Jin–Xin shock homotopy for PINNs

(1) Integration point: modify a standard one-dimensional PINN so its MLP outputs $(\hat{u},\hat{v})$ rather than only $\hat{u}$. Use automatic differentiation to obtain $u_t,v_t,u_x,v_x$. For Burgers, set $f(u)=u^2/2$ and use a scalar relaxation speed $A=a$.

(2) Pseudocode: choose $\varepsilon_0>\varepsilon_1>\cdots>\varepsilon_K$, for example $\varepsilon_{k+1}=q\varepsilon_k$ with $q=0.5$. Initialize the network and impose $v(0,x)=f(u_0(x))$. For stage $k$, sample collocation points $(t_j,x_j)$, compute derivatives, form $R_1=u_t+v_x$ and $R_2=\varepsilon_k(v_t+a^2u_x)-f(u)+v$, and minimize $$L_k=\operatorname{MSE}(R_1)+\operatorname{MSE}(R_2)+\lambda_{IC}L_{IC}+\lambda_{BC}L_{BC}.$$ Stop when validation loss stagnates, retain the weights, reduce $\varepsilon$, and continue from those weights. Use the same collocation distribution and optimizer budget for every baseline. Because derivatives can grow in the sharp-layer stages, optionally clip gradient norm and increase collocation density near locations where $|u_x|$ is large.

(3) Compute from the paper's mathematics: the coupled residuals, equilibrium flux initialization, continuation direction, and the admissibility requirement on $a$. Estimate the Burgers sub-characteristic speed from the current state range using $a=(1+\delta)\max_j|u_j|$, with $\delta$ between $0.1$ and $0.5$; freeze or update $a$ only between stages to avoid introducing discontinuous optimization targets. Estimate the limiting error by comparing the final prediction to a high-resolution TVD or WENO finite-volume entropy solution. Fit the observed error over several final values of $\varepsilon$ to test whether it decreases approximately like $\varepsilon^{1/4}$.

(4) First cheap experiment: solve Burgers' equation on $t\in[0,0.4]$, $x\in[-1,1]$ with Riemann initial data using a four-layer tanh MLP. Compare direct training at $\varepsilon=10^{-3}$, fixed-parameter relaxation training, and the schedule $1,0.3,0.1,0.03,0.01,0.003$. Keep total gradient evaluations and collocation counts equal. Measure $L^1$ and $L^2$ error, shock-location error, residual variance in a window around the shock, and the number of optimizer divergences. The method succeeds if it gives lower shock error and smoother loss descent at equal compute, particularly when direct PINN training produces oscillations, excessive smoothing, or convergence to an incorrect averaged discontinuity.