Learning dynamically consistent flow reconstructions from limited observations
A core inverse problem in the experimental sciences is the inference of a hidden dynamical state from sparse or indirect measurements. There is a natural opportunity for deep learning methods here, but machine-learnt reconstruction methods typically require full state data for training. We present Trajectory-Consistent Network Training (TraCTra), a label-free framework that trains reconstruction networks using only partial observation sequences and a differentiable forward model. TraCTra requires the network reconstruction and dynamical evolution to be mutually consistent: network-predicted states are marched forward in time to match subsequent observations and to agree in the full state space with independent reconstructions at later times. Across four fluid systems, the same objective reconstructs three-dimensional turbulence from coarse-grained fields, velocity from observations of density fluctuations, and three-dimensional density and velocity from sequences of projected two-dimensional shadowgraphs, while also recovering global vorticity from observations confined to a small spatial window. TraCTra outperforms assimilation-only and physics-informed neural approaches, preserves dynamically important multiscale structure, and remains accurate beyond the optimisation window. It transfers to held-out times in the three-dimensional shadowgraph problem and, when trained across trajectories, generalises to unseen flows in the two-dimensional problem. The results establish trajectory consistency as a general supervision principle for reconstructing hidden dynamical states without full state training targets.