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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

arXiv:2610.06348v1math.NAcs.LGcs.NA

TL;DR

IGA provides exact NURBS geometry and Galerkin error control, while neural PDE solvers may be difficult to optimize and lack error control. IGA-KAN post-processes an IGA solution with local closed-form models and a safeguard; across eight benchmarks, it reduces error at unchanged Galerkin unknowns.

  • Problem

    IGA's accuracy at a given degree requires finer meshes and larger linear systems, while neural PDE solvers can involve non-convex optimization without error control.

    Source: §1 · p. 1

  • Method

    IGA-KAN fits local ridge models to the equation, exact boundary data and IGA solution, blends them, and uses a safeguard to retain the IGA solution where needed.

    Source: §1 · p. 2

  • Results

    Across eight benchmarks, IGA-KAN reduces IGA error at unchanged Galerkin unknowns by 4.2 to 90 in the relative L2 norm and 4.1 to 220 in the H1 norm on reference meshes.

    Source: §1 · p. 2, §6 · p. 25

  • Takeaways & Limitations

    The reported gains are attributed to superconvergent local averages of the Galerkin solution recovered with a kernel chosen by the equation.

    Source: §6 · p. 25

    On coarse meshes or with unresolved wave oscillations, the safeguard can decline corrections, limiting or eliminating the gain.

    Source: §4.2 · p. 13

Abstract

from arXiv · show

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.

1. Introduction

IGA offers exact NURBS geometry and predictable Galerkin error control, while neural PDE solvers can be difficult to train and lack error control. IGA-KAN instead post-processes an IGA solution with local closed-form models and a safeguard, improving accuracy across the reported benchmarks.

  • The paper addresses the gap between neural PDE solvers and classical methods by adding accuracy to the IGA solution rather than replacing it.The benchmarks cover curved and complex NURBS geometries, including a domain fitted to a brain slice segmented from MRI.Source: §1 · pp. 1–3
  • The analysis attributes the gain to superconvergent local averages of the Galerkin solution, recovered by a kernel chosen by the equation.The error bound uses local-model residuals and patch-boundary errors, with overlapping-patch consensus as a computable surrogate.Source: §1 · pp. 2–3
  • A Galerkin solve produces u_h, then local ridge KANs fit the equation, exact boundary data and u_h before blending by IGA hat functions.Fixed inner functions make the fit one batched linear least-squares problem.Source: §1 · p. 2
  • Across eight benchmarks, IGA-KAN reduces IGA error at unchanged Galerkin unknowns by 4.2 to 90 in the relative L2 norm and 4.1 to 220 in the H1 norm on reference meshes.The largest factors are on the disk, whose quadratic solution lies in the local space.Source: §1 · p. 2
  • Its L2 error is 6 to 6 × 10^4 times smaller than the best from-scratch KAN under the fixed training protocol.The reported factors describe that protocol, not the best accuracy such networks can reach.Source: §1 · p. 3
  • Four computable tests decide patch by patch whether to use a local model, while the IGA solution remains the fallback.For elliptic operators, the residual indicator uses a maximum-principle bound.Source: §1 · pp. 2–3

2. Related work

The related work frames IGA-KAN against exact-geometry Galerkin methods, physics-informed and Kolmogorov–Arnold neural solvers, and classical local recovery methods. It positions the contribution as a safeguarded local correction of an IGA solution, grounded in superconvergence and fitted without nonlinear optimization.

2.1. Isogeometric analysis

  • IGA uses the NURBS geometry as the Galerkin trial space, representing the geometry exactly and yielding smooth Cp−1 discrete functions across knot spans.That smoothness makes the Galerkin error amenable to local averaging.Source: §2.1 · p. 3

2.2. Physics-informed neural networks and their failure modes

  • PINNs train on equation residuals, but their optimization can be ill-conditioned, and randomized least-squares alternatives remain stand-alone solvers.The cited randomized approaches use random features and couple their least-squares problems over the whole domain.Source: §2.2 · pp. 3–4

2.3. Kolmogorov–Arnold networks

  • PDE KAN methods mainly replace an MLP inside physics-informed losses or variational forms, using the network as the discretization.The surveyed methods do not retain an IGA solution as a substrate for local correction.Source: §2.3 · p. 4

2.4. Classical solver plus learned correction

  • Earlier learned corrections include FEM-LSSVR, while other correction and surrogate schemes use reference data or reproduce an IGA solution.FEM-LSSVR uses a fixed feature map and one linear least-squares solve with the equation in the rows.Source: §2.4 · p. 4

2.5. Post-processing and superconvergence

  • Local averaging can achieve O(h^2p) interior accuracy for degree-p elements, versus O(h^p+1) for the solution, because the error superconverges in negative norms.For p = 1, the rates coincide; the paper combines recovery with equation and boundary rows and an acceptance filter.Source: §2.5 · p. 4

2.6. Positioning of the present method

  • IGA-KAN keeps the IGA solve and fits local models by linear least squares, with a safeguard that accepts, refits or discards corrections patch by patch.Because its mechanism relies on superconvergent local averages, an L2 gain that grows under refinement requires p ≥2.Source: §2.6 · p. 5

3. The hybrid IGA–KAN method

IGA–KAN starts with a classical Galerkin solve on exact NURBS geometry, then locally fits and blends ridge models while retaining the Galerkin solution wherever a computable safeguard rejects a patch. The fitting is a batched linear least-squares problem, and the post-process applies to several operator and problem classes.

