Abstract
Partial differential equations, essential for modeling dynamic systems, persistently confront computational complexity bottlenecks in high-dimensional problems, yet DNA-based parallel computing architectures, leveraging their discrete mathematics merits, provide transformative potential by harnessing inherent molecular parallelism. This research introduces an augmented matrix-based DNA molecular neural network to achieve molecular-level solving of biological Brusselator PDEs. Two crucial innovations address existing technological constraints: (i) an augmented matrix-based error-feedback DNA molecular neural network, enabling multidimensional parameter integration through DNA strand displacement cascades and iterative weight optimization; (ii) incorporating membrane diffusion theory with division operation principles into DNA circuits to develop partial differential calculation modules. Simulation results demonstrate that the augmented matrix-based DNA neural network efficiently and accurately learns target functions; integrating the proposed partial derivative computation strategy, this architecture solves the biological Brusselator PDE numerically with errors below 0.02 within 12,500 s. This work establishes a novel intelligent non-silicon-based computational framework, providing theoretical foundations and potential implementation paradigms for future bio-inspired computing and unconventional computing devices in life science research.