---
title: 'Weak-form EvoKAN: Scalable Neural PDE Solver'
url: https://www.emergentmind.com/topics/weak-form-evolutionary-kolmogorov-arnold-network-kan
type: topic
---

# Weak-form EvoKAN: Scalable Neural PDE Solver

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 [2602.18515]. 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 [2602.18515]. 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 [2503.01618].

## 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 [2602.18515]. 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 $P$ and $K$, and is independent of the number of quadrature points $N_x$ used to evaluate the integrals [2602.18515]. 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
$$
f(x)=\sum_{q=1}^{2n+1}\Phi_q\!\left(\sum_{p=1}^n \phi_{q,p}(x_p)\right),
$$
and a deep KAN generalizes this with layers $\ell=0,\dots,L-1$ and edge-wise trainable univariate mappings $\phi_{\ell,j,i}$ [2602.18515]. The layer update is
$$
x_{\ell+1,j}=\sum_{i=1}^{n_\ell}\phi_{\ell,j,i}(x_{\ell,i}).
$$

In the 2025 energy-dissipative EvoKAN formulation, a KAN of depth $L$ has layer sizes $[n_0,n_1,\dots,n_L]$, where $n_0=d$ is the spatial dimension and $n_L=m$ is the number of PDE solution components [2503.01618]. With coordinates $x_{0,i}=x_i$, the forward recursion is
$$
x_{\ell+1,j}=\sum_{i=1}^{n_\ell}\phi_{\ell,j,i}(x_{\ell,i}),
$$
and the output
$$
\hat u(x;K)=x_{L,1:m}
$$
approximates the PDE solution $u(x)$.

The parameterization of the edge functions differs slightly across the two formulations. In the 2025 paper, all spline functions $\phi_{\ell,j,i}$ are represented in a local B-spline basis, and the vector of spline coefficients and biases is denoted
$$
K(t)=\{\kappa_{\ell,j,i,\alpha}\},
$$
where $\alpha$ indexes B-spline basis functions on the reference interval [2503.01618]. The same paper sometimes writes $w_{\ell,j,i}(x)=\phi_{\ell,j,i}(x)$ 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 $\phi_{\ell,j,i}(z)$ with a Gaussian radial-basis expansion
$$
\hat\phi_{\ell,j,i}(z)=\sum_{k=1}^g w^k_{\ell,j,i}\exp\!\left(-\frac{|z-c_k|^2}{2h^2}\right),
$$
and the full network ansatz is written as
$$
\hat u(x;W)=(\Psi_{L-1}\circ\cdots\circ\Psi_0)(x).
$$

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 [2503.01618].

## 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
$$
J(\gamma)=\frac12\sum_k (J_k\cdot \gamma + N_k)^2,
$$
whose stationarity yields the normal equations
$$
J^TJ\cdot\gamma=-J^TN.
$$
All spatial integrals are evaluated by quadrature:
$$
J_{kj}\approx\sum_{i=1}^{N_x}w_i\left[\frac{\partial \hat u}{\partial W_j}(x_i)\,v_k(x_i)\right],\qquad
N_k\approx\sum_{i=1}^{N_x}w_i\left[\mathcal{N}(\hat u(x_i))\,v_k(x_i)\right].
$$
The reported remarks are twofold: the linear system size is fixed by $P$ and $K$, independent of $N_x$, and integration by parts both reduces differentiation order and enforces Neumann boundary conditions [2602.18515].

The 2025 EvoKAN paper presents the same idea in variational language for parameter-space evolution. Writing
$$
\hat u(x,t)=\mathrm{KAN}(x;K(t)),
$$
the chain rule gives
$$
\partial_t \hat u=\sum_j \left(\frac{\partial \hat u}{\partial K_j}\right)\cdot\frac{dK_j}{dt}.
$$
For a vector PDE
$$
u_t+\mathcal{N}_x(u)=0,
$$
the weak variational condition is stated as: find $\gamma(x)\approx dK/dt$ such that for all test-directions $\delta K$,
$$
\int_\Omega \left(\frac{\partial \hat u}{\partial K}\right)^T\left(\frac{\partial \hat u}{\partial K}\gamma-\mathcal{N}(\hat u)\right)v\,dx=0.
$$
An equivalent normal-equation form is
$$
\int_\Omega (\partial_t w_{\ell,j,i})\,v\,dx + a(w,v)=0,
$$
where $w$ collects the spline-weight functions and
$$
a(w,v)=\int_\Omega \left(\frac{\partial \hat u}{\partial K}\right)^T\mathcal{N}(\hat u)\,v\,dx.
$$
The same source states that boundary conditions, including periodic and Dirichlet, enter through the choice of test space and integration by parts [2503.01618].

