{"ID":2825043,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.22084","arxiv_id":"2512.22084","title":"A Frobenius-Optimal Projection for Enforcing Linear Conservation in Learned Dynamical Models","abstract":"We consider the problem of restoring linear conservation laws in data-driven linear dynamical models. Given a learned operator $\\widehat{A}$ and a full-rank constraint matrix $C$ encoding one or more invariants, we show that the matrix closest to $\\widehat{A}$ in the Frobenius norm and satisfying $C^\\top A = 0$ is the orthogonal projection $A^\\star = \\widehat{A} - C(C^\\top C)^{-1}C^\\top \\widehat{A}$. This correction is uniquely defined, low rank and fully determined by the violation $C^\\top \\widehat{A}$. In the single-invariant case it reduces to a rank-one update. We prove that $A^\\star$ enforces exact conservation while minimally perturbing the dynamics, and we verify these properties numerically on a Markov-type example. The projection provides an elementary and general mechanism for embedding exact invariants into any learned linear model.","short_abstract":"We consider the problem of restoring linear conservation laws in data-driven linear dynamical models. Given a learned operator $\\widehat{A}$ and a full-rank constraint matrix $C$ encoding one or more invariants, we show that the matrix closest to $\\widehat{A}$ in the Frobenius norm and satisfying $C^\\top A = 0$ is the...","url_abs":"https://arxiv.org/abs/2512.22084","url_pdf":"https://arxiv.org/pdf/2512.22084v1","authors":"[\"John M. Mango\",\"Ronald Katende\"]","published":"2025-12-26T17:11:16Z","proceeding":"math.DS","tasks":"[\"math.DS\",\"cs.LG\",\"math.NA\"]","methods":"[]","has_code":false}