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With the aid of neural networks, this article develops two data-driven designs of fault detection (FD) for dynamic systems. The first neural network is constructed for generating residual signals in the so-called finite impulse response (FIR) filter-based form, and the second one is designed for recursively generating residual signals. By theoretical analysis, we show that two proposed neural networks via self-organizing learning can find their optimal architectures, respectively, corresponding to FIR filter and recursive observer for FD purposes. Additional contributions of this study lie in that we establish bridges that link model- and neural-network-based methods for detecting faults in dynamic systems. An experiment on a three-tank system is adopted to illustrate the effectiveness of two proposed neural network-aided FD algorithms.