Frobenius-Optimal Projection for Conserving Linear Dynamical Models
Analysis
This paper addresses a crucial problem in data-driven modeling: ensuring physical conservation laws are respected by learned models. The authors propose a simple, elegant, and computationally efficient method (Frobenius-optimal projection) to correct learned linear dynamical models to enforce linear conservation laws. This is significant because it allows for the integration of known physical constraints into machine learning models, leading to more accurate and physically plausible predictions. The method's generality and low computational cost make it widely applicable.
Key Takeaways
- •Proposes a Frobenius-optimal projection method to enforce linear conservation laws in learned linear dynamical models.
- •The method is computationally efficient and guarantees exact conservation.
- •It minimally perturbs the learned dynamics.
- •Applicable to any learned linear model and provides a general mechanism for embedding exact invariants.
“The matrix closest to $\widehat{A}$ in the Frobenius norm and satisfying $C^ op A = 0$ is the orthogonal projection $A^\star = \widehat{A} - C(C^ op C)^{-1}C^ op \widehat{A}$.”