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

Source: arXiv - 2512.22084v1

Overview

The paper tackles a common pain point in data‑driven modeling of dynamical systems: learned linear models often drift away from known physical invariants (e.g., mass, charge, probability). The authors present a simple, closed‑form “Frobenius‑optimal” projection that forces any learned linear operator to satisfy linear conservation constraints while staying as close as possible to the original model.

Key Contributions

• Closed‑form optimal correction – Derivation of the unique matrix

  [ A^\star = \widehat{A} - C(C^\top C)^{-1}C^\top \widehat{A} ]

  that minimizes the Frobenius distance to the learned operator (\widehat{A}) under the linear constraint (C^\top A = 0).

• Low‑rank, rank‑one special case – Shows that when there is a single invariant the correction reduces to a rank‑one update, making it computationally cheap.

• Theoretical guarantees – Proof that the projected operator enforces exact conservation and is the minimal perturbation in the Frobenius norm sense.

• Numerical validation – Demonstrates the method on a Markov‑type system, confirming exact invariant preservation and negligible impact on predictive performance.

• Broad applicability – The projection can be plugged into any pipeline that yields a linear model (e.g., system identification, Koopman operator learning, linearized neural ODEs).

Methodology

1. Problem setup – Assume a learned linear dynamics matrix (\widehat{A}) (e.g., from regression, DMD, or a neural network linearization). A full‑rank constraint matrix (C\in\mathbb{R}^{n\times m}) encodes (m) linear invariants such that any admissible dynamics must satisfy (C^\top A = 0).
2. Optimization formulation – Find the matrix (A) that (i) satisfies the constraint and (ii) minimizes (|A-\widehat{A}|_F). This is a classic constrained least‑squares problem.
3. Derivation of the projector – Using Lagrange multipliers and the fact that (C) has full column rank, the optimal solution is obtained analytically as the orthogonal projection of (\widehat{A}) onto the subspace ({A\mid C^\top A = 0}). The resulting expression is the simple matrix subtraction shown above.
4. Implementation notes – Computing (C(C^\top C)^{-1}C^\top) is a one‑time cost; after that the correction is a matrix‑multiply with (\widehat{A}). For a single invariant, the term collapses to an outer product, i.e., a rank‑one update.

Results & Findings

• Exact conservation – After projection, the invariants encoded by (C) are satisfied to machine precision, eliminating drift that would otherwise accumulate over long simulations.
• Minimal disturbance – The Frobenius norm of the difference (|A^\star-\widehat{A}|_F) equals the norm of the violation (C^\top\widehat{A}), confirming the theoretical claim of minimal perturbation.
• Empirical test – On a synthetic Markov chain with a known probability‑mass conservation law, the uncorrected model slowly leaked probability, while the projected model preserved total probability exactly, with virtually identical prediction error on short‑term forecasts.

Practical Implications

• Safety‑critical simulations – Engineers can enforce energy, mass, or probability conservation in learned controllers or simulators without redesigning the learning algorithm.
• Model‑based reinforcement learning – Embedding conservation constraints can improve stability of learned dynamics used for planning, especially in robotics where momentum or volume preservation matters.
• Koopman operator learning – Many recent works approximate nonlinear dynamics with linear operators in lifted spaces; the projection offers a plug‑and‑play step to guarantee that linear invariants (e.g., total charge) survive the lifting.
• Data‑efficient system identification – When only a few trajectories are available, imposing known invariants can regularize the identification problem, leading to more physically plausible models.
• Low computational overhead – The correction is a simple matrix multiplication (or rank‑one update), making it suitable for real‑time pipelines or large‑scale problems where re‑training would be prohibitive.

Limitations & Future Work

• Linear invariants only – The method enforces linear conservation laws; extending the idea to nonlinear invariants (e.g., quadratic energy) would require more sophisticated projections.
• Assumes full‑rank constraint matrix – If the invariants are linearly dependent, the current derivation breaks down; handling rank‑deficient (C) is an open question.
• Static correction – The projection is applied post‑hoc to a learned matrix. Integrating the constraint directly into the learning objective (e.g., via differentiable projection layers) could yield better data efficiency.
• Scalability to very high dimensions – While the rank‑one case is cheap, forming (C(C^\top C)^{-1}C^\top) may be costly for millions of states; future work could explore randomized or iterative approximations.

Bottom line: The Frobenius‑optimal projection offers a mathematically clean, computationally light way to inject exact linear conservation into any learned linear dynamical model—turning a common source of simulation error into a non‑issue for developers building physics‑aware AI systems.

Authors

• John M. Mango
• Ronald Katende

Paper Information

• arXiv ID: 2512.22084v1
• Categories: math.DS, cs.LG, math.NA
• Published: December 26, 2025
• PDF: Download PDF