Weak-form EvoKAN: Scalable Neural PDE Solver

• Weak-form EvoKAN is an evolutionary neural network method that approximates time-dependent PDE solutions using weak-residual projections for scalable parameter updates.
• It rigorously enforces boundary conditions—Dirichlet, periodic, and Neumann—through trial-space constructions and integration by parts, enhancing stability.
• The approach decouples the update system size from quadrature resolution, yielding significant computational speedups and improved conditioning compared to strong-form methods.

Weak-form Evolutionary Kolmogorov–Arnold Networks, denoted VEvoKAN in the 2026 exposition following Kim et al., are a class of evolutionary neural-network methods for time-dependent partial differential equations in which the PDE solution is represented by a Kolmogorov–Arnold Network (KAN) whose parameters evolve in time according to a weak residual projection rather than repeated retraining (Kim et al., 19 Feb 2026). In this framework, the pointwise residual is projected onto a test space, producing a parameter-update linear system whose size is fixed by the number of trainable parameters and test functions, rather than by the number of quadrature points. The method is presented as a response to limitations of strong-form evolutionary approaches, which can yield ill-conditioned linear systems due to pointwise residual discretization and whose computational cost scales unfavorably with the number of training samples (Kim et al., 19 Feb 2026). The weak-form formulation also supports rigorous treatment of Dirichlet, periodic, and Neumann boundary conditions through boundary-constrained KAN trial spaces and integration by parts. In the closely related energy-dissipative EvoKAN framework, the same evolutionary KAN perspective is coupled with the scalar auxiliary variable (SAV) method to obtain unconditional energy stability for gradient-flow PDEs (Lin et al., 3 Mar 2025).

1. Conceptual position within evolutionary neural PDE solvers

VEvoKAN belongs to the broader family of evolutionary neural networks in which temporal dynamics are captured by parameter evolution rather than by solving a fresh optimization problem at each time step (Kim et al., 19 Feb 2026). The basic ansatz is a parametric network approximation

$\hat u(x;W(t)),$

with time-dependent parameters $W(t)\in\mathbb{R}^P$. Its temporal derivative is written as

$$\frac{\partial \hat u}{\partial t}=\frac{\partial \hat u}{\partial W}\cdot \gamma,\qquad \frac{\partial W}{\partial t}=\gamma.$$

For a general time-dependent PDE on $\Omega\subset\mathbb{R}^d$,

$$\frac{\partial u}{\partial t}(x,t)+\mathcal{N}(u(x,t))=0,\qquad u(x,0)=u_0(x),$$

the strong-form residual is defined by

$$R(x;W,\gamma)\equiv \frac{\partial \hat u}{\partial t}+\mathcal{N}(\hat u) =\frac{\partial \hat u}{\partial W}(x,W)\cdot\gamma+\mathcal{N}(\hat u(x,W)).$$

The weak-form step consists of projecting this residual against test functions $\{v_k\}_{k=1}^K$:

$$\int_\Omega R(x;W,\gamma)\,v_k(x)\,dx \simeq 0,\qquad k=1,\dots,K.$$

This leads to weak sensitivities and residual projections,

$$J_{kj}\equiv \int_\Omega \frac{\partial \hat u}{\partial W_j}(x,W)\,v_k(x)\,dx,\qquad N_k\equiv \int_\Omega \mathcal{N}(\hat u(x,W))\,v_k(x)\,dx,$$

and a least-squares problem whose optimality condition is

$J^TJ\,\gamma=-J^TN.$

The distinguishing claim of the weak formulation is that the size of this normal-equation system is determined by $W(t)\in\mathbb{R}^P$0 and $W(t)\in\mathbb{R}^P$1, and is independent of the number of quadrature points $W(t)\in\mathbb{R}^P$2 used to evaluate the integrals (Kim et al., 19 Feb 2026). This decoupling is the central scalability argument of VEvoKAN. A plausible implication is that the method targets the regime in which spatial resolution must increase without proportionally enlarging the update-system dimension.

