The core challenge of the Unique Continuity (UC) problem lies in inferring solutions across an entire domain using limited observational data, holding significant practical implications for multiphysics coupled models. Recently, physics-informed neural networks (PINNs) have shown considerable promise in addressing the UC problem. However, the reliance on a fixed activation function and a fixed weighted loss function prevents PINNs from adequately representing the multiphysics characteristics embedded in coupled models. To overcome these limitations, we propose a novel dual adaptive neural network (DANN) algorithm. This approach integrates trainable adaptive activation functions and an adaptively weighted loss scheme, enabling the network to dynamically balance the observational data and governing physics. Our method is applicable not only to the UC problem but also to general forward problems governed by partial differential equations. Furthermore, we provide a theoretical foundation for the algorithm by deriving a generalization error estimate, discussing the potential causes of neural networks solving this problem. Extensive numerical experiments including 3D demonstrate the superior accuracy and effectiveness of the proposed DANN framework in solving the UC problem compared to standard PINNs.