For gradient-flow PDEs, the 2025 paper gives the Allen–Cahn equation as the model example:
$$
u_t=\epsilon^2\Delta u-G'(u),\qquad G(u)=\frac{(u^2-1)^2}{4\epsilon^2},
$$
with energy
$$
E[u]=\int_\Omega \frac12 |\nabla u|^2+G(u)\,dx.
$$
More generally, it states that for a vector PDE $u_t+\mathcal{N}_x(u)=0$, an energy $E[u]$ is identified so that the evolution is a gradient flow
$$
\partial_t u=-\mathcal{G}\,\frac{\delta E}{\delta u}.
$$
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 [2602.18515]. For homogeneous Dirichlet boundary conditions, the first-layer RBF outputs are multiplied by
$$
h_1(x)=\prod_{i=1}^d (x_i-k_i)^{p_i},\qquad 0<p_i\le 1,
$$
so that $\hat u|_{\partial\Omega}=0$. Subsequent layers use
$$
h_2(z)=z^q,\qquad 0<q\le 1,
$$
to preserve zeros. For nonhomogeneous Dirichlet data $g(x,t)$, a lifting $l(x,t)$ is introduced and the trial function becomes
$$
u(x;W(t))=\hat u(x;W(t))+l(x,t).
$$

Periodic boundary conditions are enforced by Fourier-feature embedding,
$$
x_i\mapsto (\sin(\omega x_i),\cos(\omega x_i),\dots,\sin(m\omega x_i),\cos(m\omega x_i)),\qquad \omega=\frac{2\pi}{L},
$$
so that any composition is periodic of order $\ge k$ [2602.18515]. 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
$$
\int_\Omega \Delta u\cdot v\,dx
= -\int_\Omega \nabla u\cdot\nabla v\,dx + \int_{\partial\Omega}\frac{\partial u}{\partial n}\cdot v\,ds
= -\int_\Omega \nabla u\cdot\nabla v\,dx,
$$
for the case $\partial u/\partial n=0$ on $\partial\Omega$.

The numerical examples instantiate these mechanisms for distinct boundary types. The 1D Allen–Cahn benchmark on $[-1,1]$ uses homogeneous Dirichlet conditions and test functions
$$
v_k(x)=\sin\!\left(\frac{k\pi(x+1)}{2}\right),
$$
with weak form
$$
\int u_t v_k\,dx+\int u_x v_k'\,dx-\int \epsilon^{-2}u(u^2-1)v_k\,dx=0
$$
after integration by parts and $v_k(\pm1)=0$ [2602.18515]. The 2D Burgers problem on $\Omega=(-1,1)^2$ uses $u|_{\partial\Omega}=0$. The 2D heat problem uses Neumann conditions. The porous-medium-plus-drift example is periodic on $(-1,1)^2$ with weak form
$$
\int u_t v_k = -\int \nabla(u^2)\cdot \nabla v_k + \int (Vu)\cdot \nabla v_k.
$$

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 [2602.18515].

## 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 [2602.18515]. At each time step $t_n\to t_{n+1}=t_n+\Delta t$:

1. $\hat u_i=\hat u(x_i,W_n)$ is evaluated at quadrature or collocation points $x_i$, $i=1,\dots,N_x$.
2. Weak sensitivities are computed:
   $$
   J_{k,j}^{(n)}=\sum_{i=1}^{N_x}w_i\left[\frac{\partial \hat u_i}{\partial W_j}\right]v_k(x_i).
   $$
3. Weak residuals are computed:
   $$
   N_k^{(n)}=\sum_{i=1}^{N_x}w_i\,[\mathcal{N}(\hat u_i)]\,v_k(x_i).
   $$
4. The system
   $$
   J^{(n)T}J^{(n)}\gamma^{(n)}=-J^{(n)T}N^{(n)}
   $$
   is solved.
5. Parameters are updated by
   $$
   W_{n+1}=W_n+\Delta t\,\gamma^{(n)}.
   $$

