The solution of large, sparse linear systems often dominates the computational effort of scientific applications and is a frequent optimization target. Modern libraries provide numerous solver and preconditioner configurations, but their performance varies significantly across problem instances. Previous works have addressed the selection of an optimal solver, but are typically limited by the problem set addressed (e.g., only symmetric positive definite matrices), the use of expensive matrix features, or the complexity of the approach. This work proposes a modular, low-cost embedding-based framework for solver selection that decouples performance modeling from feature representation and downstream prediction. Solver-problem relationships are learned directly from observed performance data, while inexpensive numerical features are used to project unseen problems into the learned embedding space. The framework focuses on multilabel prediction and evaluation using user-centric metrics, such as MAPE and nDCG, which better reflect relative performance. Experiments on 621 matrices from the SuiteSparse matrix collection across 101 PETSc solver configurations demonstrate a 17% increase in top-prediction accuracy over classical feature-based models when expensive numerical features are included, along with reductions of 37% in mean average percentage error (MAPE) and 46% in top-prediction error (1-error). When restricted to a reduced feature set, the embedding approach remains competitive, while still consistently achieving ca. 24% lower MAPE and 1-error across a broad range of problems.

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