Weak-form modified sparse identification of nonlinear dynamics

The paper derives the weak-form residual, defines B and G, and replaces the mSINDy derivative-fit term with the weak residual to obtain L_WmSINDy(Ξ, x̃) = ||B(x̃) − G(x̃)Ξ||^2 + e_s, matching Eq. (18a,b) in the paper. It also states that the Ξ-update is performed by sparse regression using the weak convolution form (Algorithm 1, step 8), i.e., minimizing ||B(x̃) − G(x̃)Ξ|| subject to sparsity, which aligns with the candidate solution’s reduction of the Ξ-subproblem to sparse least-squares for fixed x̃. Minor implementation details (norm conventions, quadrature specifics) differ but are non-substantive. Citations: weak-form and linear system B=GΞ (Eqs. (7)–(13)) , WmSINDy loss (Eq. (18a,b)) , mSINDy loss (Eq. (4)) and flow-map usage (Eq. (5)) , and statement about sparse-regression Ξ-update in WmSINDy (text around Algorithm 1) .