Inverse problems in physical or biological sciences often involve recovering an unknown parameter that is random. The sought-after quantity is a probability distribution of the unknown parameter, that produces data that aligns with measurements. Consequently, these problems are naturally framed as stochastic inverse problems. In this paper, we explore three aspects of this problem: direct inversion, variational formulation with regularization, and optimization via gradient flows, drawing parallels with deterministic inverse problems. A key difference from the deterministic case is the space in which we operate. Here, we work within probability space rather than Euclidean or Sobolev spaces, making tools from measure transport theory necessary for the study. Our findings reveal that the choice of metric -- both in the design of the loss function and in the optimization process -- significantly impacts the stability and properties of the optimizer.

The intersection of machine learning and dynamical systems has generated considerable interest recently. Neural Ordinary Differential Equations (NODEs) represent a rich overlap between these fields. In this paper, we develop a continuous time neural network approach based on Delay Differential Equations (DDEs). Our model uses the adjoint sensitivity method to learn the model parameters and delay directly from data. Our approach is inspired by that of NODEs and extends earlier neural DDE models, which have assumed that the value of the delay is known a priori. We perform a sensitivity analysis on our proposed approach and demonstrate its ability to learn DDE parameters from benchmark systems. We conclude our discussion with potential future directions and applications.