Introducing the Jani Transform: A Novel Integral Approach for Modeling Differential Equations and Dynamical Systems in AI

Jani, Haresh; Singh, Twinkle

doi:10.22080/cjms.2026.30834.1789

Introducing the Jani Transform: A Novel Integral Approach for Modeling Differential Equations and Dynamical Systems in AI

Document Type : Research Articles

Authors

Haresh Jani ¹
Twinkle Singh ²

¹ Shri Sad Vidya Mandal Institute of Technology

² Sardar Vallabhbhai National Institute of Technology

10.22080/cjms.2026.30834.1789

Abstract

This work introduces the Jani Transform (JT), a novel integral transform with a gamma-distribution-inspired kernel, designed for solving complex differential equations and dynamical systems in Artificial Intelligence (AI) modeling. Unlike classical Laplace and Fourier transforms, JT is tailored for nonlinear, memory-dependent, and time-localized systems frequently encountered in Neural ODEs, reinforcement learning, spiking neural networks, and time-series forecasting. The paper develops the theoretical framework of JT, including its definition, properties, inverse formulation, and transform tables for common functions. Derivative properties are established to facilitate operational calculus. Several illustrative examples demonstrate how JT simplifies the solution of ODEs and PDEs in AI contexts, providing interpretable, closed-form solutions. Comparative analysis highlights JT’s adaptability to nonlinearities, probabilistic interpretability, and superior integration into AI workflows. Applications in signal decomposition, denoising, and feature extraction for machine learning pipelines are discussed, along with future research directions involving fractional dynamics, fuzzy systems, hybrid transforms, quantum AI, and energy-efficient modeling. The results position the Jani Transform as a mathematically elegant and practically powerful tool for advancing interpretable and efficient AI models.

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