Physics-Informed Kolmogorov-Arnold Networks

• PIKAN is a physics-informed learning model where a Kolmogorov-Arnold Network (KAN) replaces the conventional MLP to parameterize solutions of differential equations.
• It employs learnable univariate functions along network edges, enhancing local approximations to better capture oscillatory, multiscale, and sharp-gradient phenomena.
• Despite demonstrating improved accuracy and faster convergence in benchmarks, PIKAN's sensitivity to optimizer settings and training protocols highlights ongoing challenges.

Physics-Informed Kolmogorov-Arnold Networks (PIKANs) are physics-informed learning models in which the unknown solution $u_\theta$ is parameterized by a Kolmogorov-Arnold Network (KAN) rather than the multilayer perceptron (MLP) backbone used in standard Physics-Informed Neural Networks (PINNs). In the now-standard formulation, the governing equations, boundary or initial conditions, and any optional observational data enter through the same residual-based objective used by PINNs; what changes is the representation model. A unified benchmark states this distinction explicitly as “PINN = physics-informed loss + MLP surrogate” and “PIKAN = physics-informed loss + KAN surrogate,” and reports that, under identical physics-informed formulations and matched parameter budgets, PIKANs “consistently achieve more accurate solutions, converge in fewer iterations, and yield superior gradient estimates” (Dzimah et al., 14 Feb 2026). The review literature places PIKAN as a recent representation-model extension inside the broader physics-informed machine learning framework rather than as a separate training philosophy (Toscano et al., 2024).

1. Definition, origin, and conceptual scope

In the terminology used across the PIKAN literature, the broader physics-informed machine learning framework employs a representation model to approximate the solution of an ODE, PDE, or related scientific system. When that representation model is an MLP, the model is called a PINN; when it is a KAN, it is called a PIKAN (Toscano et al., 2024). This framing is important because it locates the novelty of PIKAN at the architectural level. The governing equations, residual construction, collocation strategy, and optimization loop are usually inherited directly from PINNs.

The KAN component is motivated by the Kolmogorov-Arnold representation idea. Instead of an affine map followed by a fixed scalar activation at each node, KANs place learnable univariate functions on edges. The review literature writes a KAN layer as

$$\bm{z}^{(l)}=\sum_{i=1}^{H}\Phi_i\!\left(\sum_{j=1}^{H}\phi_{i,j}(z_j^{(l-1)})\right),$$

where $\phi_{i,j}$ are inner univariate functions and $\Phi_i$ are outer univariate functions (Toscano et al., 2024). The unified benchmark summarizes the architectural difference more compactly: KANs use “learnable univariate transformations along each edge, providing richer local approximations and improved expressivity,” whereas MLPs retain fixed activations and a more global approximation bias (Dzimah et al., 14 Feb 2026).

This architectural shift is presented as especially relevant for oscillatory, multiscale, and sharp-gradient solutions. The central claim is not that PIKAN changes the philosophy of physics-informed learning, but that it changes the hypothesis class in a way that can better match difficult solution structure. The review literature treats this as one branch of a broader movement in representation-model design, and cites early PIKAN-oriented work by Shukla et al., Howard et al., Rigas et al., Zhang et al., Shuai and Li, Patra et al., Wang et al., and Koenig et al. (Toscano et al., 2024).

2. Mathematical formulation

The mathematical core of PIKAN is the standard physics-informed residual framework. The review literature writes the governing problem as

$$\mathcal{F}_\tau[\hat{u}](x)=f(x), \quad x\in\Omega,$$

with boundary conditions

$$\mathcal{B}_\tau[\hat{u}](x)=b(x), \quad x\in\Omega_B,$$

and approximates the true solution by a representation model $u(\theta,x)$ (Toscano et al., 2024). In a PIKAN, $u(\theta,x)$ is a KAN rather than an MLP.

The residuals used in training are not PIKAN-specific. The same review defines

$$r_E(x,\theta)=\mathcal{F}_{\tau}[u](x,\theta)-f(x),$$

$$r_B(x,\theta)=\mathcal{B}_{\tau}[u](x,\theta)-b(x),$$

and, when inverse-problem data are present,

$$\bm{z}^{(l)}=\sum_{i=1}^{H}\Phi_i\!\left(\sum_{j=1}^{H}\phi_{i,j}(z_j^{(l-1)})\right),$$0

with a continuous or discretized multi-objective loss formed from equation, boundary, and data residuals (Toscano et al., 2024). The benchmark study makes the same point in methodological terms: PINNs and PIKANs are compared under “identical physics-informed formulations and matched parameter budgets to isolate the architectural effect” (Dzimah et al., 14 Feb 2026).

