IGA-KAN: Isogeometric Analysis with Physics-Informed Closed-Form Kolmogorov-Arnold Networks for Forward and Inverse PDEs

Sima Naraghi; Kourosh Parand; Amirhossein Sadr; Dara Rahmati

math.NAOct 5, 2026

IGA-KAN: Isogeometric Analysis with Physics-Informed Closed-Form Kolmogorov-Arnold Networks for Forward and Inverse PDEs

Authors: Sima Naraghi, Kourosh Parand, Amirhossein Sadr, Dara Rahmati

Organizations: Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran · Department of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran · Department of Cognitive Modeling, Institute for Cognitive and Brain Sciences, Shahid Beheshti University, Tehran, Iran · Department of Computer Science and Engineering, Shahid Beheshti University, Tehran, Iran

Abstract

Isogeometric analysis (IGA) solves partial differential equations accurately on exact NURBS geometry, whereas neural solvers are mesh-free but often orders of magnitude less accurate and typically trained by non-convex optimization without error control. We propose IGA-KAN, which uses local Kolmogorov-Arnold networks, fitted in closed form, to improve the IGA solution instead of replacing it. An IGA Galerkin solve produces u_h; on every knot-vertex patch a Kolmogorov-Arnold ridge model is fitted to the strong form of the equation, the exact boundary data and u_h, and the models are blended by IGA hat functions. With fixed inner functions the fit is one batched linear least-squares problem, without optimizer, learning rate or initialization. An a posteriori safeguard, motivated by a maximum-principle bound, decides where local models are used, keeping the IGA solution elsewhere. On eight benchmarks with exact solutions, five from the literature and one also posed on a domain fitted to a brain slice from MRI, the method reduces the error of IGA, at an unchanged number of Galerkin unknowns, by factors of 4.2 to 90 in L^2 and 4.1 to 220 in H^1 on the reference meshes, and its L^2 error is 6 to 6x10^4 times smaller than that of the best Kolmogorov-Arnold network trained from scratch on the same equations with a fixed budget. In an inverse problem it recovers an unknown constant source from one noise-free observation 167 times more accurately than IGA. The gain is attributed to the superconvergence of local averages of the Galerkin solution.

Figures & tables

Figure 1: What the hybrid adds to a Galerkin solve, read across four rows: what each method sees of the domain, what it solves for, how the unknowns are found and what it returns. (a) Classical isogeometric analysis, whose unknowns are the control variables

c_{A}

. (b) A stand-alone Kolmogorov–Arnold network of the KINN type: one global

u_{\theta}

on sampled points, its weights found by a non-convex descent and its boundary data imposed by a penalty. (c) The method of this section: on every knot-vertex patch the ridge model ( 2 ), with fixed directions

\mathbf{a}_{q}

and outer functions

\Phi_{q}

in the Bernstein basis, fitted by one batched linear least-squares solve to the physics, data and boundary rows ( 4 ) and screened by the four tests ( 8 )–( 11 ). The two lanes below show how the hybrid depends on IGA:

u_{h}

enters the data rows of every patch, and a patch that fails the tests returns

K_{v}:=u_{h}

(red).

Figure 2: Left: the knot-vertex patches on the physical mesh, that is, the support of one hat function

N_{v}

(

2\times 2

elements) and its vertex patch

\omega_{v}

, that support enlarged by the halo, with the collocation and data points and the exact boundary points marked. Right: the local model ( 2 ), with fixed linear inner functions

z_{q}=\mathbf{a}_{q}\cdot(\mathbf{x}-\mathbf{x}_{v})/\rho_{v}

and learnable degree-

d_{\mathrm{loc}}

outer B-splines.

Figure 3: The hybrid as five stages (the steps are listed in Algorithm 3.5 ): (1) the isogeometric solve; (2) one fitting patch

\omega_{v}

per knot vertex, the support of the hat

N_{v}

enlarged by a halo of

\ell

element widths; (3) the local model ( 2 ) with fixed directions and Bernstein outer functions ( 3 ), fitted to the three row blocks ( 4 ) by one batched least-squares solve; (4) the four tests ( 8 )–( 11 ); (5) the partition-of-unity blend. Red: a patch failing a test is refitted on a smaller halo and, if it still fails at

\ell=0

, set to

K_{v}:=u_{h}

. Violet: test (a) is re-checked against the final blend.