The computational-complexity comparison in the 2026 paper is formulated against strong-form EvoKAN. For strong-form EvoKAN, the system matrix has size $J\in\mathbb{R}^{N_{\hat u}\times P}$, whereas for weak-form VEvoKAN it has size $J\in\mathbb{R}^{K\times P}$ [2602.18515]. The reported per-step costs are:

| Quantity | Strong-Form EvoKAN | Weak-Form VEvoKAN |
|---|---:|---:|
| System size | $J\in\mathbb{R}^{N_{\hat u}\times P}$ | $J\in\mathbb{R}^{K\times P}$ |
| Residual eval. | $\mathcal{O}(N_{\hat u}C_{\rm PDE}^{\rm SF})$ | $\mathcal{O}(N_xC_{\rm PDE}^{\rm WF}+K N_x)$ |
| Jacobian (AD) | $\mathcal{O}(N_{\hat u}P\,C_{\rm AD})$ | $\mathcal{O}(N_xP\,C_{\rm AD}+K N_xP)$ |
| LSQ formation | $\mathcal{O}(N_{\hat u}P^2)$ | $\mathcal{O}(K P^2)$ |

The total-over-$M$-steps expressions are likewise given explicitly in the source, and the stated conclusion is that because typically $N_{\hat u}\gg K$, VEvoKAN decouples system size from data resolution, yielding far better scaling as $N_x$ grows [2602.18515]. 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 $N_{\hat u}$” [2602.18515]. 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 [2503.01618]. 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
$$
E_1[\phi]=\int_\Omega g(\phi)\,dx\ge 0,
$$
the SAV variable is defined by
$$
r(t)=\sqrt{E_1[\hat u]+C_0},
$$
where $C_0$ is chosen so the radicand stays positive. The gradient-flow system is then rewritten as
$$
\phi_t=\mathcal{G}\!\left(\mathcal{L}\phi+\frac{r}{\sqrt{E_1[\phi]}}\,U[\phi]\right),\qquad
r_t=\frac{1}{2\sqrt{E_1[\phi]}}\int_\Omega U[\phi]\phi_t\,dx,
$$
where $U[\phi]=\delta E_1/\delta\phi$, $\mathcal{L}$ is the linear part of $\mu$, and $\mathcal{G}$ is the mobility [2503.01618].

The first-order SAV time discretization is
$$
\frac{w^{n+1}-w^n}{\Delta t}
+\mathcal{G}\!\left[\mathcal{L}w^{n+1}
+\frac{r^{n+1}}{\sqrt{E_1[w^n]}}U(w^n)\right]=0,
$$
$$
\frac{r^{n+1}-r^n}{\Delta t}
=\frac{1}{2\sqrt{E_1[w^n]}}
\int_\Omega U(w^n)\left(\frac{w^{n+1}-w^n}{\Delta t}\right)\,dx.
$$
The paper states that by taking inner products with $\mu$ and $2r$, the discrete energy
$$
E^{n+1}=(\phi^{n+1},\mathcal{L}\phi^{n+1})+(r^{n+1})^2
$$
satisfies
$$
E^{n+1}\le E^n
$$
unconditionally [2503.01618].

Its numerical algorithm uses the mass matrix
$$
M_{ij}=\int_\Omega \psi_i(x)\psi_j(x)\,dx
$$
and stiffness matrix
$$
S_{ij}=\int_\Omega \nabla \psi_i\cdot \nabla \psi_j\,dx
$$
for spline basis functions $\psi_i$, then forms
$$
b=Mw^n-\Delta t\left(\frac{r^n}{\sqrt{E_1[w^n]}}\right)M\,U(w^n),
$$
solves
$$
(M+\Delta t\,\mathcal{L}_{\rm sav})w^{n+1}=b,\qquad \mathcal{L}_{\rm sav}=M^{-1}S,
$$
and updates $r^{n+1}$ 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 $N_K$, the total number of spline coefficients [2503.01618].

