Equilibria for Networks of Linear Translational Springs

Luke Oeding; Ethan Clayton; Jackson Elsea; Nicholas Wang; Adam Rutkowski

math.AGSep 2, 2026

Equilibria for Networks of Linear Translational Springs

Authors: Luke Oeding, Ethan Clayton, Jackson Elsea, Nicholas Wang, Adam Rutkowski

Organizations: Auburn University, Department of Mathematics and Statistics, Auburn, AL, USA · New York University, Courant Institute of Mathematical Sciences, New York, NY, USA · Air Force Research Laboratory, Eglin Air Force Base, Florida, USA

Abstract

We use tools from nonlinear algebra to study the equilibria of small linear translational spring networks. Specifically we use the techniques of homotopy continuation, monodromy, and parameter homotopy (a.k.a. cheater homotopy) to solve all rigid linear translational spring networks up to $5$ nodes in both $2$ and $3$ dimensions. We describe a method of implementing parameter homotopy that arises naturally from the physical structure of the system. We give precise total degree bounds on the maximum number of solutions for general planar spring networks. We discuss further efficiency gains obtained from polyhedral homotopy methods. We compare the computation efficiency of these techniques against a baseline of Newton's method.

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Dual-Network PINNs for Optimal Control: A Reproducible Benchmark on the Mass-Spring-Damper System

Abdeladhim Tahimi, Rinaldo Vieira da Silva Junior

This work presents a transparent and reproducible benchmark study of a direct dual-network Physics-Informed Neural Network (PINN) formulation for the optimal control of a mass-spring-damper system. The classical linear-quadratic optimal control problem is solved by two independent classical methods -- Pontryagin's Minimum Principle with single shooting, and direct transcription through trapezoidal collocation -- and recast as a constrained optimization problem solved by two feedforward neural networks: a state network whose boundary conditions are enforced exactly through a composite cubic-and-mask ansatz, and an unconstrained control network. The composite loss combines the physics residual at the collocation points with a trapezoidal approximation of the cost functional, weighted by a single scalar hyperparameter. On the benchmark considered, the PINN reproduces the classical optimal cost to four significant digits, satisfies the terminal state constraints exactly by construction, and produces pointwise state and control errors that fall within the spread of the two classical references. Training is approximately two orders of magnitude slower than classical shooting on this benchmark, which is honestly reported. The contribution is methodological clarity rather than methodological novelty: the formulation and the accompanying Google Colab implementation are intended to lower the barrier to entry for practitioners exploring PINN-based optimal control without prior exposure to adjoint methods or two-point boundary value problems.

Conventional Physics-Informed Neural Networks Optimal Control Neural Control Boundary Value Problems Hyperparameter Optimization Reproducibility

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Abstract

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