Figure 4: Poisson equation on the quarter annulus, $p=2$ on $16\times 16$ elements: (a) the exact solution; (b)–(d) the pointwise absolute errors of the KAN, of IGA and of IGA–KAN, each on its own color bar. The upper end of each error color bar is the maximum pointwise absolute error $\max|u-u^{\ast}|$ of that panel; the relative errors $e_{L^{2}}$ and $e_{H^{1}}$ are given in Table 1 .

method	size	$e_{L^{2}}$	$e_{H^{1}}$	time [s]
KAN	1300 parameters	$1.22\times 10^{-3}$	$4.75\times 10^{-3}$	49
KINN	1430 parameters	$1.50\times 10^{-3}$	$4.77\times 10^{-3}$	538
IGA	256 unknowns	$6.57\times 10^{-5}$	$1.51\times 10^{-3}$	0.006
IGA–KAN	256 unknowns	$\mathbf{3.46\times 10^{-6}}$	$\mathbf{1.67\times 10^{-5}}$	1.18
gain of IGA–KAN over IGA		$19.0\times$	$90.0\times$
gain of IGA–KAN over KINN		$434\times$	$285\times$

Table 1: Poisson equation on the quarter annulus: comparison of the methods on the reference mesh (

p=2

16\times 16

Figure 5: Unit disk, forward problem, $p=2$ on $3\times 3$ elements ( $25$ control points): (a) the analytical solution with the observation point of the inverse problem; (b) the IGA–KAN solution; (c)–(f) the pointwise errors of IGA, the KAN, KINN and IGA–KAN, each on its own color bar (color map and layout as in PI-GGN’s Figs. 2–3). The upper end of each error color bar is the maximum pointwise absolute error $\max|u-u^{\ast}|$ of that panel; the relative errors $e_{L^{2}}$ and $e_{H^{1}}$ are given in Table 2 .

Figure 6: Unit disk: (a) relative

L^{2}

error

e_{L^{2}}

of the forward problem against the number of control points, for IGA and IGA–KAN at

p=2,3

, KINN and PI-GGN’s published value (a nodal error, as in Table 2 , not

e_{L^{2}}

); (b) relative error of the inferred source during the training of KINN (penalty and hard data), against the one-division results of IGA and IGA–KAN on

25

control points and PI-GGN’s reported bound; (c) relative error of the inferred source against the number of control points, the band marking the two KINN results.

method	size	$e_{L^{2}}$ (forward)	$e_{H^{1}}$ (forward)	source error (inverse)	time [s]
PI-GGN [ 24 ]	25 nodes	$5\times 10^{-4}$ (nodal)	—	$<1\%$	—
KAN	1300 parameters	$1.12\times 10^{-4}$	$5.02\times 10^{-4}$	—	105
KINN, forward	1430 parameters	$9.75\times 10^{-5}$	$4.17\times 10^{-4}$	—	196
KINN, penalty data	1430 parameters	—	—	$1.55\times 10^{-4}$	200
KINN, hard data	1430 parameters	—	—	$5.39\times 10^{-5}$	205
IGA	25 control points	$1.43\times 10^{-3}$	$7.07\times 10^{-3}$	$4.14\times 10^{-3}$	0.006

Table 2: Unit disk: comparison of the methods on

25

control points (

p=2

3\times 3

), the size of PI-GGN’s graph. Forward: relative errors of

u

for

f=1

. Inverse: relative error of the identified source

|\hat{f}-f^{\ast}|/f^{\ast}

. The times of IGA and IGA–KAN are those of the forward problem; identifying the source adds

0.002

s to each.

Figure 7: Thick ring, $p=2$ on $16^{3}$ elements, on two interior sections: the mid-height section $z=\tfrac{1}{2}$ (a–c) and the diagonal section $\theta=\pi/4$ (d–f). (a), (d) The section inside the ring, colored by the exact solution; (b), (c), (e), (f) the pointwise absolute errors $|u-u^{\ast}|$ of IGA and IGA–KAN on that section, the diagonal section drawn in the $(r,z)$ plane. The solution vanishes on four of the six faces, so the errors are shown on interior sections. The upper end of each error color bar is the maximum pointwise absolute error of that panel; the relative errors are given in Table 3 .

