We develop a data-driven framework for identifying non-Markovian equations of motion for open quantum systems, demonstrated here for qubit-environment dynamics. Starting from the Nakajima–Zwanzig formalism, we vectorize the reduced density matrix into a four-dimensional state vector and cast the dynamics as a Volterra integro-differential equation with an operator-valued memory kernel. The learning task is then formulated as a constrained optimization problem over the admissible operator space, where correlation functions are approximated by rational functions using Padé approximants. We establish well-posedness of the learning problem, ensuring existence of minimizers. To assess performance, we construct synthetic datasets from representative test problems of increasing complexity: (i) exactly solvable pure dephasing, with correlation functions expressed in terms of special functions, (ii) a damped Jaynes–Cummings model with an analytic coherence kernel, (iii) a transverse Born model with frequency-resolved bath integrals and population–coherence coupling, and (iv) a nonrotating-wave quantum Rabi model whose memory kernel has no closed form. Numerical experiments demonstrate that Padé captures nontrivial temporal structures such as oscillatory memory, algebraic tails, and phase-sensitive coherence transfer and that the learned models generalize across ensembles of physically admissible initial states. We perform a parametrization-invariant sensitivity analysis and show that the trajectories are insensitive to the unrecoverable parts of the kernel, so the learned models stay predictive despite severe ill-conditioning in kernel recovery. These results together illustrate that data-driven rational approximation provides an effective route to identifying non-Markovian kernels of practical relevance in quantum technologies.

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• Open quantum systems & decoherence