In the talk, we will present a method for constructing a special type of shallow neural network that learns univariate meromorphic functions with pole-type singularities. In contrast to known research, our neural network approximant is a discontinuous function itself, and singularities of the neural network capture singularities of the function under consideration. To achieve these properties, we introduce a concept of "unsafe" PAUs (Padé Activation Units): an adaptive construction of rational activation functions as meromorphic functions with a single pole each, situated within the domain of investigation. Furthermore, we present a novel backpropagation-free method for determining the weights and biases of the hidden layer from the parameters of rational Laurent-Padé approximation. Application of exactly the Laurent-Padé method ensures locally uniform convergence of our neural network approximant. Employing the weights and biases of the hidden layer, we then scale and shift, respectively, the pole of the activation function to find the estimated locations of the singularities. While the weights and biases of the hidden layer are tuned so as to capture the singularities, the least-squares fitting for the computation of weights and biases of the output layer ensures approximation of the function in the rest of the domain. Our method is based on using a finite set of Laurent coefficients as input information, which we compute by FFT, employing values of the investigated function on some closed contour in the complex plane. We will illustrate the effectiveness of our method through numerical experiments, including the construction of extensions of the time-dependent solutions of nonlinear PDEs into the complex plane, and study the dynamics of their singularities.