A useful feature of this shared formulation is that hard-constraint constructions carry over unchanged. The review gives the constrained ansatz

$$\bm{z}^{(l)}=\sum_{i=1}^{H}\Phi_i\!\left(\sum_{j=1}^{H}\phi_{i,j}(z_j^{(l-1)})\right),$$1

where $$\bm{z}^{(l)}=\sum_{i=1}^{H}\Phi_i\!\left(\sum_{j=1}^{H}\phi_{i,j}(z_j^{(l-1)})\right),$$2 is a generic representation model and may therefore be either an MLP or a KAN (Toscano et al., 2024). In other words, exact Dirichlet enforcement, input transforms, and output transforms belong to the physics-informed layer of the method, not to the choice between PINN and PIKAN.

Because the residual depends on derivatives of the learned solution, automatic differentiation is structurally central. The benchmark paper emphasizes that PIKAN’s empirical advantage is not limited to function values; it explicitly reports superior gradient reconstruction accuracy, which is especially consequential when first- and higher-order derivatives drive the PDE or ODE residual (Dzimah et al., 14 Feb 2026).

3. Architectural families and training variants

Within the general PIKAN template, several architectural subfamilies have appeared. One early line uses Chebyshev-parameterized univariate functions. The review discusses cPIKANs, in which the inner and outer univariate functions are represented by Chebyshev polynomials, and describes these models as improving performance with fewer parameters and being more robust to noise in the cited studies (Toscano et al., 2024). A related systems-pharmacology study introduces tanh-cPIKAN, which adds extra $$\bm{z}^{(l)}=\sum_{i=1}^{H}\Phi_i\!\left(\sum_{j=1}^{H}\phi_{i,j}(z_j^{(l-1)})\right),$$3 contractions around Chebyshev blocks to keep activations in a numerically favorable range and to smooth the loss landscape; that paper reports that PIKAN performance is tightly coupled to optimizer choice, learning-rate schedule, warm-up, and floating-point precision (Daryakenari et al., 10 Apr 2025).

Another family replaces spline-based edge functions with orthogonal polynomial expansions tailored to PDE problems. “Physics-informed Kolmogorov-Arnold Network with Chebyshev Polynomials for Fluid Mechanics” proposes ChebPIKAN, which combines a KAN-style backbone with Chebyshev-polynomial edge functions and physics-informed residuals for Allen-Cahn, nonlinear Burgers, Helmholtz, Kovasznay flow, cylinder wake flow, and cavity flow (Guo et al., 2024). The same general idea appears in geometry-aware and point-cloud settings: PI-KAN-PointNet replaces PointNet’s shared MLP modules with shared KAN modules, using Jacobi polynomial parameterizations to solve inverse problems over many irregular geometries in a single training run (Kashefi et al., 8 Apr 2025).

A separate line of work focuses on training mechanisms rather than basis choice. Grid-dependent PIKANs preserve the adaptive-grid character of spline-based KANs and introduce regular grid adaptation, grid extension, and adaptive optimizer-state transition across grid changes (Rigas et al., 2024). Separable PIKANs (SPIKANs) impose a low-rank separable ansatz,

$$\bm{z}^{(l)}=\sum_{i=1}^{H}\Phi_i\!\left(\sum_{j=1}^{H}\phi_{i,j}(z_j^{(l-1)})\right),$$4

so that each dimension is handled by a separate univariate KAN; multi-resolution MR-PIKAN alternates or sequences low- and high-resolution training batches for multiscale PDEs; HWF-PIKAN adds hybrid Fourier-wavelet input embeddings before the KAN core for advection and collisionless Boltzmann problems; and a modified hybrid parallel KAN-MLP architecture learns a convex mixture of KAN and MLP branches under overlapping domain decomposition (Jacob et al., 2024, Yang et al., 26 Jul 2025, Heravifard et al., 12 Dec 2025, Huang et al., 14 Nov 2025).

These variants suggest that “PIKAN” is best understood as a family of physics-informed architectures organized around a KAN surrogate, not as a single fixed network design.

4. Benchmark evidence and empirical behavior

The clearest benchmark-level statement currently available is the 2026 unified comparison of MLP-based PINNs and KAN-based PIKANs over a representative collection of ODEs and PDEs. With identical physics-informed formulations and matched parameter budgets, that study reports that PIKANs “consistently achieve more accurate solutions, converge in fewer iterations, and yield superior gradient estimates,” with the strongest motivation arising in oscillatory, multiscale, and sharp-gradient regimes (Dzimah et al., 14 Feb 2026).

