# A Frobenius-Optimal Projection for Enforcing Linear Conservation in Learned Dynamical Models

arXiv: https://arxiv.org/abs/2512.22084 · HTML: https://mlsanity.com/paper/2512.22084/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
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- First posted on arXiv 26 Dec 2025.

## Abstract (arXiv)

We consider the problem of restoring linear conservation laws in data-driven linear dynamical models. Given a learned operator widehatA and a full-rank constraint matrix C encoding one or more invariants, we show that the matrix closest to widehatA in the Frobenius norm and satisfying C^top A = 0 is the orthogonal projection A^star = widehatA - C(C^top C)-1C^top widehatA. This correction is uniquely defined, low rank and fully determined by the violation C^top widehatA. 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.