The ability of neural networks to process information intelligently has allowed them to be successfully applied in the fields of information processing, controls, engineering, medicine, and economics. The brain-like working mode of a neural network gives it incomparable advantages in solving complex nonlinear problems compared with other methods. In this paper, we propose a feedforward DNA neural network framework based on an enzyme-free, entropy-driven DNA reaction network that uses a modular design. A multiplication gate, an addition gate, a subtraction gate, and a threshold gate module based on the DNA strand displacement principle are cascaded into a single DNA neuron, and the neuron cascade is used to form a feedforward transfer neural network. We use this feedforward neural network to realize XOR logic operation and full adder logic operation, which proves that the molecular neural network system based on DNA strand displacement can carry out complex nonlinear operation and reflects the powerful potential of building these molecular neural networks. A feedforward neural network framework is proposed based on enzyme-free, entropy-driven DNA strand replacement. The framework is completed by means of modular design. The cascade of operation modules realizes a single neuron module, and the cascade of multiple single neuron modules realizes a feedforward neural network. The operation mode of the framework is demonstrated by taking the logic operation of XOR and three input full adders as an example.