2. KAN representation and functional parameterization

The underlying network architecture is a KAN. In the canonical description, the Kolmogorov–Arnold theorem is invoked in the form

$W(t)\in\mathbb{R}^P$3

and a deep KAN generalizes this with layers $W(t)\in\mathbb{R}^P$4 and edge-wise trainable univariate mappings $W(t)\in\mathbb{R}^P$5 (Kim et al., 19 Feb 2026). The layer update is

$W(t)\in\mathbb{R}^P$6

In the 2025 energy-dissipative EvoKAN formulation, a KAN of depth $W(t)\in\mathbb{R}^P$7 has layer sizes $W(t)\in\mathbb{R}^P$8, where $W(t)\in\mathbb{R}^P$9 is the spatial dimension and $$\frac{\partial \hat u}{\partial t}=\frac{\partial \hat u}{\partial W}\cdot \gamma,\qquad \frac{\partial W}{\partial t}=\gamma.$$0 is the number of PDE solution components (Lin et al., 3 Mar 2025). With coordinates $$\frac{\partial \hat u}{\partial t}=\frac{\partial \hat u}{\partial W}\cdot \gamma,\qquad \frac{\partial W}{\partial t}=\gamma.$$1, the forward recursion is

$$\frac{\partial \hat u}{\partial t}=\frac{\partial \hat u}{\partial W}\cdot \gamma,\qquad \frac{\partial W}{\partial t}=\gamma.$$2

and the output

$$\frac{\partial \hat u}{\partial t}=\frac{\partial \hat u}{\partial W}\cdot \gamma,\qquad \frac{\partial W}{\partial t}=\gamma.$$3

approximates the PDE solution $$\frac{\partial \hat u}{\partial t}=\frac{\partial \hat u}{\partial W}\cdot \gamma,\qquad \frac{\partial W}{\partial t}=\gamma.$$4.

The parameterization of the edge functions differs slightly across the two formulations. In the 2025 paper, all spline functions $$\frac{\partial \hat u}{\partial t}=\frac{\partial \hat u}{\partial W}\cdot \gamma,\qquad \frac{\partial W}{\partial t}=\gamma.$$5 are represented in a local B-spline basis, and the vector of spline coefficients and biases is denoted

$$\frac{\partial \hat u}{\partial t}=\frac{\partial \hat u}{\partial W}\cdot \gamma,\qquad \frac{\partial W}{\partial t}=\gamma.$$6

where $$\frac{\partial \hat u}{\partial t}=\frac{\partial \hat u}{\partial W}\cdot \gamma,\qquad \frac{\partial W}{\partial t}=\gamma.$$7 indexes B-spline basis functions on the reference interval (Lin et al., 3 Mar 2025). The same paper sometimes writes $$\frac{\partial \hat u}{\partial t}=\frac{\partial \hat u}{\partial W}\cdot \gamma,\qquad \frac{\partial W}{\partial t}=\gamma.$$8 to emphasize that each edge “weight” is itself a function of its input. In the 2026 weak-form paper, the canonical KAN uses cubic B-splines, while an RBF-KAN variant replaces each $$\frac{\partial \hat u}{\partial t}=\frac{\partial \hat u}{\partial W}\cdot \gamma,\qquad \frac{\partial W}{\partial t}=\gamma.$$9 with a Gaussian radial-basis expansion

$\Omega\subset\mathbb{R}^d$0

and the full network ansatz is written as

$\Omega\subset\mathbb{R}^d$1

These descriptions are compatible at the level of the evolutionary idea: the network is not retrained repeatedly; rather, the parameters themselves are advanced in time. In the 2025 formulation this point is explicit: EvoKAN encodes only the PDE’s initial state during an initial learning phase, after which the network parameters evolve numerically, governed by the same PDE, without any additional optimization (Lin et al., 3 Mar 2025).

3. Weak formulation and variational parameter evolution

The weak-form derivation starts from the parametric residual and replaces pointwise enforcement with projection onto a test space. The least-squares functional is

$\Omega\subset\mathbb{R}^d$2

whose stationarity yields the normal equations

$\Omega\subset\mathbb{R}^d$3

All spatial integrals are evaluated by quadrature:

$\Omega\subset\mathbb{R}^d$4

The reported remarks are twofold: the linear system size is fixed by $\Omega\subset\mathbb{R}^d$5 and $\Omega\subset\mathbb{R}^d$6, independent of $\Omega\subset\mathbb{R}^d$7, and integration by parts both reduces differentiation order and enforces Neumann boundary conditions (Kim et al., 19 Feb 2026).

