The study of chaotic dynamical systems, marked by their unpredictable behavior and sensitivity to initial conditions, stands at the forefront of mathematical and physical sciences. Traditional forecasting methods often fall short in capturing the essence of chaos, leading to a significant gap in our understanding and ability to predict such phenomena accurately. Addressing this gap, we introduce the Discrete-Temporal Sobolev Network (DTSN), specifically designed to enhance the accuracy of forecasting chaotic systems. The DTSN revolutionizes the analysis of chaotic systems by employing a unique loss function inspired by the temporal Sobolev norm. We demonstrate that DTSN has capability in forecasting the intricate dynamics of chaotic systems such as the Lorenz-63 system, Standard Map and the Chua circuit.