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
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 cA . (b) A stand-alone Kolmogorov–Arnold network of the KINN type: one global uθ 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 aq and outer functions Φq 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: uh enters the data rows of every patch, and a patch that fails the tests returns Kv:=uh (red).
Figure 2: Left: the knot-vertex patches on the physical mesh, that is, the support of one hat function Nv ( 2×2 elements) and its vertex patch ωv , 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 zq=aq⋅(x−xv)/ρv and learnable degree- dloc 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 ωv per knot vertex, the support of the hat Nv enlarged by a halo of ℓ 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 ℓ=0 , set to Kv:=uh . Violet: test (a) is re-checked against the final blend.
Figure 4: Poisson equation on the quarter annulus, p=2 on 16×16 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∗∣ of that panel; the relative errors eL2 and eH1 are given in Table 1 .
method
size
eL2
eH1
time [s]
KAN
1300 parameters
1.22×10−3
4.75×10−3
49
KINN
1430 parameters
1.50×10−3
4.77×10−3
538
IGA
256 unknowns
6.57×10−5
1.51×10−3
0.006
IGA–KAN
256 unknowns
3.46×10−6
1.67×10−5
1.18
gain of IGA–KAN over IGA
19.0×
90.0×
gain of IGA–KAN over KINN
434×
285×
Table 1: Poisson equation on the quarter annulus: comparison of the methods on the reference mesh ( p=2 , 16×16 ).
Figure 5: Unit disk, forward problem, p=2 on 3×3 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∗∣ of that panel; the relative errors eL2 and eH1 are given in Table 2 .
Figure 6: Unit disk: (a) relative L2 error eL2 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 eL2 ); (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
eL2 (forward)
eH1 (forward)
source error (inverse)
time [s]
PI-GGN [ 24 ]
25 nodes
5×10−4 (nodal)
—
<1%
—
KAN
1300 parameters
1.12×10−4
5.02×10−4
—
105
KINN, forward
1430 parameters
9.75×10−5
4.17×10−4
—
196
KINN, penalty data
1430 parameters
—
—
1.55×10−4
200
KINN, hard data
1430 parameters
—
—
5.39×10−5
205
IGA
25 control points
1.43×10−3
7.07×10−3
4.14×10−3
0.006
Table 2: Unit disk: comparison of the methods on 25 control points ( p=2 , 3×3 ), the size of PI-GGN’s graph. Forward: relative errors of u for f=1 . Inverse: relative error of the identified source ∣f^−f∗∣/f∗ . 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 163 elements, on two interior sections: the mid-height section z=21 (a–c) and the diagonal section θ=π/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∗∣ 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 L2 error eL2 against the number of unknowns for IGA (dashed, open markers) and IGA–KAN (solid, filled markers) at p=1,2,3 , local degree dloc=8 ; (b) L2 gain of IGA–KAN over IGA against the local degree dloc , one line per degree p and mesh; (c) L2 gain against the fraction of patches the safeguard accepts, for all 24 runs. Color: degree p ; marker: mesh.
method
size
eL2
eH1
time [s]
KAN [3,10,10,1]
1400 parameters
5.78×10−3
1.82×10−2
14 299
IGA
4096 unknowns
3.24×10−5
7.16×10−4
6.3
IGA–KAN
4096 unknowns
1.13×10−6
7.76×10−6
80.9
gain of IGA–KAN over IGA
28.7×
92.1×
gain of IGA–KAN over KAN
5100×
2340×
Table 3: Thick ring: comparison of the methods on the reference mesh ( p=2 , 16×16×16 ).
Figure 9: Flower, p=2 on 80×16 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∗∣ of that panel, 1.1×10−3 for IGA–KAN; Table 4 gives the relative errors, eL2=4.80×10−5 for IGA–KAN.
our measures
KINN’s measure
method
size
eL2
eH1
∥u−u∗∥2/∥u∗∥2
time [s]
published
KINN-BINN (KAN) [ 64 ]
975 parameters
—
—
4.73×10−3
—
BINN (MLP) [ 64 ]
2911 parameters
—
—
2.06×10−4
—
PG-KINN [ 54 ]
[2,5,5,5,1]
—
—
4.73×10−3
—
this work
Table 4: Flower: comparison of the methods on the reference mesh ( p=2 , 80×16 elements, 1361 unknowns). Columns 3 and 4 are the relative errors eL2 and eH1 of ( 15 ), computed by Gauss quadrature over the whole flower. Column 5 is KINN’s error measure, ∥u−u∗∥2/∥u∗∥2 with discrete 2 -norms over KINN’s 52264 evaluation points (a uniform 300×300 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, λ=1 , p=2 on 16×16 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∗∣ of that panel; the relative errors eL2 and eH1 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
eL2
eH1
time [s]
KAN
1300 parameters
6.94×10−4
3.50×10−3
690
KINN
1430 parameters
2.95×10−4
9.96×10−4
1618
IGA
256 unknowns
4.74×10−6
1.48×10−4
0.22
IGA–KAN
256 unknowns
2.60×10−7
1.75×10−6
1.87
gain of IGA–KAN over IGA
18.3×
84.3×
gain of IGA–KAN over KINN
1140×
568×
Table 5: Bratu equation, λ=1 : comparison of the methods on the reference mesh ( p=2 , 16×16 , 256 unknowns). IGA takes 11 Picard iterations.