The 2025 EvoKAN paper presents the same idea in variational language for parameter-space evolution. Writing

$\Omega\subset\mathbb{R}^d$8

the chain rule gives

$\Omega\subset\mathbb{R}^d$9

For a vector PDE

$$\frac{\partial u}{\partial t}(x,t)+\mathcal{N}(u(x,t))=0,\qquad u(x,0)=u_0(x),$$0

the weak variational condition is stated as: find $$\frac{\partial u}{\partial t}(x,t)+\mathcal{N}(u(x,t))=0,\qquad u(x,0)=u_0(x),$$1 such that for all test-directions $$\frac{\partial u}{\partial t}(x,t)+\mathcal{N}(u(x,t))=0,\qquad u(x,0)=u_0(x),$$2,

$$\frac{\partial u}{\partial t}(x,t)+\mathcal{N}(u(x,t))=0,\qquad u(x,0)=u_0(x),$$3

An equivalent normal-equation form is

$$\frac{\partial u}{\partial t}(x,t)+\mathcal{N}(u(x,t))=0,\qquad u(x,0)=u_0(x),$$4

where $$\frac{\partial u}{\partial t}(x,t)+\mathcal{N}(u(x,t))=0,\qquad u(x,0)=u_0(x),$$5 collects the spline-weight functions and

$$\frac{\partial u}{\partial t}(x,t)+\mathcal{N}(u(x,t))=0,\qquad u(x,0)=u_0(x),$$6

The same source states that boundary conditions, including periodic and Dirichlet, enter through the choice of test space and integration by parts (Lin et al., 3 Mar 2025).

For gradient-flow PDEs, the 2025 paper gives the Allen–Cahn equation as the model example:

$$\frac{\partial u}{\partial t}(x,t)+\mathcal{N}(u(x,t))=0,\qquad u(x,0)=u_0(x),$$7

with energy

$$\frac{\partial u}{\partial t}(x,t)+\mathcal{N}(u(x,t))=0,\qquad u(x,0)=u_0(x),$$8

More generally, it states that for a vector PDE $$\frac{\partial u}{\partial t}(x,t)+\mathcal{N}(u(x,t))=0,\qquad u(x,0)=u_0(x),$$9, an energy $$R(x;W,\gamma)\equiv \frac{\partial \hat u}{\partial t}+\mathcal{N}(\hat u) =\frac{\partial \hat u}{\partial W}(x,W)\cdot\gamma+\mathcal{N}(\hat u(x,W)).$$0 is identified so that the evolution is a gradient flow

$$R(x;W,\gamma)\equiv \frac{\partial \hat u}{\partial t}+\mathcal{N}(\hat u) =\frac{\partial \hat u}{\partial W}(x,W)\cdot\gamma+\mathcal{N}(\hat u(x,W)).$$1

This suggests that, in the energy-dissipative setting, the weak-form parameter dynamics are intended to inherit a variational structure from the underlying PDE rather than merely approximate its pointwise residual.

4. Boundary treatment and trial-space construction

A central technical feature of VEvoKAN is the explicit construction of boundary-constrained KAN trial spaces (Kim et al., 19 Feb 2026). For homogeneous Dirichlet boundary conditions, the first-layer RBF outputs are multiplied by

$$R(x;W,\gamma)\equiv \frac{\partial \hat u}{\partial t}+\mathcal{N}(\hat u) =\frac{\partial \hat u}{\partial W}(x,W)\cdot\gamma+\mathcal{N}(\hat u(x,W)).$$2

so that $$R(x;W,\gamma)\equiv \frac{\partial \hat u}{\partial t}+\mathcal{N}(\hat u) =\frac{\partial \hat u}{\partial W}(x,W)\cdot\gamma+\mathcal{N}(\hat u(x,W)).$$3. Subsequent layers use

$$R(x;W,\gamma)\equiv \frac{\partial \hat u}{\partial t}+\mathcal{N}(\hat u) =\frac{\partial \hat u}{\partial W}(x,W)\cdot\gamma+\mathcal{N}(\hat u(x,W)).$$4

