  Characterizing the memory properties of the environment has become critical
for the high-fidelity control of qubits and other advanced quantum systems.
However, current non-Markovian tomography techniques are either limited to
discrete superoperators, or they employ machine learning methods, neither of
which provide physical insight into the dynamics of the quantum system. To
circumvent this limitation, we design learning architectures that explicitly
encode physical constraints like the properties of completely-positive
trace-preserving maps in a differential form. This method preserves the
versatility of the machine learning approach without sacrificing the efficiency
and fidelity of traditional parameter estimation methods. Our approach provides
the physical interpretability that machine learning and opaque superoperators
lack. Moreover, it is aware of the underlying continuous dynamics typically
disregarded by superoperator-based tomography. This paradigm paves the way to
noise-aware optimal quantum control and opens a path to exploiting the bath as
a control and error mitigation resource.
