Tracking the Discontinuity: Explicit Front Tracking for Neural Euler Solvers

Physics-informed neural networks smear shocks because of their objective, not their optimiser: across a sharp shock whose two states satisfy the Rankine-Hugoniot conditions, the pointwise strong-form residual scales as the inverse of the width, so its squared norm is minimised by spreading the jump. We develop explicit front tracking for the compressible Euler equations, in which smooth networks are separated by fronts that move by their jump conditions and the residual is never evaluated at a discontinuity. Four components make this work for a system with several interacting wave families. The number, characteristic family and admissibility of the fronts are derived from the exact star state of the Riemann data, with a contact kept only if it carries strength. Two-dimensional fronts are level sets anchored to the initial discontinuity, with the jump conditions imposed along the learned normal at points carried onto the front by Newton projection. The plateaus beside a shock, which lie in the null space of the residual, are anchored through the jump conditions to the exact far-field states rather than to neighbouring learned ones. And the best of several runs is selected by an uncalibrated physics score, without ground truth. On Riemann problems with discontinuities the method is 8 to 14 times more accurate than strong-form and relaxation networks at the same budget on Sod and 386 to 879 times on a problem of two shocks and a contact, anchoring improves the latter a further sixfold, and selection finds the most accurate of three seeds in 11 of 13 settings, against one in three by chance. The method reproduces a planar oblique shock in two dimensions and, on a nonconservative system, realises the jump condition of each prescribed path, where a standard path-conservative scheme responds with the wrong sign. It does not yet handle a curved front, and a classical solver remains three to four orders of magnitude cheaper whenever a rarefaction is present.