to preserve zeros. For nonhomogeneous Dirichlet data $$R(x;W,\gamma)\equiv \frac{\partial \hat u}{\partial t}+\mathcal{N}(\hat u) =\frac{\partial \hat u}{\partial W}(x,W)\cdot\gamma+\mathcal{N}(\hat u(x,W)).$$5, a lifting $$R(x;W,\gamma)\equiv \frac{\partial \hat u}{\partial t}+\mathcal{N}(\hat u) =\frac{\partial \hat u}{\partial W}(x,W)\cdot\gamma+\mathcal{N}(\hat u(x,W)).$$6 is introduced and the trial function becomes

$$R(x;W,\gamma)\equiv \frac{\partial \hat u}{\partial t}+\mathcal{N}(\hat u) =\frac{\partial \hat u}{\partial W}(x,W)\cdot\gamma+\mathcal{N}(\hat u(x,W)).$$7

Periodic boundary conditions are enforced by Fourier-feature embedding,

$$R(x;W,\gamma)\equiv \frac{\partial \hat u}{\partial t}+\mathcal{N}(\hat u) =\frac{\partial \hat u}{\partial W}(x,W)\cdot\gamma+\mathcal{N}(\hat u(x,W)).$$8

so that any composition is periodic of order $$R(x;W,\gamma)\equiv \frac{\partial \hat u}{\partial t}+\mathcal{N}(\hat u) =\frac{\partial \hat u}{\partial W}(x,W)\cdot\gamma+\mathcal{N}(\hat u(x,W)).$$9 (Kim et al., 19 Feb 2026). Neumann conditions are not built into the trial space in the same way; instead, they are incorporated through the weak form via boundary integrals. The 2D heat example is given as

$\{v_k\}_{k=1}^K$0

for the case $\{v_k\}_{k=1}^K$1 on $\{v_k\}_{k=1}^K$2.

The numerical examples instantiate these mechanisms for distinct boundary types. The 1D Allen–Cahn benchmark on $\{v_k\}_{k=1}^K$3 uses homogeneous Dirichlet conditions and test functions

$\{v_k\}_{k=1}^K$4

with weak form

$\{v_k\}_{k=1}^K$5

after integration by parts and $\{v_k\}_{k=1}^K$6 (Kim et al., 19 Feb 2026). The 2D Burgers problem on $\{v_k\}_{k=1}^K$7 uses $\{v_k\}_{k=1}^K$8. The 2D heat problem uses Neumann conditions. The porous-medium-plus-drift example is periodic on $\{v_k\}_{k=1}^K$9 with weak form

$$\int_\Omega R(x;W,\gamma)\,v_k(x)\,dx \simeq 0,\qquad k=1,\dots,K.$$0

A common misconception is that weak-form enforcement merely softens boundary conditions. The boundary-constrained trial-space construction in VEvoKAN is specifically described as rigorously enforcing Dirichlet and periodic conditions, while derivative boundary conditions are incorporated directly into the weak formulation for Neumann conditions (Kim et al., 19 Feb 2026).

5. Time stepping, linear systems, and computational scaling

The VEvoKAN update algorithm is an explicit sequence of quadrature evaluation, weak sensitivity assembly, residual projection, linear solve, and parameter update (Kim et al., 19 Feb 2026). At each time step $$\int_\Omega R(x;W,\gamma)\,v_k(x)\,dx \simeq 0,\qquad k=1,\dots,K.$$1:

1. $$\int_\Omega R(x;W,\gamma)\,v_k(x)\,dx \simeq 0,\qquad k=1,\dots,K.$$2 is evaluated at quadrature or collocation points $$\int_\Omega R(x;W,\gamma)\,v_k(x)\,dx \simeq 0,\qquad k=1,\dots,K.$$3, $$\int_\Omega R(x;W,\gamma)\,v_k(x)\,dx \simeq 0,\qquad k=1,\dots,K.$$4.
2. Weak sensitivities are computed:

$$\int_\Omega R(x;W,\gamma)\,v_k(x)\,dx \simeq 0,\qquad k=1,\dots,K.$$5

1. Weak residuals are computed:

$$\int_\Omega R(x;W,\gamma)\,v_k(x)\,dx \simeq 0,\qquad k=1,\dots,K.$$6

2. The system

$$\int_\Omega R(x;W,\gamma)\,v_k(x)\,dx \simeq 0,\qquad k=1,\dots,K.$$7

is solved.