Figure 8: Thick ring. (a) Relative

L^{2}

error

e_{L^{2}}

against the number of unknowns for IGA (dashed, open markers) and IGA–KAN (solid, filled markers) at

p=1,2,3

, local degree

d_{\mathrm{loc}}=8

; (b)

L^{2}

gain of IGA–KAN over IGA against the local degree

d_{\mathrm{loc}}

, one line per degree

p

and mesh; (c)

L^{2}

gain against the fraction of patches the safeguard accepts, for all

24

runs. Color: degree

p

; marker: mesh.

method	size	$e_{L^{2}}$	$e_{H^{1}}$	time [s]
KAN $[3,10,10,1]$	1400 parameters	$5.78\times 10^{-3}$	$1.82\times 10^{-2}$	14 299
IGA	4096 unknowns	$3.24\times 10^{-5}$	$7.16\times 10^{-4}$	6.3
IGA–KAN	4096 unknowns	$\mathbf{1.13\times 10^{-6}}$	$\mathbf{7.76\times 10^{-6}}$	80.9
gain of IGA–KAN over IGA		$28.7\times$	$92.1\times$
gain of IGA–KAN over KAN		$5100\times$	$2340\times$

Table 3: Thick ring: comparison of the methods on the reference mesh (

p=2

16\times 16\times 16

Figure 9: Flower, $p=2$ on $80\times 16$ elements: (a) the exact solution; (b)–(d) the pointwise absolute errors of the KAN, of IGA and of IGA–KAN, each on its own color bar (color map as in KINN’s Fig. 25). The upper end of each error color bar is the maximum pointwise absolute error $\max|u-u^{\ast}|$ of that panel, $1.1\times 10^{-3}$ for IGA–KAN; Table 4 gives the relative errors, $e_{L^{2}}=4.80\times 10^{-5}$ for IGA–KAN.

		our measures		KINN’s measure
method	size	$e_{L^{2}}$	$e_{H^{1}}$	$\\|u-u^{\ast}\\|_{2}/\\|u^{\ast}\\|_{2}$	time [s]
published
KINN-BINN (KAN) [ 64 ]	975 parameters	—	—	$4.73\times 10^{-3}$	—
BINN (MLP) [ 64 ]	2911 parameters	—	—	$2.06\times 10^{-4}$	—
PG-KINN [ 54 ]	$[2,5,5,5,1]$	—	—	$4.73\times 10^{-3}$	—
this work

Table 4: Flower: comparison of the methods on the reference mesh (

p=2

80\times 16

elements,

1361

unknowns). Columns 3 and 4 are the relative errors

e_{L^{2}}

and

e_{H^{1}}

of ( 15 ), computed by Gauss quadrature over the whole flower. Column 5 is KINN’s error measure,

\|u-u^{\ast}\|_{2}/\|u^{\ast}\|_{2}

with discrete

2

-norms over KINN’s

52\,264

evaluation points (a uniform

300\times 300

grid restricted to the flower, without a band about

0.03

wide along the boundary); it is the only measure in which the published boundary-integral neural network (BINN; KINN’s Table 6) and PG-KINN (its Table 5) numbers exist, so only this column compares all methods. A dash means not reported .

Figure 10: Bratu equation, $\lambda=1$ , $p=2$ on $16\times 16$ elements: (a) the exact solution; (b), (c) the pointwise absolute errors of IGA and of IGA–KAN, each on its own color bar. The upper end of each error color bar is the maximum pointwise absolute error $\max|u-u^{\ast}|$ of that panel; the relative errors $e_{L^{2}}$ and $e_{H^{1}}$ are given in Table 5 . The network runs of that table stored their errors but not the networks, so they have no panel.

