Laplacian Eigenfunction-Based Neural Operator for Learning Nonlinear Reaction-Diffusion Dynamics.

Area of Science:

• Scientific computing
• Mathematical physics
• Data-driven modeling

Background:

• Reaction-diffusion equations are crucial in diverse fields like fluid dynamics, materials science, and biology.
• Learning these complex systems often faces challenges with computational cost and data requirements.

Purpose of the Study:

• To develop a novel framework for efficiently learning nonlinear reaction terms in reaction-diffusion equations.
• To address limitations in operator learning, such as data scarcity and large model sizes.

Main Methods:

• Proposed the Laplacian Eigenfunction-Based Neural Operator (LE-NO) framework.
• Utilized Laplacian eigenfunctions as a spectral basis for modeling nonlinear operators.
• Leveraged direct matrix inversion for computational efficiency.

Main Results:

• LE-NO demonstrated efficient approximation of nonlinear terms.
• The framework showed reduced computational complexity compared to traditional methods.
• LE-NO generalized well across different boundary conditions and provided interpretable dynamics.

Conclusions:

• LE-NO offers a powerful and robust tool for discovering and predicting reaction-diffusion dynamics.
• The spectral approach effectively captures complex nonlinear behaviors in mathematical physics.
• This method alleviates common challenges in operator learning, enhancing applicability.