SODAs: Sparse Optimization for the Discovery of Differential and Algebraic Equations

The paper introduces SODAs and explicitly claims that the approach leads to a sequence of convex optimization problems while discovering algebraic relations first and then dynamics; this is supported by the per-column sparse-fitting formulation and the use of convex choices like LASSO in practice . However, the same section and Algorithm 1 also display an ℓ0 penalty in the loss and a sequential-thresholding heuristic, which are non-convex globally (even if each LASSO subproblem is convex) . The paper provides no formal correctness theorem about exact algebraic recovery or nullspace dimension reduction; it offers an algorithmic pipeline (including equation reduction, factor removal, and SVD-based stopping) and empirical demonstrations instead . The candidate model solution presents an idealized proof sketch asserting: (A) convexity of each subproblem under convex penalties; (B) strict nullspace shrinkage via factor removal, yielding exact algebraic recovery after K iterations; and (C) exact dynamic recovery under full rank and standard sparsity conditions. While (A) is conditionally correct (for convex penalties), (B)–(C) rely on strong assumptions (e.g., correct selection of relations from the true algebraic span, uniqueness via full-rank dynamic columns, and sparse-identifiability conditions) that are not justified by the paper nor proven by the model. As a result, both the paper (no formal proof) and the model (missing and unverified hypotheses) are incomplete.