method	size	$e_{L^{2}}$	$e_{H^{1}}$	time [s]
KAN	1300 parameters	$6.94\times 10^{-4}$	$3.50\times 10^{-3}$	690
KINN	1430 parameters	$2.95\times 10^{-4}$	$9.96\times 10^{-4}$	1618
IGA	256 unknowns	$4.74\times 10^{-6}$	$1.48\times 10^{-4}$	0.22
IGA–KAN	256 unknowns	$\mathbf{2.60\times 10^{-7}}$	$\mathbf{1.75\times 10^{-6}}$	1.87
gain of IGA–KAN over IGA		$18.3\times$	$84.3\times$
gain of IGA–KAN over KINN		$1140\times$	$568\times$

Table 5: Bratu equation,

\lambda=1

: comparison of the methods on the reference mesh (

p=2

16\times 16

256

unknowns). IGA takes

11

Picard iterations.

Figure 11: Fisher–Kolmogorov equation on the unit square at $t=T$ , $p=2$ on $16\times 16$ elements: (a) the exact solution; (b)–(e) the pointwise absolute errors of the KAN, of KINN, of IGA and of IGA–KAN, each on its own color bar (networks: the seed with the median $L^{2}$ error). The upper end of each error color bar is the maximum pointwise absolute error of that panel; the relative errors are given in Table 6 a.

Figure 12: Fisher–Kolmogorov equation on a brain slice from MRI, all panels drawn on the $T_{1}$ -weighted slice of the MRI2FEM data set [ 41 ] (subject abby , CC BY 4.0) in the coordinates of the unit square. (a) The slice with its FreeSurfer segmentation. (b) The two spline curves that bound the computational domain and the reference mesh, $p=2$ on $32\times 8$ elements. (c) The exact solution at $t=T$ . (d)–(f) The pointwise absolute errors at $t=T$ of the KAN (the seed with the median $L^{2}$ error), of IGA and of IGA–KAN, each on its own color bar, whose upper end is the maximum pointwise absolute error of that panel; the relative errors are given in Table 6 b.

method	size	$e_{L^{2}}$	$e_{H^{1}}$	time [s]
(a) unit square
KAN $[3,10,10,1]$	1400 parameters	$1.35\times 10^{-1}$	$4.41\times 10^{-1}$	1883
KINN	1540 parameters	$2.24\times 10^{-2}$	$1.51\times 10^{-1}$	3820
IGA	324 coefficients	$1.51\times 10^{-5}$	$1.06\times 10^{-3}$	0.49
IGA–KAN	324 coefficients	$\mathbf{5.57\times 10^{-7}}$	$\mathbf{7.14\times 10^{-6}}$	2.07
gain of IGA–KAN over IGA		$27.1\times$	$148\times$

Table 6: Fisher–Kolmogorov equation: comparison of the methods at

t=T

on the reference meshes, (a) on the unit square (

p=2

16\times 16

) and (b) on the brain slice from MRI (

p=2

32\times 8

). The times of the networks are wall-clock times of single training runs, the median of three seeds, and vary between seeds (KINN:

3069

4970

s).

Figure 13: High-frequency heat equation,

F=50

: (a) the exact solution; (b)–(e) the pointwise absolute errors of KINN’s perceptron, of the KAN built from KINN’s layer, of IGA and of IGA–KAN at

p=3

256\times 8

elements, each on its own linear color bar, whose upper end is the maximum pointwise absolute error

\max|u-u^{\ast}|

of that panel; (f), (g) the relative

L^{2}

error

e_{L^{2}}

of IGA and of IGA–KAN over the number of elements in

x

and the frequency

F

(

p=3

8

elements in

t

), on one shared logarithmic scale. The numbers printed in (f) and (g) are relative

L^{2}

errors, the measure of Table 7 ; the color-bar maxima of (b)–(e) are not.

method	size	$e_{L^{2}}$	$e_{H^{1}}$	time [s]
MLP $[2,30,30,30,30,1]$ (KINN’s)	2911 parameters	$1.00$	$1.00$	844
KAN $[2,5,5,1]$ , grid 100, our layer	4160 parameters	$1.00$	$1.00$	12 177
KINN: the same KAN, KINN’s layer	4200 parameters	$8.09\times 10^{-3}$	$8.46\times 10^{-3}$	11 440
IGA	2849 coefficients	$1.51\times 10^{-4}$	$1.46\times 10^{-3}$	0.11
IGA–KAN	2849 coefficients	$\mathbf{9.76\times 10^{-6}}$	$\mathbf{4.51\times 10^{-5}}$	36.0
gain of IGA–KAN over IGA		$15.5\times$	$32.4\times$

Table 7: High-frequency heat equation,

F=50

: comparison of the methods on the reference mesh (

p=3

256\times 8

space–time elements).

