Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equations

Bongseok Kim; Jiahao Zhang; Guang Lin

Figure 1: Comparison between the Kolmogorov–Arnold Network (KAN) and a conventional fully connected DNN. (a) The KAN architecture employs functional mappings on each edge, enabling richer nonlinear representations and adaptive feature extraction. (b) The DNN architecture is composed of fully connected layers with activation functions.

Figure 2: Evolutionary Kolmogorov–Arnold network with Gaussian RBFs : Strong-form formulation. At each time step, the parameters

{W}(t)

evolve by minimizing the PDE residual functional

\displaystyle\mathcal{J}(\gamma)=\tfrac{1}{2}\!\int_{\Omega}\!\big\|\tfrac{\partial\hat{u}}{\partial{W}}\gamma+\mathcal{N}(\hat{u})\big\|_{2}^{2}\,\mathrm{d}\mathbf{x}

, where the optimal update direction satisfies

\displaystyle\frac{\partial{W}}{\partial t}=\arg\min_{\gamma}\mathcal{J}(\gamma)

. After obtaining

\gamma_{\text{opt}}

, the network parameters are explicitly updated in time using the forward Euler scheme

\displaystyle{W}(t_{n+1})={W}(t_{n})+\gamma_{\text{opt}}\Delta t

, yielding a discrete-time parameter evolution associated with the strong-form PDE residual minimization.

Figure 3: Weak-form evolutionary Kolmogorov–Arnold network. At each time step, the parameters

W(t)

are updated by minimizing the weak–form residual functional

\mathcal{J}(\gamma)=\tfrac{1}{2}\sum_{k=1}^{K}\!\left(\int_{\Omega}\big(\tfrac{\partial\hat{u}}{\partial W}\gamma+\mathcal{N}(\hat{u})\big)v_{k}(\mathbf{x})\,\mathrm{d}\mathbf{x}\right)^{2}

, where

\{v_{k}\}_{k=1}^{K}

denote the test functions. The optimal update direction is given by

\gamma_{\text{opt}}=\arg\min_{\gamma}\mathcal{J}(\gamma)

, and the network parameters are advanced in time via the forward Euler step

W(t_{n+1})=W(t_{n})+\gamma_{\text{opt}}\Delta t

, resulting in a discrete-time parameter evolution associated with the weak-form residual minimization.

	Strong form	Weak form
System to solve	$J^{T\,(P\times N_{\hat{u}})}J^{(N_{\hat{u}}\times P)}\gamma_{\text{opt}}=-J^{T\,(P\times N_{\hat{u}})}N^{(N_{\hat{u}}\times 1)}$	$J^{T\,(P\times K)}J^{(K\times P)}\gamma_{\text{opt}}=-J^{T\,(P\times K)}N^{(K\times 1)}$
Data–system coupling	Coupled : system size scales with $N_{\hat{u}}$	Decoupled : system size fixed by $K$
$J$ evaluation	$\mathcal{O}(N_{\hat{u}}\,P\,C_{\mathrm{AD}})$	$\mathcal{O}(N_{x}\,P\,C_{\mathrm{AD}}+KN_{x}P)$
$N$ evaluation	$\mathcal{O}(N_{\hat{u}}\,C_{\mathrm{PDE}}^{\mathrm{SF}})$	$\mathcal{O}(N_{x}\,C_{\mathrm{PDE}}^{\mathrm{WF}}+KN_{x})$
Least-squares system formation	$\mathcal{O}(N_{\hat{u}}P^{2})$	$\mathcal{O}(KP^{2})$
Total cumulative cost	$\mathcal{O}\!\left(M\left(N_{\hat{u}}P^{2}+N_{\hat{u}}PC_{\mathrm{AD}}+N_{\hat{u}}C_{\mathrm{PDE}}^{\mathrm{SF}}\right)\right)$	$\mathcal{O}\!\left(M\left(KP^{2}+N_{x}PC_{\mathrm{AD}}+N_{x}C_{\mathrm{PDE}}^{\mathrm{WF}}\right.\right.$ $\left.\left.+KN_{x}P+KN_{x}\right)\right)$

Table 1: Computational complexity comparison of strong-form and weak-form evolutionary solvers over

M

time steps. Here

J=\partial\hat{u}/\partial W

and

N=\mathcal{N}(\hat{u})

denote the parameter Jacobian and PDE residual.