3.1. Model problem and the isogeometric solve

  • The unchanged IGA solve on exact NURBS geometry supplies u_h as the baseline for the post-process.The discrete solution is computed in the push-forward NURBS basis, with Dirichlet data imposed by projection onto the trace space.Source: §3.1 · p. 5

3.2. Local Kolmogorov–Arnold ridge models

  • The local ridge model uses fixed inner functions and Bernstein outer functions; with the paper’s settings it parametrizes P9 rather than extending that polynomial space.The paper reports 120 coefficients for a 55-dimensional space in two dimensions; the fit uses the equation, exact boundary data, and u_h.Source: §3.2 · pp. 6–7

3.3. The fit is one linear least-squares problem

  • With fixed inner functions, the equation, data, and boundary rows are linear in the outer coefficients and are solved together by batched least squares.The closed-form fit uses a minimum-norm solution for the rank-deficient matrix, avoiding an optimizer, learning rate, initialization, or back-propagation through the operator.Source: §3.3 · p. 8

3.4. Blending

  • Degree-one hat functions blend the local models, and setting a rejected patch’s model to u_h makes the fallback exact.The partition of unity also preserves continuity across periodic seams and collapsed sides by identifying the corresponding hats and fitting a shared model.Source: §3.4 · p. 8

3.5. The a posteriori safeguard

  • Four computable tests screen each patch; rejected models are refitted on smaller halos and ultimately replaced by u_h, with acceptance re-checked against the final blend.The tests use correction size, disagreement with neighboring patches, residual, and a lower-degree consistency check; the latter is used where the right-hand side is zero. The residual indicator is calibrated on a maximum-principle bound, not the bound itself.Source: §3.5 · pp. 8–9

3.6. Operators and problem classes

  • The fitting covers second-order elliptic and space–time parabolic operators, frozen or linearized nonlinear terms, and triangularly coupled systems.Fourth-order operators and systems such as elasticity are not examined because the residual test’s maximum-principle basis does not carry over as stated.Source: §3.6 · pp. 9–10

3.7. Cost

  • On reference meshes, the post-process costs 4 to about 500 times the IGA solve; on the quarter annulus, refinement matches its L2 error faster when only L2 accuracy is wanted.For the quarter annulus, IGA on 50 × 50 elements matches the hybrid on 16 × 16 in 0.036 s, versus 1.17 s for the hybrid in that timing run.Source: §3.7 · p. 10

4. Analysis: why a smooth local model can improve the Galerkin solution it is fitted to

The post-process exploits accurate local averages in the Galerkin solution: smooth local models fit to the equation, boundary data, and that solution can discard element-scale oscillations. A computable safeguard screens the models, while error bounds and failure regimes clarify what the screening can—and cannot—establish.

4. Analysis: why a smooth local model can improve the Galerkin solution it is fitted to

  • A weak fit to oscillatory u_h preserves its accurate local averages, while equation rows restore fine structure among smooth functions near those averages.The Galerkin error oscillates at the element scale, whereas local averages are more accurate for degree p ≥2.Source: §4 · p. 10
  • At p=2, hybrid L2 rates are 3.9 and 4.0 on Fisher–Kolmogorov and 3.8 and 3.7 on the quarter annulus, versus IGA’s 3.0.At p=3, the hybrid does not converge faster than IGA on these mesh pairs; its gain is a factor of 9.7 to 39.Source: §4 · p. 11

4.1. Error bounds for the blended solution

  • Nonnegative hat functions blend local models so the result is pointwise no worse than the least accurate model meeting there; local error bounds depend on residual and patch-boundary error.The local bound follows from a maximum-principle argument with an explicit quadratic barrier.Source: §4.1 · pp. 11–12
  • When rows use an approximated right-hand side, the bound includes its discrepancy from the true right-hand side, which depends on u_h error and is not computable.This qualification applies to frozen or linearized nonlinear terms, semi-discrete time derivatives, and super-resolved second fields.Source: §4.1 · p. 12
  • The safeguard is a computable filter, not a guarantee that the error bound is small: it uses RMS residuals and consensus in place of supremum residuals and unknown interior boundary error.The consensus test acts as a surrogate because overlapping models estimate the same solution.Source: §4.1 · p. 13

