用化学反应网络实现神经学习，更紧凑且天然支持梯度更新。

Molecular circuits capable of autonomous learning could unlock novel applications in fields such as bioengineering and synthetic biology. To this end, existing chemical implementations of neural computing have mainly relied on emulating discrete-layered neural architectures using steady-state computations of mass action kinetics. In contrast, we propose an alternative dynamical systems-based approach in which neural computations are modeled as the time evolution of molecular concentrations. The analog nature of our framework naturally aligns with chemical kinetics-based computation, leading to more compact circuits. We present the advantages of our framework through three key demonstrations. First, we assemble an end-to-end supervised learning pipeline using only two sequential phases, the minimum required number for supervised learning. Then, we show (through appropriate simplifications) that both linear and nonlinear modeling circuits can be implemented solely using unimolecular and bimolecular reactions, avoiding the complexities of higher-order chemistries. Finally, we demonstrate that first-order gradient approximations can be natively incorporated into the framework, enabling nonlinear models to scale linearly rather than combinatorially with input dimensionality. All the circuit constructions are validated through training and inference simulations across various regression and classification tasks. Our work presents a viable pathway toward embedding learning behaviors in synthetic biochemical systems.