First-Order Flux Formulations Enable Hard-Spiking Physics-Informed Neural Networks

Direct physics-informed training of spiking networks with genuinely binary internal spikes is widely reported to be infeasible. Surrogate gradients are blamed, but the literature does not show how they fail. We show the mechanism on the viscous Burgers equation with a hard-threshold quadratic integrate-and-fire (QIF) backbone. The classical strong-form loss built on u_xx is unstable, with the training loss climbing seventeen orders of magnitude, yet the identical u_xx formed by central differences of forward passes trains without incident. The instability therefore belongs to the surrogate second derivative, not to the binary spikes. The surrogate first derivative is not blameless either. Applied once, it trains stably while biasing the shock speed. Guided by this mechanism, we rewrite the PDE as two first-order flux equations coupled by a pointwise closure. The reformulation restores trainability under unchanged surrogate autodiff, and replacing autodiff with conservative central differences of the network's own forward passes improves accuracy further. Conservative differencing matches autodiff on shock formation, trains from random initialization on a single CPU, and, warm-started, reduces error six-fold on shock propagation over three seeds, where autodiff places the front more than a full shock width off at comparable residuals. We also prove that the conservation residual is blind to the gauge q -> q + h(t). Removing the closure drives it 300x below the honest optimum while the solution is badly wrong, so, in the two-field form, the closure is a correctness requirement rather than a modeling choice.