N_{\hat{u}}

and

N_{x}

are the numbers of collocation and quadrature points,

K

the number of test functions, and

P

the number of trainable parameters.

C_{\mathrm{AD}}

denotes parameter automatic differentiation cost, while

C_{\mathrm{PDE}}

denotes spatial differentiation cost for residual evaluation. Due to integration by parts,

C_{\mathrm{PDE}}^{\mathrm{WF}}

is lower than

C_{\mathrm{PDE}}^{\mathrm{SF}}

.

	EvoKAN-WF	EvoKAN-SF	PINN-SF
Hidden layers	[3, 3, 3, 3]	[3, 3, 3, 3]	[15, 15, 15]
Activation functions	Gaussian RBFs/SiLU	Gaussian RBFs/SiLU	Tanh
Grid points number of activation functions	4	4	-
Number of trainable parameters	162	195	526
Optimizer	Adam	Adam	Adam/L-BFGS-B
Timestep	1e-07	1e-07	-

Table 2: Training configuration for the 1D Allen-Cahn equation (Eq. ( 41 )).

Figure 4: Comparison of predicted solutions for the one-dimensional Allen–Cahn equation (Eq. ( 41 )) at

t=2\times 10^{-5}

, trained with 100 data points. Both EvoKAN–WF and EvoKAN–SF closely match the ground-truth solution at this time, whereas the vanilla PINN does not reproduce the correct profile.

Figure 5: Comparison of predicted solutions for the one-dimensional Allen–Cahn equation (Eq. ( 41 )) at

t=5\times 10^{-5}

, trained with 100 data points. As the solution profile becomes steeper, EvoKAN–WF maintains good accuracy, whereas EvoKAN–SF shows small deviations near the center and boundaries, and the vanilla PINN fails to recover the correct solution.

Figure 6: Strong-form (EvoKAN-SF) versus weak-form (EvoKAN-WF) evolutionary KAN solvers for the 1D Allen–Cahn equation (Eq. ( 41 )) with respect to the number of collocation points: (a)

L_{2}

relative error, (b) condition number, and (c) wall-clock time. The weak-form approach achieves lower errors and substantially better conditioning, while avoiding the rapid growth in computational cost observed in the strong-form solver.

	$L_{2}$ relative error		Wall clock time
Number of collocation points	EvoKAN-SF	EvoKAN-WF	EvoKAN-SF [s]	EvoKAN-WF [s]	Speedup
64	4.2814e-02	7.1881e-03	7.4396 †	11.8412	0.63
100	2.6065e-02	7.1722e-03	8.5188 †	12.0379	0.71
1,000	8.3525e-03	7.1633e-03	16.1555	12.0761	1.34
10,000	8.9389e-03	7.1644e-03	33.5028	12.1654	2.76
50,000	9.0494e-03	6.8493e-03	340.6359	12.1727	27.99

Table 3:

L_{2}

relative error and wall clock time for the one-dimensional Allen–Cahn equation (Eq. ( 41 )) with varying the number of collocation points. Speedup is defined as

T_{\text{SF}}/T_{\text{WF}}

. Entries marked with

\dagger

correspond to EvoKAN-SF solutions whose

L_{2}

relative error remains above

\mathcal{O}(10^{-2})

, indicating insufficient accuracy.

	EvoKAN-WF	EvoKAN-SF	PINN-SF
Hidden layers	[16, 16, 16]	[16, 16, 16]	[25, 25, 25]
Activation functions	Gaussian RBFs/SiLU	Gaussian RBFs/SiLU	Tanh
Grid points number of activation functions	5	5	-
Number of trainable parameters	3392	3472	3670
Optimizer	Adam	Adam	Adam/L-BFGS-B
Timestep	1e-03	1e-03	-

Table 4: Training configuration for the 2D Burgers equation (Eq. ( 47 )).

Figure 7: Comparison of two-dimensional Burgers’ equation (Eq. ( 47 )) solutions at

t=0.1

. All models are trained using

900

collocation points. The top row shows the ground-truth FDM solution together with predictions from EvoKAN-WF, EvoKAN-SF, and PINN-SF. The bottom row presents the corresponding absolute error fields with respect to the ground truth. At this early time, EvoKAN-WF yields smaller errors than the other models while providing comparable representations of the overall flow structure.

