Programmable DNA‐Based Molecular Neural Network Biocomputing Circuits for Solving Partial Differential Equations

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.