<jats:p>Photonic systems excel at performing linear computations, such as matrix-vector multiplications, in a highly parallel and energy-efficient manner. However, implementing nonlinear computations in photonic systems remains challenging without relying on optoelectronic conversions or nonlinear/active materials, both of which are energy-intensive. Here, we present a nonlinear computing approach for time series processing. This approach enables energy-efficient and nonlinear computations of large-scale optical networks within a single linear (passive) microcavity by leveraging the interplay between cavity modes and an optical phase-encoded input signal and facilitates an on-chip implementation on a silicon photonic platform. We experimentally demonstrate higher-order nonlinear computational capacity using a silicon photonic microcavity, achieving superior performance in time-dependent processing tasks, such as chaotic time-series prediction and nonlinear observer tasks.</jats:p>