Figure 8: Comparison of two-dimensional Burgers’ equation (Eq. ( 47 )) solutions at

t=0.4

. All models are trained using

900

collocation points. The top row shows the ground-truth FDM solution together with predictions from EvoKAN-WF, EvoKAN-SF, and PINN-SF. The bottom row presents the corresponding absolute error fields with respect to the ground truth. Despite the limited number of collocation samples, EvoKAN-WF provides more accurate predictions than the other models and better captures the flow structure across the domain.

Figure 9: Strong-form (EvoKAN-SF) versus weak-form (EvoKAN-WF) evolutionary KAN solvers for the two-dimensional Burgers’ equation (Eq. ( 47 )) with respect to the number of collocation points: (a)

L_{2}

relative error, (b) condition number, and (c) wall-clock time. The weak-form approach achieves lower errors and improved numerical conditioning, while avoiding the rapid growth in computational cost observed in the strong-form solver.

	$L_{2}$ relative error		Wall clock time
Number of collocation points	EvoKAN-SF	EvoKAN-WF	EvoKAN-SF [s]	EvoKAN-WF [s]	Speedup
900	5.67383e-02	2.27452e-02	10.7565 †	25.3241	0.42
1,600	4.01466e-02	2.25431e-02	28.1469	26.4252	1.07
2,500	4.27184e-02	2.17294e-02	57.7473	26.1492	2.21
10,000	2.53424e-02	2.15498e-02	617.0453	26.1723	23.58
40,000	2.17728e-02	2.16729e-02	2139.5068	29.4292	72.69

Table 5:

L_{2}

relative error and wall clock time for the two-dimensional Burgers’ equation (Eq. ( 47 )) with varying the number of collocation points. Speedup is defined as

T_{\text{SF}}/T_{\text{WF}}

. Entries marked with

\dagger

correspond to EvoKAN-SF solutions whose

L_{2}

relative error exceeds

5\%

, indicating insufficient accuracy.

	EvoKAN-WF	EvoKAN-SF	PINN-SF
Hidden layers	[4, 4, 4]	[4, 4, 4]	[15, 15, 15]
Activation functions	Gaussian RBFs/SiLU	Gaussian RBFs/SiLU	Tanh
Grid points number of activation functions	8	5	-
Number of trainable parameters	388	388	526
Optimizer	Adam	Adam	Adam/L-BFGS-B
Timestep	1e-03	1e-03	-

Table 6: Training configuration for the 2D Heat equation with nonlinear forcing term (Eq. ( 51 )).

Figure 10: Comparison of two-dimensional heat equation (Eq. ( 51 )) solutions at

t=0.20

. All models are trained using

400

collocation points. The top row shows the ground-truth solution together with predictions from EvoKAN-WF, EvoKAN-SF, and PINN-SF. The bottom row presents the corresponding absolute error fields with respect to the ground truth. The weak-form solver (EvoKAN-WF) exhibits smoother error distributions and reduced boundary artifacts compared to the strong-form approaches under the same training budget.

Figure 11: Comparison of two-dimensional heat equation solutions at

t=0.50

. All models are trained using

400

collocation points. The top row presents the ground-truth solution along with predictions obtained by EvoKAN-WF, EvoKAN-SF, and PINN-SF. The bottom row displays the corresponding absolute error fields relative to the ground truth. Even at this later time, the weak-form solver (EvoKAN-WF) maintains higher accuracy, exhibiting consistently lower errors and smoother spatial error patterns than the strong-form approaches under the same training budget.

Figure 12: Strong-form (EvoKAN-SF) versus weak-form (EvoKAN-WF) evolutionary KAN solvers for the two-dimensional heat equation with respect to the number of collocation points: (a)

L_{2}

relative error, (b) condition number, and (c) wall-clock time. The weak-form approach achieves lower errors and substantially better conditioning, while avoiding the rapid growth in computational cost observed in the strong-form solver.

