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

TL;DR

This paper introduces a Frobenius-optimal projection method to enforce linear conservation laws in learned linear dynamical models, ensuring invariants are exactly preserved with minimal perturbation.

Contribution

The authors derive a unique, low-rank correction to a learned operator that enforces linear invariants exactly, generalizing to multiple invariants and reducing to a rank-one update in the single-invariant case.

Findings

The correction enforces exact conservation laws in learned models.

The method minimally perturbs the original dynamics.

Numerical experiments confirm the theoretical properties.

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.