Outer–Inner Layer Separation for Third-Order Mixed-Derivative PINNs with Bilinear Lagging

Higher-order bilinear PDE residuals pose two coupled challenges for physics-informed neural networks (PINNs). First, high-order derivatives entangle parameters across layers. Second, nonlinear products typically destroy the least-squares structure that makes Layer Separation (LySep) computationally attractive. We introduce an outer–inner LySep framework that addresses both. At each outer step, one bilinear factor is frozen, yielding a linear variable-coefficient PDE. An inner third-order LySep solver then reconstructs mixed derivatives through order three using an auxiliary hierarchy \(a\!\to\!d\!\to\!q\!\to\!r\), while retaining a conditional least-squares output block. We establish exact auxiliary recovery and state-dependent residual consistency, and characterize the quartic obstruction that motivates outer lagging. Experiments span an intrinsically third-order forced KdV-type equation, hard-solenoidal Navier–Stokes, and resistive MHD. On KdV, the direct third-order route yields lower solution and original-PDE residual errors in all five paired seeds than both a matched low-order LySep route and a MIM-style first-order mixed-residual PINN, with median comparator-to-direct ratios of 1.95–2.97 for solution error and 2.47–3.61 for original-PDE residual. On hard-solenoidal Navier–Stokes, all five paired initializations give lower velocity, nonlinear-momentum, and pressure-gradient errors with LySep than with a matched end-to-end PINN. The MHD comparison likewise yields lower field and nonlinear-residual errors for LySep in all five pairs.