A data-driven model reduction approach for backward fractional diffusion-wave equations

Referee: [Abstract] Abstract: the central claim that the observation system solution is one-to-one with the forward-problem solution (and therefore shares the same linear structure in finite-dimensional space) is asserted without an explicit proof that the mapping commutes with the nonlocal time-fractional operator and with the subsequent discretization or projection step. Because the backward problem already amplifies instability, any loss of exact linearity under reduction would invalidate the transfer of theoretical guarantees to the forward problem.

Authors: We thank the referee for highlighting this important aspect. The one-to-one mapping is established in Section 3 of the manuscript by constructing the observation system directly from the data such that it satisfies the same governing equation as the forward problem but with observed quantities. This construction ensures that the mapping commutes with the fractional operator by design, as the observation system is derived to replicate the linear structure. However, to make this more explicit and address concerns about discretization, we will add a new subsection providing a commutation diagram and a rigorous proof of preservation of the linear structure under the finite-dimensional projection. This will include error bounds to ensure stability despite the backward problem's ill-posedness. revision: yes

Referee: [Theoretical results] Theoretical results (as summarized in the abstract): the statement that 'model reduction approaches constructed for the observation system also work well for the forward problem' rests on the unverified assumption that the finite-dimensional linear structure is identical and preserved; no explicit verification (e.g., commutation diagram or error bound) is supplied showing that the chosen reduction operator (POD, DMD, etc.) commutes with the fractional derivative after discretization.

Authors: We acknowledge that the current presentation of the theoretical results could benefit from more explicit verification. The manuscript derives that the observation system and forward problem share the same finite-dimensional linear operator due to the one-to-one mapping. To strengthen this, we will include an explicit commutation diagram in the revised version and provide error bounds demonstrating that the reduction operators (POD and DMD) commute with the discretized fractional derivative operator. This will clarify how the theoretical guarantees transfer. revision: yes

Referee: [Numerical examples] Numerical examples: the examples are said to 'support our finding,' yet the manuscript provides no details on the construction of the observation system from data, the fractional orders tested, the discretization parameters, or quantitative error metrics comparing the reduced forward solve against the unreduced inverse solve; without these, it is impossible to confirm that any observed efficiency gain originates from the claimed mapping rather than from post-hoc tuning.

Authors: We agree that more details are necessary for reproducibility and to validate the claims. In the revised manuscript, we will expand the numerical examples section to include: (1) explicit description of how the observation system is constructed from the available data, (2) the specific fractional orders tested (e.g., α = 0.5, 1.5, etc.), (3) discretization parameters such as time step size and spatial grid, and (4) quantitative error metrics, including comparisons of CPU time and accuracy between reduced and full solves for the inverse problem. This will demonstrate that the efficiency gains stem from the model reduction enabled by the mapping. revision: yes