FI-KAN: Fractal Interpolation Kolmogorov-Arnold Networks

Kolmogorov-Arnold Networks (KAN) employ B-spline bases on a fixed grid, providing no intrinsic multi-scale decomposition for non-smooth function approximation. We introduce Fractal Interpolation KAN (FI-KAN), which incorporates learnable fractal interpolation function (FIF) bases from iterated function system (IFS) theory into KAN. Two variants are presented: Pure FI-KAN (Barnsley, 1986) replaces B-splines entirely with FIF bases; Hybrid FI-KAN (Navascues, 2005) retains the B-spline path and adds a learnable fractal correction. The IFS contraction parameters give each edge a differentiable fractal dimension that adapts to target regularity during training. On a Holder regularity benchmark ($α\in [0.2, 2.0]$), Hybrid FI-KAN outperforms KAN at every regularity level (1.3x to 33x). On fractal targets, FI-KAN achieves up to 6.3x MSE reduction over KAN, maintaining 4.7x advantage at 5 dB SNR. On non-smooth PDE solutions (scikit-fem), Hybrid FI-KAN achieves up to 79x improvement on rough-coefficient diffusion and 3.5x on L-shaped domain corner singularities. Pure FI-KAN's complementary behavior, dominating on rough targets while underperforming on smooth ones, provides controlled evidence that basis geometry must match target regularity. A fractal dimension regularizer provides interpretable complexity control whose learned values recover the true fractal dimension of each target. These results establish regularity-matched basis design as a principled strategy for neural function approximation.

Radial Müntz-Szász Networks: Neural Architectures with Learnable Power Bases for Multidimensional Singularities

Radial singular fields, such as $1/r$, $\log r$, and crack-tip profiles, are difficult to model for coordinate-separable neural architectures. We show that any $C^2$ function that is both radial and additively separable must be quadratic, establishing a fundamental obstruction for coordinate-wise power-law models. Motivated by this result, we introduce Radial Müntz-Szász Networks (RMN), which represent fields as linear combinations of learnable radial powers $r^μ$, including negative exponents, together with a limit-stable log-primitive for exact $\log r$ behavior. RMN admits closed-form spatial gradients and Laplacians, enabling physics-informed learning on punctured domains. Across ten 2D and 3D benchmarks, RMN achieves 1.5$\times$--51$\times$ lower RMSE than MLPs and 10$\times$--100$\times$ lower RMSE than SIREN while using 27 parameters, compared with 33,537 for MLPs and 8,577 for SIREN. We extend RMN to angular dependence (RMN-Angular) and to multiple sources with learnable centers (RMN-MC); when optimization converges, source-center recovery errors fall below $10^{-4}$. We also report controlled failures on smooth, strongly non-radial targets to delineate RMN's operating regime.

Discovering Scaling Exponents with Physics-Informed Müntz-Szász Networks

Physical systems near singularities, interfaces, and critical points exhibit power-law scaling, yet standard neural networks leave the governing exponents implicit. We introduce physics-informed M"untz-Sz'asz Networks (MSN-PINN), a power-law basis network that treats scaling exponents as trainable parameters. The model outputs both the solution and its scaling structure. We prove identifiability, or unique recovery, and show that, under these conditions, the squared error between learned and true exponents scales as $O(|μ- α|^2)$. Across experiments, MSN-PINN achieves single-exponent recovery with 1--5% error under noise and sparse sampling. It recovers corner singularity exponents for the two-dimensional Laplace equation with 0.009% error, matches the classical result of Kondrat'ev (1967), and recovers forcing-induced exponents in singular Poisson problems with 0.03% and 0.05% errors. On a 40-configuration wedge benchmark, it reaches a 100% success rate with 0.022% mean error. Constraint-aware training encodes physical requirements such as boundary condition compatibility and improves accuracy by three orders of magnitude over naive training. By combining the expressiveness of neural networks with the interpretability of asymptotic analysis, MSN-PINN produces learned parameters with direct physical meaning.

