KF-SPINN: Separable Physics-informed Neural Networks for Kinetic-Fluid regimes

Kinetic equations, such as the BGK model of the Boltzmann equation, describe particle dynamics across both continuum and rarefied regimes within a single partial differential equation (PDE), but the additional microscopic velocity dimension makes classical grid-based solvers computationally expensive. Physics-informed neural network (PINN)-based approaches, such as separable PINNs, offer a promising mesh-free alternative for high-dimensional PDEs. However, existing PINNs for BGK solvers struggle in the low-Knudsen-number ($\mathrm{Kn}$) regime, where rapid kinetic relaxation and sharp, evolving shock layers make training challenging. In this work, we present $\text{KF-SPINN}$, a framework that addresses these difficulties through complementary solution and input representations. Specifically, KF-SPINN (i) structurally encodes fast relaxation through the Chapman–Enskog-inspired solution ansatz, (ii) incorporates multiscale, time-modulated lifting features aligned with shock geometry to capture evolving shock profiles, and (iii) enhances training through conservation and entropy-transport residuals combined with a self-adaptive Charbonnier penalty. KF-SPINN improves robustness across the tested low-Kn regime and reduces mean macroscopic prediction errors by up to two orders of magnitude on 1D3V shock benchmarks relative to the strongest baseline evaluated, while keeping GPU memory requirements and training times modest.