3. Parameters are updated by

$$\int_\Omega R(x;W,\gamma)\,v_k(x)\,dx \simeq 0,\qquad k=1,\dots,K.$$8

The computational-complexity comparison in the 2026 paper is formulated against strong-form EvoKAN. For strong-form EvoKAN, the system matrix has size $$\int_\Omega R(x;W,\gamma)\,v_k(x)\,dx \simeq 0,\qquad k=1,\dots,K.$$9, whereas for weak-form VEvoKAN it has size $$J_{kj}\equiv \int_\Omega \frac{\partial \hat u}{\partial W_j}(x,W)\,v_k(x)\,dx,\qquad N_k\equiv \int_\Omega \mathcal{N}(\hat u(x,W))\,v_k(x)\,dx,$$0 (Kim et al., 19 Feb 2026). The reported per-step costs are:

Quantity	Strong-Form EvoKAN	Weak-Form VEvoKAN
System size	$$J_{kj}\equiv \int_\Omega \frac{\partial \hat u}{\partial W_j}(x,W)\,v_k(x)\,dx,\qquad N_k\equiv \int_\Omega \mathcal{N}(\hat u(x,W))\,v_k(x)\,dx,$$1	$$J_{kj}\equiv \int_\Omega \frac{\partial \hat u}{\partial W_j}(x,W)\,v_k(x)\,dx,\qquad N_k\equiv \int_\Omega \mathcal{N}(\hat u(x,W))\,v_k(x)\,dx,$$2
Residual eval.	$$J_{kj}\equiv \int_\Omega \frac{\partial \hat u}{\partial W_j}(x,W)\,v_k(x)\,dx,\qquad N_k\equiv \int_\Omega \mathcal{N}(\hat u(x,W))\,v_k(x)\,dx,$$3	$$J_{kj}\equiv \int_\Omega \frac{\partial \hat u}{\partial W_j}(x,W)\,v_k(x)\,dx,\qquad N_k\equiv \int_\Omega \mathcal{N}(\hat u(x,W))\,v_k(x)\,dx,$$4
Jacobian (AD)	$$J_{kj}\equiv \int_\Omega \frac{\partial \hat u}{\partial W_j}(x,W)\,v_k(x)\,dx,\qquad N_k\equiv \int_\Omega \mathcal{N}(\hat u(x,W))\,v_k(x)\,dx,$$5	$$J_{kj}\equiv \int_\Omega \frac{\partial \hat u}{\partial W_j}(x,W)\,v_k(x)\,dx,\qquad N_k\equiv \int_\Omega \mathcal{N}(\hat u(x,W))\,v_k(x)\,dx,$$6
LSQ formation	$$J_{kj}\equiv \int_\Omega \frac{\partial \hat u}{\partial W_j}(x,W)\,v_k(x)\,dx,\qquad N_k\equiv \int_\Omega \mathcal{N}(\hat u(x,W))\,v_k(x)\,dx,$$7	$$J_{kj}\equiv \int_\Omega \frac{\partial \hat u}{\partial W_j}(x,W)\,v_k(x)\,dx,\qquad N_k\equiv \int_\Omega \mathcal{N}(\hat u(x,W))\,v_k(x)\,dx,$$8

The total-over-$$J_{kj}\equiv \int_\Omega \frac{\partial \hat u}{\partial W_j}(x,W)\,v_k(x)\,dx,\qquad N_k\equiv \int_\Omega \mathcal{N}(\hat u(x,W))\,v_k(x)\,dx,$$9-steps expressions are likewise given explicitly in the source, and the stated conclusion is that because typically $J^TJ\,\gamma=-J^TN.$0, VEvoKAN decouples system size from data resolution, yielding far better scaling as $J^TJ\,\gamma=-J^TN.$1 grows (Kim et al., 19 Feb 2026). The abstract further states that strong-form evolutionary approaches can yield ill-conditioned linear systems due to pointwise residual discretization, whereas the weak form improves scalability and stability.