Figure 14: Coupled advection–diffusion–reaction system at $t=1$ , $p=3$ on $16\times 16$ elements; top row $u$ , bottom row $v$ : the exact fields and the pointwise absolute errors of the KAN, of KINN, of IGA and of IGA–KAN, each on its own color bar. The upper end of each error color bar is the maximum pointwise absolute error, $\max|u-u^{\ast}|$ or $\max|v-v^{\ast}|$ , of that panel; the relative errors $e_{L^{2}}$ and $e_{H^{1}}$ of both fields are given in Table 8 .

		field $u$		field $v$
method	size	$e_{L^{2}}$	$e_{H^{1}}$	$e_{L^{2}}$	$e_{H^{1}}$	time [s]
KAN $[3,10,10,2]$	1500 parameters	$3.66\times 10^{-1}$	$4.14\times 10^{-1}$	$3.74\times 10^{-1}$	$4.24\times 10^{-1}$	4068
KINN	1650 parameters	$4.28\times 10^{-1}$	$4.96\times 10^{-1}$	$4.41\times 10^{-1}$	$5.06\times 10^{-1}$	10 233
IGA	361 coefficients	$3.19\times 10^{-5}$	$2.90\times 10^{-4}$	$3.19\times 10^{-5}$	$2.90\times 10^{-4}$	0.99
IGA–KAN	361 coefficients	$\mathbf{5.92\times 10^{-6}}$	$\mathbf{1.06\times 10^{-5}}$	$\mathbf{6.48\times 10^{-6}}$	$\mathbf{2.36\times 10^{-5}}$	9.24
gain of IGA–KAN over IGA		$5.39\times$	$27.3\times$	$4.92\times$	$12.3\times$

Table 8: Coupled advection–diffusion–reaction system: comparison of the methods at

t=1

on the reference mesh (

p=3

16\times 16

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PG-KINN: A Physics-Informed Petrov-Galerkin Kolmogorov-Arnold Network for Solving Forward and Inverse PDEs

Amirhossein Sadr, Nima Soltani, Vahideh Moghtadaiee, Aida Pakniyat+2

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Variational Formulation Partial Differential Equations Kolmogorov-Arnold Networks System Identification

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Physics-informed learning of partial differential equations (PDEs) has been dominated by multilayer perceptrons (MLPs), whose spectral bias and dense parameterization limit both accuracy and interpretability. Kolmogorov Arnold Networks (KANs) mitigate these limitations because their learnable spline activations are structurally aligned with the piecewise-polynomial bases of classical discretizations. However, the way a PDE is cast into a loss functional is as decisive as the choice of approximator: strong-form residual minimization requires high-order derivatives and heavily weighted losses, the energy (Bubnov-Galerkin) form is restricted to self-adjoint operators and, as we show, collapses to a trivial solution for parameter-identification problems, and boundary integral forms require a known fundamental solution. We propose PG-KINN, a physics-informed KAN built on a Petrov-Galerkin formulation in which the trial space is a KAN and the test space is an independent, compactly supported, piecewise-polynomial space evaluated with Gauss-Legendre quadrature. Integration by parts lowers the differentiation order while retaining applicability to general non-self-adjoint, nonlinear, and inverse problems; the localized test functions turn the global residual into a set of element-wise weak residuals with favorable conditioning. On a suite of benchmarks spanning crack singularities, stress concentration, Neo-Hookean hyperelasticity, inverse parameter identification in heterogeneous media, and complex geometries, PG-KINN consistently outperforms legacy MLP baselines and state-of-the-art KAN-based strong/energy/inverse formulations (PIKAN). These results position the Petrov-Galerkin coupling of KAN trial spaces and polynomial test spaces as a robust and accurate route for AI-based computational mechanics.

Abstract

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