Subsequent work has refined the training side of the picture. For grid-dependent PIKANs, a JAX-based implementation reports up to 84 times faster training than the original KAN implementation and substantial improvements in relative $$\bm{z}^{(l)}=\sum_{i=1}^{H}\Phi_i\!\left(\sum_{j=1}^{H}\phi_{i,j}(z_j^{(l-1)})\right),$$5 error under adaptive schedules. The reported benchmark reductions are from $$\bm{z}^{(l)}=\sum_{i=1}^{H}\Phi_i\!\left(\sum_{j=1}^{H}\phi_{i,j}(z_j^{(l-1)})\right),$$6 to $$\bm{z}^{(l)}=\sum_{i=1}^{H}\Phi_i\!\left(\sum_{j=1}^{H}\phi_{i,j}(z_j^{(l-1)})\right),$$7 on diffusion, from $$\bm{z}^{(l)}=\sum_{i=1}^{H}\Phi_i\!\left(\sum_{j=1}^{H}\phi_{i,j}(z_j^{(l-1)})\right),$$8 to $$\bm{z}^{(l)}=\sum_{i=1}^{H}\Phi_i\!\left(\sum_{j=1}^{H}\phi_{i,j}(z_j^{(l-1)})\right),$$9 on Helmholtz, from $\phi_{i,j}$0 to $\phi_{i,j}$1 on Burgers, and from $\phi_{i,j}$2 to $\phi_{i,j}$3 on Allen-Cahn (Rigas et al., 2024). These results support the view that representation choice and training protocol cannot be cleanly separated in practice.

SPIKANs address a different bottleneck: the cost of multivariate collocation in higher dimensions. On the reported Helmholtz benchmarks, speedups range from $\phi_{i,j}$4 to $\phi_{i,j}$5 relative to standard PIKAN while also improving accuracy in the listed cases. On steady lid-driven cavity flow, the reported speedups are $\phi_{i,j}$6 and $\phi_{i,j}$7 for the two SPIKAN settings; on the 2D+1 Klein-Gordon problem, the reported speedups are $\phi_{i,j}$8, $\phi_{i,j}$9, and $\Phi_i$0 (Jacob et al., 2024). The price is a separability assumption: performance depends on the rank $\Phi_i$1 and the degree to which the solution admits a useful low-rank factorization.

MR-PIKAN targets multiscale PDEs from a sampling-resolution perspective. On the forward Poisson benchmark, sequential multi-resolution training is reported to deliver about a $\Phi_i$2 training-time reduction, while alternating multi-resolution training yields about one-sixth training-time savings together with improved test accuracy relative to single-resolution high-resolution training. On the inverse Gray-Scott system, MR-PIKAN maintains diffusion-coefficient errors within about $\Phi_i$3 while reducing training time by roughly $\Phi_i$4, and on hidden elasticity of two-phase random materials it requires over $\Phi_i$5 less time to reach the stated “acceptable” accuracy threshold (Yang et al., 26 Jul 2025).

For advection and collisionless Boltzmann problems, HWF-PIKAN extends this trend. On the collisionless Sod benchmark, the reported total loss at 10,000 epochs falls from $\Phi_i$6 for vanilla PINN and $\Phi_i$7 for vanilla PIKAN to $\Phi_i$8 for HWF-PIKAN, with Fourier-only and wavelet-only PIKAN variants in between (Heravifard et al., 12 Dec 2025). This result is notable because it attributes gains not merely to KAN itself but to the interaction between PIKAN and multi-resolution spectral embeddings.

5. Applications and domain-specific adaptations

PIKAN has already been used across a wide range of scientific-machine-learning settings. In power-system dynamics, the first PIKAN framework for this area models swing-equation dynamics and identifies uncertain inertia and damping coefficients. On the single-machine infinite-bus system, the paper reports 195 parameters for PIKAN versus 481 for the PINN baseline, with median trajectory error $\Phi_i$9 for PIKAN-I versus $$\mathcal{F}_\tau[\hat{u}](x)=f(x), \quad x\in\Omega,$$0 for PINN-I and $$\mathcal{F}_\tau[\hat{u}](x)=f(x), \quad x\in\Omega,$$1 for PIKAN-II versus $$\mathcal{F}_\tau[\hat{u}](x)=f(x), \quad x\in\Omega,$$2 for PINN-II. On the 4-bus, 2-generator system, the parameter counts are 720 versus 1234, and the best median trajectory error drops to $$\mathcal{F}_\tau[\hat{u}](x)=f(x), \quad x\in\Omega,$$3 for PIKAN-II versus $$\mathcal{F}_\tau[\hat{u}](x)=f(x), \quad x\in\Omega,$$4 for PINN-II; the same study reports around $$\mathcal{F}_\tau[\hat{u}](x)=f(x), \quad x\in\Omega,$$5 median relative error for inertia coefficients and around $$\mathcal{F}_\tau[\hat{u}](x)=f(x), \quad x\in\Omega,$$6 for several damping coefficients in the PIKAN-II identification setting (Shuai et al., 2024).