The 2026 numerical section also reports a condition-number comparison for the porous-medium example: “Fig. 9: condition number & time vs $J^TJ\,\gamma=-J^TN.$2” (Kim et al., 19 Feb 2026). While no numerical values are quoted in the data block, the inclusion of that comparison supports the paper’s emphasis on conditioning as a practical distinction between weak- and strong-form formulations.

6. Energy dissipation, SAV coupling, and relation to energy-stable EvoKAN

The 2025 paper introduces “Energy-Dissipative Evolutionary Kolmogorov-Arnold Networks” as a framework that builds on KANs and advances parameters by the same PDE without additional optimization, while integrating the scalar auxiliary variable method to guarantee unconditional energy stability and computational efficiency (Lin et al., 3 Mar 2025). This is not identical to VEvoKAN, but it is the closest directly related formulation in the provided literature and shares the evolutionary KAN perspective.

For an energy decomposition with

$J^TJ\,\gamma=-J^TN.$3

the SAV variable is defined by

$J^TJ\,\gamma=-J^TN.$4

where $J^TJ\,\gamma=-J^TN.$5 is chosen so the radicand stays positive. The gradient-flow system is then rewritten as

$J^TJ\,\gamma=-J^TN.$6

where $J^TJ\,\gamma=-J^TN.$7, $J^TJ\,\gamma=-J^TN.$8 is the linear part of $J^TJ\,\gamma=-J^TN.$9, and $W(t)\in\mathbb{R}^P$00 is the mobility (Lin et al., 3 Mar 2025).

The first-order SAV time discretization is

$W(t)\in\mathbb{R}^P$01

$W(t)\in\mathbb{R}^P$02

The paper states that by taking inner products with $W(t)\in\mathbb{R}^P$03 and $W(t)\in\mathbb{R}^P$04, the discrete energy

$W(t)\in\mathbb{R}^P$05

satisfies

$W(t)\in\mathbb{R}^P$06

unconditionally (Lin et al., 3 Mar 2025).

Its numerical algorithm uses the mass matrix

$W(t)\in\mathbb{R}^P$07

and stiffness matrix

$W(t)\in\mathbb{R}^P$08

for spline basis functions $W(t)\in\mathbb{R}^P$09, then forms

$W(t)\in\mathbb{R}^P$10

solves

$W(t)\in\mathbb{R}^P$11

and updates $W(t)\in\mathbb{R}^P$12 by the second SAV equation. The implementation claim is that this only involves constant-coefficient linear solves at each step and that the cost per step is dominated by two sparse solves of size $W(t)\in\mathbb{R}^P$13, the total number of spline coefficients (Lin et al., 3 Mar 2025).

The relation between VEvoKAN and SAV-EvoKAN is therefore structural rather than identical. VEvoKAN emphasizes weak residual projection, decoupled system size, and rigorous boundary handling (Kim et al., 19 Feb 2026), whereas the 2025 formulation emphasizes energy dissipation, unconditional stability, and linear constant-coefficient solves for gradient-flow systems (Lin et al., 3 Mar 2025). This suggests two complementary lines within evolutionary KAN methods: one centered on weak-form scalability and one centered on energy-stable temporal integration.

7. Empirical behavior, benchmarks, and interpretive context

The 2026 VEvoKAN paper reports experiments on four PDE classes: 1D Allen–Cahn, 2D Burgers with Dirichlet conditions, 2D Heat with Neumann conditions, and 2D Porous Medium plus Drift with periodic conditions (Kim et al., 19 Feb 2026). The comparisons are made against strong-form EvoKAN (EvoKAN-SF) and vanilla PINN-SF.

For 1D Allen–Cahn,

$W(t)\in\mathbb{R}^P$14

the network settings are listed as layers $W(t)\in\mathbb{R}^P$15 for both EvoKAN-WF and EvoKAN-SF, activations RBF/SiLU, and parameter counts $W(t)\in\mathbb{R}^P$16 and $W(t)\in\mathbb{R}^P$17, respectively; PINN-SF uses layers $W(t)\in\mathbb{R}^P$18, tanh, and $W(t)\in\mathbb{R}^P$19 parameters. The reported table shows that the weak-form error remains approximately $W(t)\in\mathbb{R}^P$20 across $W(t)\in\mathbb{R}^P$21 to $W(t)\in\mathbb{R}^P$22, while the speedup rises from $W(t)\in\mathbb{R}^P$23 to $W(t)\in\mathbb{R}^P$24 and reaches $W(t)\in\mathbb{R}^P$25 at $W(t)\in\mathbb{R}^P$26 (Kim et al., 19 Feb 2026).

