A Frobenius-Optimal Projection for Enforcing Linear Conservation in Learned Dynamical Models
This provides a general mechanism for embedding exact invariants into learned linear models, which is incremental as it builds on existing projection methods for conservation laws.
The paper tackles the problem of restoring linear conservation laws in data-driven linear dynamical models by proposing a Frobenius-optimal projection to enforce exact invariants, showing that it minimally perturbs the dynamics and verifying this numerically on a Markov-type example.
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.