned deformations for pdes on general geometries, Journal of Machine Learning Research 24 (388) (2023) 1–26.
  * [14] L. Lu, P. Jin, G. Pang, Z. Zhang, G. E. Karniadakis, Learning nonlinear operators via deeponet based on the universal approximation theorem of operators, Nature machine intelligence 3 (3) (2021) 218–229.
  * [15] L. Lu, X. Meng, Z. Mao, G. E. Karniadakis, Deepxde: A deep learning library for solving differential equations, SIAM review 63 (1) (2021) 208–228.
  * [16] L. Lu, R. Pestourie, S. G. Johnson, G. Romano, Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport, Physical Review Research 4 (2) (2022) 023210.
  * [17] S. Goswami, M. Yin, Y. Yu, G. E. Karniadakis, A physics-informed variational deeponet for predicting crack path in quasi-brittle materials, Computer Methods in Applied Mechanics and Engineering 391 (2022) 114587.
  * [18] K. Kobayashi, J. Daniell, S. B. Alam, Improved generalization with deep neural operators for engineering systems: Path towards digital twin, Engineering Applications of Artificial Intelligence 131 (2024) 107844.
  * [19] S. Garg, S. Chakraborty, Vb-deeponet: A bayesian operator learning framework for uncertainty quantification, Engineering Applications of Artificial Intelligence 118 (2023) 105685.
  * [20] L. Lu, X. Meng, S. Cai, Z. Mao, S. Goswami, Z. Zhang, G. E. Karniadakis, A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data, Computer Methods in Applied Mechanics and Engineering 393 (2022) 114778.
  * [21] J. He, S. Koric, S. Kushwaha, J. Park, D. Abueidda, I. Jasiuk, Novel deeponet architecture to predict stresses in elastoplastic structures with variable complex geometries and loads, Computer Methods in Applied Mechanics and Engineering 415 (2023) 116277.
  * [22] J. He, S. Kushwaha, J. Park, S. Koric, D. Abueidda, I. Jasiuk, Sequential deep operator networks (s-deeponet) for predicting full-field solutions under time-dependent loads, Engineering Applications of Artificial Intelligence 127 (2024) 107258.
  * [23] J. He, S. Kushwaha, J. Park, S. Koric, D. Abueidda, I. Jasiuk, Predictions of transient vector solution fields with sequential deep operator network, arXiv preprint arXiv:2311.11500.
  * [24] Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, A. Anandkumar, Neural operator: Graph kernel network for partial differential equations, arXiv preprint arXiv:2003.03485.
  * [25] J. He, S. Koric, D. Abueidda, A. Najafi, I. Jasiuk, Geom-deeponet: A point-cloud-based deep operator network for field predictions on 3d parameterized geometries, arXiv preprint arXiv:2403.14788.
  * [26] W. Zhong, H. Meidani, Physics-informed mesh-independent deep compositional operator network, arXiv preprint arXiv:2404.13646.
  * [27] G. Cybenko, Approximation by superpositions of a sigmoidal function, Mathematics of control, signals and systems 2 (4) (1989) 303–314.
  * [28] K. Hornik, M. Stinchcombe, H. White, Multilayer feedforward networks are universal approximators, Neural networks 2 (5) (1989) 359–366.
  * [29] Z. Liu, Y. Wang, S. Vaidya, F. Ruehle, J. Halverson, M. Soljačić, T. Y. Hou, M. Tegmark, Kan: Kolmogorov-arnold networks, arXiv preprint arXiv:2404.19756.
  * [30] N. Kolmogorov, On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables, American Mathematical Society, 1961.
  * [31] V. I. Arnol’d, On the representation of continuous functions of three variables by superpositions of continuous functions of two variables, Matematicheskii Sbornik 90 (1) (1959) 3–74.
  * [32] J. Braun, M. Griebel, On a constructive proof of kolmogorov’s superposition theorem, Constructive approximation 30 (2009) 653–675.
  * [33] D. A. Sprecher, S. Draghici, Space-filling curves and kolmogorov superposition-based neural networks, Neural Networks 15 (1) (2002) 57–67.
  * [34] M. Köppen, On the training of a kolmogorov network, in: Artificial Neural Networks—ICANN 2002: International Conference Madrid, Spain, August 28–30, 2002 Proceedings 12, Springer, 2002, pp. 474–479.
  * [35] J.-N. Lin, R. Unbehauen, On the realization of a Kolmogorov network, Neural Computation 5 (1) (1993) 18–20.
  * [36] M.-J. Lai, Z. Shen, The Kolmogorov superposition theorem can break the curse of dimensionality when approximating high dimensional functions, arXiv preprint arXiv:2112.09963.
  * [37] D. Fakhoury, E. Fakhoury, H. Speleers, Exsplinet: An interpretable and expressive spline-based neural network, Neural Networks 152 (2022) 332–346.
  * [38] R. Genet, H. Inzirillo, Tkan: Temporal kolmogorov-arnold networks, arXiv preprint arXiv:2405.07344.
  * [39] Z. Li, Kolmogorov-arnold networks are radial basis function networks, arXiv preprint arXiv:2405.06721.
  * [40] S. Satapathy, S. Dehuri, A. Jagadev, S. Mishra, Empirical study on the performance of the classifiers in eeg classification, EEG Brain Signal Classification for Epileptic Seizure Disorder Detection (2019) 45–65.