In fluid and thermal inverse problems over irregular domains, PI-KAN-PointNet generalizes PIKAN from one-domain-per-training-run to many-domain learning over 135 geometries. With approximately the same trainable-parameter count and comparable training time per epoch, the paper reports that PI-KAN-PointNet improves the average relative error for the unknown inner-boundary temperature from $$\mathcal{F}_\tau[\hat{u}](x)=f(x), \quad x\in\Omega,$$7 in the MLP-based PointNet baseline to $$\mathcal{F}_\tau[\hat{u}](x)=f(x), \quad x\in\Omega,$$8 in the pure KAN version, while the best overall result comes from a hybrid architecture with an MLP encoder and KAN decoder, which reaches $$\mathcal{F}_\tau[\hat{u}](x)=f(x), \quad x\in\Omega,$$9 (Kashefi et al., 8 Apr 2025). This is one of the clearest demonstrations that KAN components may be most useful in specific stages of a larger architecture rather than uniformly everywhere.

For gray-box system identification in systems pharmacology, tanh-cPIKAN is applied to pharmacokinetics and pharmacodynamics models with sparse, ill-posed, and non-unique data. The key result is conditional rather than absolute: in single precision, PINNs are often more robust, but in double precision with second-order or hybrid optimization, tanh-cPIKAN reaches the best reported mean absolute errors, including $$\mathcal{B}_\tau[\hat{u}](x)=b(x), \quad x\in\Omega_B,$$0 on the pharmacokinetics problem and $$\mathcal{B}_\tau[\hat{u}](x)=b(x), \quad x\in\Omega_B,$$1 on the pharmacodynamics problem under hybrid RAdam plus BFGS-backtracking (Daryakenari et al., 10 Apr 2025). This application area makes optimizer sensitivity a first-class part of PIKAN practice.

High-index differential-algebraic equations provide another natural testbed. DAE-KAN uses separate KANs for differential and algebraic variables and compares index-1, index-2, and index-3 formulations. The paper reports that DAE-KAN reduces the absolute errors of both differential and algebraic variables by 1 to 2 orders of magnitude compared to traditional PINNs and that both PINNs and DAE-KAN outperform classical numerical methods in controlling drift-off error (Luo et al., 22 Apr 2025). In the PIKAN taxonomy, this work is best read as a DAE-specific PIKAN variant.

6. Limitations, misconceptions, and open questions

A recurrent misconception is that PIKAN represents a different kind of physics-informed learning from PINNs. In the core literature, this is incorrect. PIKAN usually preserves the same residual minimization, boundary or initial penalties, and sampling framework, changing only the representation model from MLP to KAN (Dzimah et al., 14 Feb 2026, Toscano et al., 2024). The practical distinction is architectural, not methodological.

A second misconception is that KAN backbones are uniformly superior. The evidence is benchmark-based and heterogeneous. The unified benchmark is strongly favorable to PIKAN under matched conditions (Dzimah et al., 14 Feb 2026), but other settings are less favorable. For inverse Poisson problems on infinite and semi-infinite domains, a dedicated comparison reports that PINNs provide a more accurate and computationally efficient solution, solving the inverse problem “1,000 times faster” and with lower relative error than PIKANs in the tested setting (Pérez-Bernal et al., 12 Dec 2025). Similarly, PI-KAN-PointNet shows that the best architecture on its irregular-geometry task is not pure KAN everywhere but an MLP encoder with a KAN decoder (Kashefi et al., 8 Apr 2025).

Optimization sensitivity is another persistent limitation. The systems-pharmacology study explicitly concludes that PIKAN performance is more sensitive than PINN performance to optimizer, precision, warm-up length, and learning rate, even though it can outperform PINNs under the right double-precision, second-order configuration (Daryakenari et al., 10 Apr 2025). The grid-dependent literature likewise shows that naive grid extension can produce loss spikes unless optimizer state is transferred carefully across grid changes (Rigas et al., 2024).

Terminology is also not fully stable. One UAV channel-modeling paper expands PIKAN as “Physics-Inspired Kolmogorov-Arnold Network,” not “Physics-Informed Kolmogorov-Arnold Network,” and explicitly distinguishes its architecture-level symbolic inductive biases from PINN-style loss-based physics enforcement (Tekbıyık et al., 7 Oct 2025). That usage is technically separate from the mainstream PIKAN literature discussed above, even though the acronym is the same.

Open questions remain at several levels. The review literature notes that PIKAN-specific theory is still limited, that many optimization issues remain those of physics-informed learning broadly, and that no dedicated PIKAN software ecosystem is yet identified in the same way as for mainstream PINN libraries (Toscano et al., 2024). The benchmark and extension papers together suggest a plausible research trajectory—architecture design, optimizer coupling, separability, multiresolution training, geometry encoding, and hybrid KAN-MLP decompositions—but they do not yet establish universal superiority, nor do they settle scaling to very high-dimensional PDEs, complex industrial geometries, noisy inverse settings, or fully mature software tooling (Dzimah et al., 14 Feb 2026).