Rankine–Hugoniot Front Tokens

(1) Integration point: insert the module between the encoder and decoder of a 1D Fourier neural operator or token transformer for time stepping. Given a grid state $u^n\in\mathbb{R}^N$, detect candidate fronts using $|u_{i+1}^n-u_i^n|$ above a threshold, cluster adjacent flagged cells, and create one token per cluster containing position $x_j^n$, estimated traces $u_j^-$ and $u_j^+$, and local pooled features. Use cross-attention from grid points to front tokens. (2) Pseudocode: compute front tokens; for each token evaluate $s_j=(F(u_j^+,x_j)-F(u_j^-,x_j))/(u_j^+-u_j^-)$; set $x_j\leftarrow x_j+\Delta t s_j$; use a small MLP to update $u_j^-$ and $u_j^+$ from attended context; render a piecewise front field; predict a smooth residual $R_\theta$ with the FNO; return front field plus residual. Merge or delete tokens when fronts collide or their jump falls below a threshold. (3) Computed from the mathematics: the balance-law interface and the front speed. Estimated empirically: front locations, traces, detection threshold, and the residual network. If $F$ is known, evaluate it exactly; otherwise use a differentiable supplied flux model. (4) First experiment: train on scalar Burgers data $F(u)=u^2/2$, optionally with a spatial source $S(r,u)$, using 64–256 grid points and randomly varying shock/rarefaction initial conditions. Compare against an equal-parameter FNO, a transformer without front tokens, and a standard residual physics-informed model. Equalize training steps, batch size, and rollout FLOPs. Measure $L^1/L^2$ field error, shock-position error, and error after 10–50 autoregressive steps. (5) Pre-register mechanism tests: at fixed grid resolution, the proposed model must reduce median shock-position error by at least 30% relative to the FNO while having no more than 10% worse smooth-region error; its one-step front-speed error must scale approximately linearly with $\Delta t$ over three step sizes, with log-log slope in $[0.8,1.2]$; and removing only the analytical update, replacing $x_j^{n+1}$ by a learned position head, must increase long-rollout front error by at least 25%. Falsification occurs if the analytical update gives no improvement over learned positions, violates the measured Rankine–Hugoniot speed by more than 10% on clean shocks, or improves field loss only through substantially higher compute.