Figure 11: Fisher–Kolmogorov equation on the unit square at t=T , p=2 on 16×16 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 L2 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 T1 -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 32×8 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 L2 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
eL2
eH1
time [s]
(a) unit square
KAN [3,10,10,1]
1400 parameters
1.35×10−1
4.41×10−1
1883
KINN
1540 parameters
2.24×10−2
1.51×10−1
3820
IGA
324 coefficients
1.51×10−5
1.06×10−3
0.49
IGA–KAN
324 coefficients
5.57×10−7
7.14×10−6
2.07
gain of IGA–KAN over IGA
27.1×
148×
Table 6: Fisher–Kolmogorov equation: comparison of the methods at t=T on the reference meshes, (a) on the unit square ( p=2 , 16×16 ) and (b) on the brain slice from MRI ( p=2 , 32×8 ). The times of the networks are wall-clock times of single training runs, the median of three seeds, and vary between seeds (KINN: 3069 to 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 on 256×8 elements, each on its own linear color bar, whose upper end is the maximum pointwise absolute error max∣u−u∗∣ of that panel; (f), (g) the relative L2 error eL2 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 L2 errors, the measure of Table 7 ; the color-bar maxima of (b)–(e) are not.
method
size
eL2
eH1
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
8.09×10−3
8.46×10−3
11 440
IGA
2849 coefficients
1.51×10−4
1.46×10−3
0.11
IGA–KAN
2849 coefficients
9.76×10−6
4.51×10−5
36.0
gain of IGA–KAN over IGA
15.5×
32.4×
Table 7: High-frequency heat equation, F=50 : comparison of the methods on the reference mesh ( p=3 , 256×8 space–time elements).
Figure 14: Coupled advection–diffusion–reaction system at t=1 , p=3 on 16×16 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∗∣ or max∣v−v∗∣ , of that panel; the relative errors eL2 and eH1 of both fields are given in Table 8 .
field u
field v
method
size
eL2
eH1
eL2
eH1
time [s]
KAN [3,10,10,2]
1500 parameters
3.66×10−1
4.14×10−1
3.74×10−1
4.24×10−1
4068
KINN
1650 parameters
4.28×10−1
4.96×10−1
4.41×10−1
5.06×10−1
10 233
IGA
361 coefficients
3.19×10−5
2.90×10−4
3.19×10−5
2.90×10−4
0.99
IGA–KAN
361 coefficients
5.92×10−6
1.06×10−5
6.48×10−6
2.36×10−5
9.24
gain of IGA–KAN over IGA
5.39×
27.3×
4.92×
12.3×
Table 8: Coupled advection–diffusion–reaction system: comparison of the methods at t=1 on the reference mesh ( p=3 , 16×16 ).
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.
Amirhossein Sadr, Nima Soltani, Vahideh Moghtadaiee +3
School of Computer Science, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran · Department of Computer Science and Engineering, Shahid Beheshti University, Tehran, Iran · Cyberspace Research Institute, Shahid Beheshti University, Tehran, Iran +1
We study free-boundary problems within a physics-informed framework using Kolmogorov-Arnold network (KAN) approximations. The proposed approach incorporates obstacle constraints, partial differential equation (PDE) inequalities, complementarity conditions, and boundary conditions through residual-based loss functions. We consider a linear elliptic obstacle problem, a nonlinear p-Laplacian obstacle problem, and a time-dependent one-phase Stefan problem. The proposed KAN solver is compared with physics-informed neural network (PINN) and residual-network baselines. Numerical experiments show that KANs achieve low relative L2 and L∞ errors while accurately resolving contact regions and moving interfaces. The results indicate that KAN representations provide an effective alternative for solving free-boundary PDEs.
Tan Phuong Dong Le
Department of Applied Mathematics, University of Waterloo, Canada · Department of Statistics and Actuarial Science, University of Waterloo, Canada
We introduce Geometric Kolmogorov--Arnold Networks (GeoKANs), a family of geometry-aware KAN-type models in which approximation is carried out in learned, geometry-adapted coordinates rather than in fixed Euclidean input coordinates. GeoKAN achieves this by learning a diagonal Riemannian metric that warps the input before basis expansion and feature mixing. The learned metric provides a geometric inductive bias through local length scaling and volume distortion, and in physics-informed settings it also affects the differential structure seen by the model. Within this framework, we develop three main variants, namely GeoKAN-NNMetric, GeoKAN-γ, and LM-KAN. For LM-KAN, we further consider three basis-specific versions, LM-KAN-RBF, LM-KAN-Wav, and LM-KAN-Fourier. These variants allow us to study geometry-aware KAN models both as general function approximators and as surrogates in physics-informed learning. By stretching regions with rapid variation and compressing smoother regions, GeoKAN reallocates representational resolution in a task-dependent manner, allowing the model to place capacity where it is most needed. As a result, GeoKAN is well suited to sharp, stiff, localized, and strongly non-uniform regimes arising in scientific machine learning and differential-equation problems.
Abhijit Sen, Bikram Keshari Parida, Giridas Maiti +2
Department of Physics and Engineering Physics, Tulane University, New Orleans, Louisiana 70118, USA. · Institute of Applied Geosciences, Karlsruhe Institute of Technology, Karlsruhe 76131, Germany.