Physics-Informed Regression: Parameter Estimation in Parameter-Linear Nonlinear Dynamic Models

The paper derives the PIR estimator by ordinary least squares for stacked residuals r_I(ω) = A_I ω − b_I, giving ω*_I = (A_I^T A_I)^{-1} A_I^T b_I, and, when partitioning known vs. unknown parameters, ω*_u,I = (A_u^T A_u)^{-1} A_u^T (b_I − A_k ω_k) (their eqs. (7) and (10)). It also presents a ridge-regularized closed form in the appendix. The candidate solution reproduces the same normal-equation derivations with explicit convexity/Hessian arguments and correctly notes rank conditions and an optional pseudoinverse remark. Minor differences are expository (the paper assumes full rank and refers to an appendix proof; the model spells out the gradient/Hessian and highlights that ridge ensures strict convexity even if A is rank-deficient). Overall, the two are consistent and essentially the same proof technique, with the model adding standard clarifications. Citations: equations (6)–(10) and (7) in the paper’s PIR section , single-time OLS formula (5) , and ridge-regularized solution in Appendix 11.3 .