Reaction–diffusion systems serve as a powerful mathematical framework for modeling dynamic behaviors and pattern formation, widely used to model physical, chemical, and biological systems, among others. Understanding their stability often involves examining eigenvalue problems derived by linearizing the governing equations around their time-independent solutions. In most practical cases, analytical solutions to these eigenvalue problems are difficult to obtain—especially in multi-component systems or when dealing with non‐self‐adjoint problems—making both analysis and computation more challenging. As a result, numerical methods remain the primary practical tool for investigating their spectral properties and gaining insight into system behavior. Recently, physics-informed neural networks (PINNs) have emerged as a promising alternative for solving partial differential equations by embedding physical laws and constraints directly into the training process. In this work, we use a PINN-based approach to compute multiple eigenpairs for the eigenvalue problems of reaction–diffusion systems, covering both single- and multi-component cases. This includes challenging scenarios involving non‐self‐adjoint problems, where traditional numerical methods often become less effective. In this work, we limit our focus to two one-dimensional reaction–diffusion models: the Zeldovich–Frank–Kamenetsky (ZFK) equation, a single-component system with a self-adjoint structure, and the FitzHugh–Nagumo (FHN) system, a two-component model exhibiting non‐self‐adjoint behavior. By embedding essential physical constraints—such as biorthonormality between left and right eigenfunctions and spectral ordering—directly into the loss function, the method maintains consistency and robustness during training. In both cases, the PINN framework yields accurate eigenvalue and eigenfunction approximations that agree closely with direct numerical simulations. These results highlight the capability of PINNs to serve as a flexible and effective tool for spectral analysis across a broad class of reaction–diffusion problems.
Citation: Burhan Bezekci. Spectral analysis of reaction-diffusion systems via physics-informed neural networks[J]. AIMS Mathematics, 2025, 10(8): 18156-18182. doi: 10.3934/math.2025809
Reaction–diffusion systems serve as a powerful mathematical framework for modeling dynamic behaviors and pattern formation, widely used to model physical, chemical, and biological systems, among others. Understanding their stability often involves examining eigenvalue problems derived by linearizing the governing equations around their time-independent solutions. In most practical cases, analytical solutions to these eigenvalue problems are difficult to obtain—especially in multi-component systems or when dealing with non‐self‐adjoint problems—making both analysis and computation more challenging. As a result, numerical methods remain the primary practical tool for investigating their spectral properties and gaining insight into system behavior. Recently, physics-informed neural networks (PINNs) have emerged as a promising alternative for solving partial differential equations by embedding physical laws and constraints directly into the training process. In this work, we use a PINN-based approach to compute multiple eigenpairs for the eigenvalue problems of reaction–diffusion systems, covering both single- and multi-component cases. This includes challenging scenarios involving non‐self‐adjoint problems, where traditional numerical methods often become less effective. In this work, we limit our focus to two one-dimensional reaction–diffusion models: the Zeldovich–Frank–Kamenetsky (ZFK) equation, a single-component system with a self-adjoint structure, and the FitzHugh–Nagumo (FHN) system, a two-component model exhibiting non‐self‐adjoint behavior. By embedding essential physical constraints—such as biorthonormality between left and right eigenfunctions and spectral ordering—directly into the loss function, the method maintains consistency and robustness during training. In both cases, the PINN framework yields accurate eigenvalue and eigenfunction approximations that agree closely with direct numerical simulations. These results highlight the capability of PINNs to serve as a flexible and effective tool for spectral analysis across a broad class of reaction–diffusion problems.
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Burhan Bezekci. Spectral analysis of reaction-diffusion systems via physics-informed neural networks[J]. AIMS Mathematics, 2025, 10(8): 18156-18182. doi: 10.3934/math.2025809
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