Conservation Laws / Constraint-Preserving Learning

Definition

This motif concerns systems where valid trajectories remain inside a feasibility set. A transfer is promising when the target AI problem also has quantities that should not drift under iteration, rollout, or decoding.

This motif is strongest when a violation is not merely undesirable but invalid. ISOM uses it to distinguish soft regularization from hard admissibility: some systems can tolerate drift, while others require exact bookkeeping across every update.

Mathematical Structure

The structure is a constrained dynamics problem: trajectories evolve under equations while preserving an invariant, a balance equation, or a hard feasibility region.

Physics Side

Physical systems often conserve mass, probability, charge, energy, or geometric admissibility. Those constraints do not merely regularize behavior; they define what valid behavior is.

AI Side

In AI this appears in planners, simulators, sequence models, or structured decoders where accumulated drift creates unstable rollouts. Constraint-preserving parameterizations can reduce correction cost and make failure easier to diagnose.

Candidate AI targets include simulators, planners, structured decoders, and long-horizon world models where small constraint errors compound. The transfer should be judged by rollout validity and repair cost, not only by average loss.

Failure Modes

The trap is enforcing a conserved quantity that the observed data do not truly obey, or that downstream tasks do not care about. Hard constraints can also make optimization brittle if the model class is too small.

The main editorial risk is mistaking a convenient metric for a conserved quantity. If the measured budget is only a proxy, exact preservation can lock the model into the wrong behavior and hide useful adaptation.

Open Questions

Which constraints should be exact and which should be soft? Can the model learn when to respect a conservation law and when to treat it as a broken approximation?