PALMA: A Lightweight Tropical Algebra Library for ARM-Based Embedded Systems

Tropical algebra, including max-plus, min-plus, and related idempotent semirings, provides a unifying framework in which many optimization problems that are nonlinear in classical algebra become linear. This property makes tropical methods particularly well suited for shortest paths, scheduling, throughput analysis, and discrete event systems. Despite their theoretical maturity and practical relevance, existing tropical algebra implementations primarily target desktop or server environments and remain largely inaccessible on resource-constrained embedded platforms, where such optimization problems are most acute. We present PALMA (Parallel Algebra Library for Max-plus Applications), a lightweight, dependency-free C library that brings tropical linear algebra to ARM-based embedded systems. PALMA implements a generic semiring abstraction with SIMD-accelerated kernels, enabling a single computational framework to support shortest paths, bottleneck paths, reachability, scheduling, and throughput analysis. The library supports five tropical semirings, dense and sparse (CSR) representations, tropical closure, and spectral analysis via maximum cycle mean computation. We evaluate PALMA on a Raspberry Pi 4 and demonstrate peak performance of 2,274 MOPS, speedups of up to 11.9 times over classical Bellman-Ford for single-source shortest paths, and sub-10 microsecond scheduling solves for real-time control workloads. Case studies in UAV control, IoT routing, and manufacturing systems show that tropical algebra enables efficient, predictable, and unified optimization directly on embedded hardware. PALMA is released as open-source software under the MIT license.

Müntz-Szász Networks: Neural Architectures with Learnable Power-Law Bases

Standard neural network architectures employ fixed activation functions (ReLU, tanh, sigmoid) that are poorly suited for approximating functions with singular or fractional power behavior, a structure that arises ubiquitously in physics, including boundary layers, fracture mechanics, and corner singularities. We introduce Müntz-Szász Networks (MSN), a novel architecture that replaces fixed smooth activations with learnable fractional power bases grounded in classical approximation theory. Each MSN edge computes $φ(x) = \sum_k a_k |x|^{μ_k} + \sum_k b_k \mathrm{sign}(x)|x|^{λ_k}$, where the exponents $\{μ_k, λ_k\}$ are learned alongside the coefficients. We prove that MSN inherits universal approximation from the Müntz-Szász theorem and establish novel approximation rates: for functions of the form $|x|^α$, MSN achieves error $\mathcal{O}(|μ- α|^2)$ with a single learned exponent, whereas standard MLPs require $\mathcal{O}(ε^{-1/α})$ neurons for comparable accuracy. On supervised regression with singular target functions, MSN achieves 5-8x lower error than MLPs with 10x fewer parameters. Physics-informed neural networks (PINNs) represent a particularly demanding application for singular function approximation; on PINN benchmarks including a singular ODE and stiff boundary-layer problems, MSN achieves 3-6x improvement while learning interpretable exponents that match the known solution structure. Our results demonstrate that theory-guided architectural design can yield dramatic improvements for scientifically-motivated function classes.

Localized Weather Prediction Using Kolmogorov-Arnold Network-Based Models and Deep RNNs

Weather forecasting is crucial for managing risks and economic planning, particularly in tropical Africa, where extreme events severely impact livelihoods. Yet, existing forecasting methods often struggle with the region's complex, non-linear weather patterns. This study benchmarks deep recurrent neural networks such as $\texttt{LSTM, GRU, BiLSTM, BiGRU}$, and Kolmogorov-Arnold-based models $(\texttt{KAN} and \texttt{TKAN})$ for daily forecasting of temperature, precipitation, and pressure in two tropical cities: Abidjan, Cote d'Ivoire (Ivory Coast) and Kigali (Rwanda). We further introduce two customized variants of $ \texttt{TKAN}$ that replace its original $\texttt{SiLU}$ activation function with $ \texttt{GeLU}$ and \texttt{MiSH}, respectively. Using station-level meteorological data spanning from 2010 to 2024, we evaluate all the models on standard regression metrics. $\texttt{KAN}$ achieves temperature prediction ($R^2=0.9986$ in Abidjan, $0.9998$ in Kigali, $\texttt{MSE} < 0.0014~^\circ C ^2$), while $\texttt{TKAN}$ variants minimize absolute errors for precipitation forecasting in low-rainfall regimes. The customized $\texttt{TKAN}$ models demonstrate improvements over the standard $\texttt{TKAN}$ across both datasets. Classical \texttt{RNNs} remain highly competitive for atmospheric pressure ($R^2 \approx 0.83{-}0.86$), outperforming $\texttt{KAN}$-based models in this task. These results highlight the potential of spline-based neural architectures for efficient and data-efficient forecasting.