Figure 13: Gradient error on each boundary for the 2D heat equation (Eq. ( 51 )). The weak form evolutionary KAN progressively reduces the gradient error as the time evolution proceeds and the error approaches zero along all boundaries. The weak formulation provides stable treatment of the Neumann boundary condition and enables monotonic error reduction during network updates. The strong-form evolutionary network enforces a constant-derivative boundary condition, making convergence of the boundary error to zero difficult, while the strong-form PINN exhibits oscillatory boundary errors during the time evolution.

	$L_{2}$ relative error		Wall-clock time
Number of collocation points	EvoKAN-SF	EvoKAN-WF	EvoKAN-SF [s]	EvoKAN-WF [s]	Speedup
400	4.6195e-02	1.7171e-02	24.1326	22.8757	1.06
900	3.0303e-02	1.2976e-02	37.9921	25.1838	1.51
1,600	4.5767e-02	1.5302e-02	74.4028	26.6005	2.80
10,000	1.7737e-02	1.4792e-02	142.7417	31.2263	4.57
40,000	1.5728e-02	1.2892e-02	1111.4212	54.5341	20.39

Table 7:

L_{2}

relative error and wall-clock time for the two-dimensional heat equation (Eq. ( 51 )) with varying numbers of collocation points. Speedup is defined as

T_{\text{SF}}/T_{\text{WF}}

.

	EvoKAN-WF	EvoKAN-SF	PINN-SF
Hidden layers	[7, 7, 7]	[7, 7, 7]	[15, 15, 15]
Activation functions	Gaussian RBFs/SiLU	Gaussian RBFs/SiLU	Tanh
Grid points number of activation functions	5	5	-
Number of trainable parameters	770	770	526
Optimizer	Adam	Adam	Adam/L-BFGS-B
Timestep	1e-03	1e-03	-

Table 8: Training configuration for the 2D porous medium equation with drift (Eq. ( 57 )).

Figure 14: Solution field comparison for the two-dimensional PME with drift under periodic boundary conditions (Eq. ( 57 )). The solution fields at

t=0

(top row) and

t=0.4

(bottom row) are shown for EvoKAN-WF, EvoKAN-SF, and PINN, respectively. All models are trained using

N_{c}=3600

collocation points, with a uniform time step

\Delta t=10^{-2}

up to the final time

T=0.4

. The color map represents the spatial distribution of the solution

u(x_{1},x_{2},t)

, illustrating the nonlinear diffusion and drift-induced transport dynamics.

Figure 15: Strong-form (EvoKAN-SF) versus weak-form (EvoKAN-WF) evolutionary KAN solvers for the two-dimensional porous medium equation (Eq. ( 57 )) with respect to the number of collocation points: (a) condition number and (b) wall-clock time. The weak-form approach achieves substantially better conditioning while avoiding the rapid growth in computational cost observed in the strong-form solver.

Figure 16: Energy evolution of strong-form (EvoKAN-SF) and weak-form (EvoKAN-WF) evolutionary KAN solvers for the two-dimensional porous medium equation under different numbers of collocation points. For EvoKAN-SF, monotone energy decay is observed only when a sufficiently large number of collocation points is used. At lower resolutions, EvoKAN-SF exhibits energy increase or incomplete decay over time. In contrast, EvoKAN-WF shows consistent energy decay across all tested resolutions.

	Approximation validity		Wall clock time
Number of collocation points	EvoKAN-SF	EvoKAN-WF	EvoKAN-SF [s]	EvoKAN-WF [s]	Speedup
400	$\times$	$\bigcirc$	21.0145	20.6104	1.02
1,600	$\times$	$\bigcirc$	86.8148	24.9313	3.48
3,600	$\bigcirc$	$\bigcirc$	113.3219	26.1163	4.34
10,000	$\bigcirc$	$\bigcirc$	580.5136	31.6187	18.37
40,000	$\bigcirc$	$\bigcirc$	3599.2407	49.2156	73.12

Table 9: Solution validity and wall clock time for the 2D PME (Eq. ( 57 )) with varying the number of collocation points. The solution validity indicator reports whether the numerical solution exhibits a dissipative energy evolution consistent with the governing equation. Entries marked with

\times

correspond to cases where the computed energy shows non-decreasing behavior or incomplete decay over the simulated time interval, indicating that the solution does not provide an admissible approximation under the given discretization. Speedup is defined as

T_{\text{SF}}/T_{\text{WF}}

.

Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equations

Abstract

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