Thermodynamic Reservoirs in Grid-Based Reduced Models

Definitions and mathematical statement

The supplied record asks about extending thermostat-based reduced modeling from ordinary differential equations to grid-based discretizations of partial differential equations. It does not specify a particular equation, discretization, thermostat, or approximation criterion.

For schematic coordinate bookkeeping only, let $I$ denote the index set of fine-grid unknowns, and write $x(t)=(x_i(t))_{i\in I}$ for a trajectory of the full discretized system. If $R\subseteq I$ indexes retained unknowns, their coordinate projection is \[ P_Rx(t)=(x_i(t))_{i\in R}. \] Under the usual retained/unresolved partition, the unresolved indices would be $I\setminus R$. This notation does not supply state spaces, equations of motion, or hypotheses absent from the source.

Source question. Can a thermostat-based reduced model for such a grid discretization use the unresolved fine-grid unknowns as its thermodynamic reservoir, while representing the reduced model by a subset of the fine-grid unknowns? Can the full trajectory then be approximated weakly by the resulting reduced dynamics?

Unresolved qualifications. The source wording describes the reduced model as a subset of "those unknowns," leaving unclear whether this refers to all fine-grid unknowns or specifically to the unresolved unknowns. Thus the retained/unresolved partition above is only a possible interpretation. The source also does not define what makes a collection of unknowns a thermodynamic reservoir, how the reduced dynamics or thermostat is constructed, or how a reduced trajectory is compared with a full trajectory of different dimension. No meaning of weak approximation, limiting parameter, time interval, admissible class of equations, or universal versus existential scope is specified. Consequently, the supplied evidence determines a research question but does not determine a single fully quantified mathematical proposition.

Short English statement

The question is whether unresolved fine-grid PDE variables can serve as a thermodynamic reservoir for a thermostat-based reduced model, and whether its dynamics can weakly approximate the full trajectory. The reservoir, retained variables, dynamics, and meaning of weak approximation require clarification before a precise theorem or conjecture can be stated.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001743$, source problem code catalogue entry, titled Thermodynamic Reservoirs in Grid-Based Reduced Models; collection: Oberwolfach Reports Open Problems. The record gives proposed year 2011 and names no proposer.
• Supplied source citation: Geometric Numerical Integration (2011), Oberwolfach Reports, pp. 825--900, DOI 10.4171/owr/2011/16. Source URL and supplied literature source labeled "original OWR report": https://doi.org/10.4171/owr/2011/16. No passage-specific page or named contributor is supplied.
• The supplied original statement contains the reservoir/subset question. The supplied corrected statement additionally asks about weak approximation of the full trajectory. The record labels its statement corrected_verified and reports that the official report confirms both questions in the extension from ordinary differential equations to grid-based PDE discretizations. These are supplied verification claims, not independent verification in this draft.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes the extracted sentence as an isolated model-reduction question lacking enough definitions to identify a theorem or resolved claim. Its listed source checked is the original OWR report linked above. This is supplied dated curation, not an independently verified literature assessment; time-sensitive status and cited claims require independent checking.