For 2D Burgers on $W(t)\in\mathbb{R}^P$27 with $W(t)\in\mathbb{R}^P$28, the paper states that EvoKAN-WF captures flow with lower error. The tabulated results report, for example, at $W(t)\in\mathbb{R}^P$29, $W(t)\in\mathbb{R}^P$30, $W(t)\in\mathbb{R}^P$31, $W(t)\in\mathbb{R}^P$32, $W(t)\in\mathbb{R}^P$33, and speedup $W(t)\in\mathbb{R}^P$34 (Kim et al., 19 Feb 2026).

For the 2D Heat equation

$W(t)\in\mathbb{R}^P$35

the paper states that “boundary gradient error decays smoothly only for VEvoKAN.” The corresponding table reports, at $W(t)\in\mathbb{R}^P$36, $W(t)\in\mathbb{R}^P$37, $W(t)\in\mathbb{R}^P$38, $W(t)\in\mathbb{R}^P$39, $W(t)\in\mathbb{R}^P$40, and speedup $W(t)\in\mathbb{R}^P$41 (Kim et al., 19 Feb 2026).

For the 2D Porous Medium plus Drift problem,

$W(t)\in\mathbb{R}^P$42

with periodic boundary conditions on $W(t)\in\mathbb{R}^P$43, the source states that both methods capture transport, but VEvoKAN yields smoother profiles. It also reports monotone energy decay for VEvoKAN even at low $W(t)\in\mathbb{R}^P$44, while EvoKAN-SF requires large $W(t)\in\mathbb{R}^P$45 to restore admissible energy dissipation. In the validity table, strong-form solutions are marked invalid at $W(t)\in\mathbb{R}^P$46 and $W(t)\in\mathbb{R}^P$47, whereas weak-form solutions are valid at all listed resolutions; the speedup reaches $W(t)\in\mathbb{R}^P$48 at $W(t)\in\mathbb{R}^P$49 (Kim et al., 19 Feb 2026).

The 2025 energy-dissipative EvoKAN paper provides a related benchmark set: one-dimensional and two-dimensional Allen–Cahn equations and two-dimensional Navier–Stokes equations (Lin et al., 3 Mar 2025). It reports for 1D Allen–Cahn on $W(t)\in\mathbb{R}^P$50 with periodic boundary conditions, sinusoidal initial data $W(t)\in\mathbb{R}^P$51, $W(t)\in\mathbb{R}^P$52, and final time $W(t)\in\mathbb{R}^P$53, that the error against a high-order spectral reference is $W(t)\in\mathbb{R}^P$54, stable across $W(t)\in\mathbb{R}^P$55. For 2D Allen–Cahn, it states that the method captures curved interface motion and coarsening, matching spectral-Galerkin benchmarks. For 2D Navier–Stokes on a periodic square with Taylor–Green vortex initial data and $W(t)\in\mathbb{R}^P$56 and $W(t)\in\mathbb{R}^P$57, EvoKAN reproduces benchmark velocity, pressure, and vorticity fields from a spectral DNS, while the EDNN (non-SAV) version becomes unstable and EvoKAN remains stable and accurate (Lin et al., 3 Mar 2025).

Taken together, the two papers support a technical characterization of weak-form evolutionary KANs as methods that combine edge-wise functional parameterization with time evolution in parameter space, but differ in their numerical priorities. VEvoKAN is presented as a stable and scalable weak-residual framework with rigorous boundary treatment (Kim et al., 19 Feb 2026); the energy-dissipative variant emphasizes unconditional energy stability for gradient flows through SAV reformulation and constant-coefficient linear solves (Lin et al., 3 Mar 2025). A plausible implication is that future work could seek an overview of both aims within a single weak-form, energy-stable evolutionary KAN formulation.