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 [2602.18515], whereas the 2025 formulation emphasizes energy dissipation, unconditional stability, and linear constant-coefficient solves for gradient-flow systems [2503.01618]. 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 [2602.18515]. The comparisons are made against strong-form EvoKAN (EvoKAN-SF) and vanilla PINN-SF.

For 1D Allen–Cahn,
$$
u_t=u_{xx}-\epsilon^{-2}u(u^2-1),\qquad x\in[-1,1],\qquad u(\pm1,t)=0,\qquad u(x,0)=0.08\sin(\pi x),\qquad \epsilon=0.002,
$$
the network settings are listed as layers $[3,3,3,3]$ for both EvoKAN-WF and EvoKAN-SF, activations RBF/SiLU, and parameter counts $162$ and $195$, respectively; PINN-SF uses layers $[15,15,15]$, tanh, and $526$ parameters. The reported table shows that the weak-form error remains approximately $7.16\times 10^{-3}$ across $N_{\hat u}=1{,}000$ to $10{,}000$, while the speedup rises from $1.34$ to $2.76$ and reaches $27.99$ at $N_{\hat u}=50{,}000$ [2602.18515].

For 2D Burgers on $\Omega=(-1,1)^2$ with $u|_{\partial\Omega}=0$, the paper states that EvoKAN-WF captures flow with lower error. The tabulated results report, for example, at $N_{\hat u}=40{,}000$, $\mathrm{Err}_{\rm SF}=2.18{\rm e}{-2}$, $\mathrm{Err}_{\rm WF}=2.17{\rm e}{-2}$, $\mathrm{Time}_{\rm SF}=2139.5{\rm s}$, $\mathrm{Time}_{\rm WF}=29.43{\rm s}$, and speedup $72.69$ [2602.18515].

For the 2D Heat equation
$$
u_t=\alpha\Delta u + u(1-u),\qquad \frac{\partial u}{\partial n}\big|_{\partial\Omega}=0,\qquad u(x,y,0)=\cos(\pi x)\cos(\pi y),
$$
the paper states that “boundary gradient error decays smoothly only for VEvoKAN.” The corresponding table reports, at $N_{\hat u}=40{,}000$, $\mathrm{Err}_{\rm SF}=1.57{\rm e}{-2}$, $\mathrm{Err}_{\rm WF}=1.29{\rm e}{-2}$, $\mathrm{Time}_{\rm SF}=1111.4{\rm s}$, $\mathrm{Time}_{\rm WF}=54.53{\rm s}$, and speedup $20.4$ [2602.18515].

For the 2D Porous Medium plus Drift problem,
$$
u_t=\nabla\cdot\nabla(u^2)-\nabla\cdot(Vu),\qquad V=\nabla(\sin\pi x\sin\pi y),
$$
with periodic boundary conditions on $(-1,1)^2$, the source states that both methods capture transport, but VEvoKAN yields smoother profiles. It also reports monotone energy decay for VEvoKAN even at low $N_{\hat u}$, while EvoKAN-SF requires large $N_{\hat u}$ to restore admissible energy dissipation. In the validity table, strong-form solutions are marked invalid at $N_{\hat u}=400$ and $1600$, whereas weak-form solutions are valid at all listed resolutions; the speedup reaches $73.1$ at $N_{\hat u}=40{,}000$ [2602.18515].

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 [2503.01618]. It reports for 1D Allen–Cahn on $\Omega=[-1,1]$ with periodic boundary conditions, sinusoidal initial data $u(x,0)=a\sin(\pi x)$, $\epsilon=0.005,0.01,0.02$, and final time $T=1$, that the error against a high-order spectral reference is $\|u-\hat u\|_{L^2}\approx 10^{-4}$, stable across $\epsilon$. 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 $\nu=0.05$ and $\nu=0.01$, 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 [2503.01618].

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 [2602.18515]; the energy-dissipative variant emphasizes unconditional energy stability for gradient flows through SAV reformulation and constant-coefficient linear solves [2503.01618]. A plausible implication is that future work could seek a synthesis of both aims within a single weak-form, energy-stable evolutionary KAN formulation.

Source: https://www.emergentmind.com/topics/weak-form-evolutionary-kolmogorov-arnold-network-kan