4.2. When the gain vanishes

  • Gain can vanish for phase-unresolved waves, coarse patches, or singular solutions; the benchmarks contain no singularities, so behavior there is unclaimed.For unresolved waves the equation residual may not reveal phase error; at coarse resolution the safeguard can return u_h.Source: §4.2 · p. 13
  • Across the reported benchmark meshes, including 24 three-dimensional runs, the safeguarded hybrid did not increase IGA error beyond round-off.The 24 runs used local degrees 6, 8, and 10.Source: §4.2 · p. 13

5. Numerical experiments

The experiments compare the IGA–KAN hybrid with classical IGA and physics-informed KAN variants across eight benchmark problems. The reported protocol defines the comparison scope, training budgets, and error measures, while noting that network and Galerkin sizes are not matched.

5. Numerical experiments

  • Eight benchmarks cover curved-domain Poisson problems, complex-geometry Laplace, Bratu, Fisher–Kolmogorov, high-frequency heat, and a stiff coupled system.The Fisher–Kolmogorov problem is also posed on a brain-slice domain fitted from magnetic resonance images; five benchmarks come from published work.Source: §5 · p. 13
  • The hybrid post-processes the IGA solution, whereas KAN and KINN solve from scratch under a fixed physics-informed training protocol.Where both networks are trained, they share widths, grid, loss, optimizer, and budget, differing in the layer; KINN has up to 10% more parameters.Source: §5 · p. 14
  • Network and Galerkin sizes are not matched; hybrid-to-IGA comparisons instead use the same number of Galerkin unknowns as the input solve.The post-process adds 120 local coefficients per knot vertex in two dimensions and 405 in three dimensions.Source: §5 · p. 14
  • Tables report relative L2 and H1 errors evaluated by Gauss quadrature on the exact geometry; figure error maps show pointwise absolute error on separate color bars.The maximum at a figure color-bar endpoint is not comparable with the table’s relative or domain-averaged errors.Source: §5 · p. 14

5.1. Poisson equation

Across the disk, thick-ring and quarter-annulus Poisson benchmarks, IGA–KAN improves on IGA; reported comparisons include a 167-fold reduction in disk source error and large L2 and H1 gains on the thick ring.

5.1.1. Quarter annulus

  • On the quarter annulus, IGA–KAN lowers IGA error by factors of 19 in L2 and 90 in H1, and is more than two orders of magnitude more accurate than both networks.Figure 4 shows the remaining hybrid error concentrated mostly next to the outer arc.Source: §5.1.1 · p. 15, Figure 4 · p. 15, Table 1 · p. 15

5.1.2. Unit disk: forward and inverse solution

  • At the disk observation, IGA–KAN’s inferred-source error is 167 times smaller than IGA’s; across 20 random observations, the median source-error ratio is 51.The forward error remains below IGA’s on every tested mesh at both degrees, and the inverse network result ends between IGA and IGA–KAN.Source: §5.1.2 · p. 16

5.1.3. Thick ring: a three-dimensional problem

  • On the reference thick-ring mesh, IGA–KAN reduces IGA error by factors of 29 in L2 and 92 in H1, and is over three orders more accurate than the KAN.For the polynomial case, local degree 8 gains persist across eight meshes; for a non-polynomial solution, gains on the reference mesh are 3.7 in L2 and 5.8 in H1.Source: §5.1.3 · pp. 17–18, Figure 8 · p. 18, Table 3 · p. 18

5.2. Laplace equation on a complex geometry

On the five-petal flower, IGA–KAN improves on IGA and the published KINN and PG-KINN results in KINN’s discrete error measure. The safeguard accepts most patches, while the remaining hybrid error is concentrated along a few radial lines.

5.2. Laplace equation on a complex geometry

  • On the reference flower mesh, IGA–KAN reduces IGA error by a factor of 4.2 and is 95 times more accurate than published KINN and PG-KINN results in KINN’s measure.Table 4’s KINN measure is the only measure in which the published neural-method values exist, so only that column compares all methods.Source: §5.2 · p. 18, Table 4 · p. 19
  • With the p-consistency test active, the safeguard accepts 904 of 1377 patches; without it, 1296 are accepted and the L2 and H1 gains would be 14 and 44.The reported gain with the test is 4.2 in L2.Source: §5.2 · pp. 18–19
  • The KAN error spreads across the flower, IGA error peaks at petal tips along radial element lines, and the remaining hybrid error lies along a few radial lines.The hybrid’s relative L2 error is 4.80 × 10^-5; its maximum pointwise absolute error is 1.1 × 10^-3.Source: §5.2 · p. 19, Figure 9 · p. 19

5.3. Nonlinear Bratu equation

The nonlinear Bratu equation is tested on the unit square with λ = 1. The hybrid post-process substantially reduces the IGA error on the reference mesh.

