Surrogate models, or machine learning based emulators of simulators, have been shown to be a powerful tool for accelerating simulations. However, capturing the system response of general nonlinear systems is still an open area of investigation. In this paper we propose a new surrogate architecture which is capable of capturing the input/output response of causal models to automatically replace large aspects of block model diagrams with neural-accelerated forms. We denote this technique the Nonlinear Response Continuous-Time Echo State Network (NR-CTESN) and describe a training mechanism for it to accurately predict the simulation response to exogenous inputs. We then describe a science-guided or physics-informed surrogate architecture based on Cellular Neural Networks to enable the NR-CTESN to accurately reproduce discontinuous output signals. We demonstrate this architecture on an inverter circuit and a Sky130 Digital to Analog Converter (DAC), showcasing a 9x and 300x acceleration of the respective simulations. These results showcase that the NR-CTESN can learn emulate the behavior of components within composable modeling frameworks and thus be reused in new applications without requiring retraining. Together this showcases a machine learning technique that can be used to generate nonlinear model order reductions of model components in SPICE simulators, Functional Markup Interface (FMI) representations of causal model components, and beyond.