In the field of distributed parameter systems, fault detection methods have typically relied on complete model knowledge. However, in many industrial scenarios, constructing accurate first-principle physical models is highly challenging, which limits the practical use of traditional model-based techniques. To address this issue, an adaptive neural network is developed to jointly estimate both the system states and the unknown nonlinear dynamics for a class of nonlinear distributed parameter systems with partial model information. Since full-state measurements are often impractical in real applications, the designed adaptive neural observer is built upon a reduced-order model, which also enhances computational efficiency. Fault detection is then realized through residual generation and evaluation based on the output estimation error of the proposed observer. Furthermore, to account for the neglected fast dynamics, a data-driven strategy for threshold generation is introduced. Comprehensive experimental studies are carried out to demonstrate and verify the effectiveness of the proposed approach.