TrajectoryOp: Explicit-Interface Operator Learning for Moving Shocks

Neural operators provide rapid predictions for families of PDE solutions, but moving shocks remain difficult to represent and learn from limited supervised numerical data. We introduce , an explicit-interface operator for conservation laws with a single tracked shock. TrajectoryOp consists of an FNO encoder, two state decoders, and a trajectory network. The FNO encoder extracts features from the discrete initial field, the two state decoders represent the primitive states on the two sides of the shock, and the trajectory network predicts the shock trajectory from the initial shock location. The solution is represented by the two regional states separated by the predicted shock interface. Regional PDE residuals constrain the states inside the two predicted regions, and the Rankine–Hugoniot condition couples these states to the shock motion along the predicted interface over the full time interval, including where numerical state labels are unavailable, thereby reducing the required amount of supervised numerical data. Compared with fully-supervised operator learning methods, proposed achieves almost performance with only about 3% supervised information, demonstrating significant advantages under such sparse supervised case.