BlinDNO: A Distributional Neural Operator for Dynamical System Reconstruction from Time-Label-Free data

We study an inverse problem for stochastic and quantum dynamical systems in a time-label-free setting, where only unordered density snapshots-sampled at unknown times drawn from an observation-time distribution νt-are available. These observations induce a distribution over state densities, from which we seek to recover the parameters of the underlying evolution operator. We formulate this as learning a distribution-to-function neural operator and propose BlinDNO, a permutation-invariant architecture that integrates a multiscale U-Net encoder with attention-based mixer. Numerical experiments on a wide range of stochastic and quantum systems-including a 3D protein-folding mechanism reconstruction problem in cryo-EM setting-demonstrate that BlinDNO reliably recovers governing parameters and consistently outperforms existing neural inverse operator baselines.