A Method for Linearization of Systems of Quadratic Equations

Integrating physical conservation laws into deep neural architectures remains a central SciML challenge. Soft-penalty approaches, including many PINNs, may be sensitive to loss weighting and optimization; generic DEQ and implicit-layer solvers may require instance-dependent iteration counts. We study a norm-linearization scheme for networked systems with linear continuity constraints and nonlinear quadratic or power-law energy balances. We derive sharp universal and finite-$n$ bounds on relative variance and, via the weighted Kantorovich inequality, relate them to worst-case single-step squared-norm distortion when target and operating states have a bounded multiplicative ratio. Euler's theorem shows that the scheme is Newton's method for degree-one norm differences $u_i(\mathbf{x})-v_i(\mathbf{x})=0$. Recasting raw degree-$\alpha$ balances as norm differences removes radial curvature, yields a degree-zero homogeneous Jacobian, eliminates the affine Newton term, and row-equilibrates the loop Jacobian at equilibrium. Under regularity and nonsingularity assumptions, this gives local quadratic convergence for Darcy-Weisbach ($\alpha=2$) and Hazen-Williams ($\alpha=1.852$) models. We implement dynamic sign partitioning in a FlowDEQ layer using a reduced incidence matrix and implicit-function adjoint differentiation.