What Does Exact Global Mass Balance Imply in Open-System Neural PDE Surrogates?

Exact global mass balance constrains one scalar property of a predicted field: its total mass. In an open system, sources and boundary exchanges change the required total mass from one step to the next. We develop a diagnostic framework to examine how specification of this total separates from spatial prediction error in finite-volume neural PDE surrogates. The results show that global mass balance must be assessed through both target specification and the field error that remains. A misspecified target can be satisfied to numerical precision. Even when the target is correct, the mass-changing error fraction varies widely across predictions, and fields satisfying this target can still differ substantially in spatial accuracy. Furthermore, omitting prescribed external mass increments can severely degrade autoregressive rollout stability. These distinctions appear across different mesh geometries and model constructions. Thus, a small absolute constraint residual only establishes agreement with the supplied target mass, without showing that the target was specified correctly or that the predicted field is accurate. Therefore, evaluating conservation-aware surrogates requires reporting how the supplied target mass is specified, the absolute constraint residual, spatial error, and rollout behavior together.