5.3. Nonlinear Bratu equation

  • The benchmark is the standard nonlinear Bratu equation of isogeometric analysis, posed as −∆u = λe^u on the unit square with exact Dirichlet data.The test uses λ = 1 and θ on the lower branch.Source: §5.3 · p. 19
  • 18× in L2 and 84× in H1: the hybrid reduces the IGA error on the reference mesh.KINN’s layer remains three orders of magnitude behind the hybrid in L2.Source: §5.3 · p. 19
  • The IGA solve uses Picard iteration, while the hybrid keeps its local fit linear by freezing the source at λe^uh.The solution depends only on x; the hybrid’s error lies mainly in patches next to x = 0 and x = 1.Source: §5.3 · p. 19

5.4. Time-dependent problems

Across three time-dependent problem settings, IGA–KAN improves on IGA at the reference mesh, including on an MRI-fitted brain domain. The gains depend on the problem and mesh, and the reported short-horizon Fisher–Kolmogorov test assesses spatial accuracy rather than long-time error accumulation.

5.4.1. Fisher–Kolmogorov equation

  • The Fisher–Kolmogorov test uses a short horizon over which the exact solution decays by only 0.1%, so it verifies spatial accuracy at T rather than long-time error accumulation.IGA advances the problem with Crank–Nicolson and Newton’s method; the hybrid linearizes the reaction term at the classical solution.Source: §5.4.1 · p. 20
  • 27× in L2 and 148× in H1: IGA–KAN reduces IGA error on the Fisher–Kolmogorov unit-square reference mesh.Across meshes, the gains are 11–54× in L2 and 15–265× in H1; the safeguard accepts all patches but one.Source: §5.4.1 · p. 20
  • 11× in L2 and 13.5× in H1: IGA–KAN reduces IGA error on the MRI-slice reference mesh.Across the tested meshes, gains are 7.7–25× in L2 and 13.5–23× in H1; the safeguard rejects only 13 of 11,488 patches.Source: §5.4.1 · p. 21

5.4.2. High-frequency heat equation

  • 15.5× in L2 and 32.4× in H1: IGA–KAN reduces IGA error for the F = 50 high-frequency heat equation on the reference mesh.The gains appear once the wave is resolved; on meshes with up to about three elements per wavelength, the error maps coincide.Source: §5.4.2 · p. 23

5.4.3. Coupled advection–diffusion–reaction system

  • 5× in L2 and 12–27× in H1: IGA–KAN reduces error for both fields of the coupled system on the reference mesh.On the coarsest mesh, the safeguard accepts no patch and the hybrid coincides with IGA; finer tested meshes show gains in both norms.Source: §5.4.3 · p. 24

6. Conclusions and future work

IGA–KAN post-processes a classical IGA solution with local models and a patchwise safeguard. Across the benchmarks, it reduces error without increasing Galerkin unknowns, and its gains are attributed to superconvergent local averages.

6. Conclusions and future work

  • The post-process fits local Kolmogorov–Arnold ridge models to the equation, boundary data, and IGA solution in one batched linear least-squares problem.A patchwise safeguard decides whether each local model may replace IGA; otherwise, IGA is retained.Source: §6 · p. 25
  • Across eight benchmarks, IGA–KAN reduces IGA error by factors of 4.2–90 in L2 and 4.1–220 in H1, with unchanged Galerkin unknowns.For the six benchmarks with non-polynomial solutions, the ranges are 4.2–27 in L2 and 4.1–148 in H1.Source: §6 · p. 25
  • The analysis attributes the gains to recovery of superconvergent local averages, and the safeguard did not worsen IGA error beyond round-off on any benchmark mesh.The error bound uses local residuals and a computable surrogate for patch-boundary error.Source: §6 · p. 25
  • Open directions include coarse-mesh gains, non-smooth solutions, higher-order problems, sharper guarantees, and noisy inverse data.On coarse meshes the safeguard can return IGA unchanged; the hybrid’s advantage decreases when observation noise approaches discretization error.Source: §6 · p. 25

Code and data availability

The study’s code and result files are openly available, with instructions for obtaining the MRI2FEM brain slice used in the paper.

Code and data availability

  • The study’s code and result files are openly available in the project repository.The repository includes the Python library, benchmark notebooks, table- and figure-generation scripts, and result files.Source: Code and data availability · p. 26
  • The repository includes logs and trained network weights for the flower, Fisher–Kolmogorov, heat, and coupled-system benchmarks.Result files include CSV, JSON, and NPZ formats.Source: Code and data availability · p. 26
  • The brain slice is from the public MRI2FEM dataset and is not redistributed; the repository provides instructions to obtain it.The cited dataset is version 2.0 and licensed CC BY 4.0.Source: Code and data availability · p. 26

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