The DNA toehold mediated strand displacement reaction is one of the semi-synthetic biology technologies for next-generation computers. In this article, we present a framework for a novel nonlinear neural network based on an engineered biochemical circuit, which is constructed by several reaction modules including catalysis, degradation and adjustment reaction modules. The proposed neural network possesses an architecture that is similar to that of an error back propagation neural network, and is built of an input layer, hidden layer and output layer. As a proof of concept, we utilize this nonlinear neural network based on an analog DNA toehold mediated strand displacement reaction circuit to learn the standard quadratic form function and analyze the robustness of the nonlinear neural network toward DNA strand concentration detection, DNA strand displacement reaction rate and noise. Unlike in error back propagation neural networks, the adaptive behavior of this DNA molecular neural network system endows it with supervised learning capability. This investigation will highlight the potential of analog DNA displacement reaction circuits for implementing artificial intelligence.