A Method for Linearization of Systems of Quadratic Equations
Abstract
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 ui(x)−vi(x)=0. Recasting raw degree-α 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 (α=2) and Hazen-Williams (α=1.852) models. We implement dynamic sign partitioning in a FlowDEQ layer using a reduced incidence matrix and implicit-function adjoint differentiation.
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BibTeX
@misc{pak2026method,
title = {{A Method for Linearization of Systems of Quadratic Equations}},
author = {A. A. Pak and A. S. Pak},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25506},
eprint = {2610.25506},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25506},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A Method for Linearization of Systems of Quadratic Equations
AU - A. A. Pak
AU - A. S. Pak
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25506
UR - https://arxiv.org/abs/2610.25506
AB - 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(x)−v_i(x)=0. Recasting raw degree-α 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 (α=2) and Hazen-Williams (α=1.852) models. We implement dynamic sign partitioning in a FlowDEQ layer using a reduced incidence matrix and implicit-function adjoint differentiation.
ER -