Learning Compact Lanczos Continuations for Efficient PDE Response Evaluation

Accurate evaluation of scalar responses of partial differential equations (PDEs) can be computationally demanding, particularly in high-contrast heterogeneous media. Limiting the number of Lanczos iterations reduces this computational cost, but leaves unresolved contributions that compromise response accuracy. We address this trade-off by learning a compact continuation of the Lanczos model, improving its response approximation without additional Lanczos iterations. For a class of linear second-order elliptic PDEs, we establish an existence guarantee for finite-dimensional continuation: accurate energy response approximation over an entire prescribed high contrast interval is possible with a continuation dimension independent of the full discretization size. At the same reduced dimension, we derive an explicit improvement factor over the classical Lanczos response bound under stated high-contrast conditions. Experiments on heterogeneous conductivity problems demonstrate lower computational cost at comparable attained response accuracy on the evaluated benchmarks, alongside improved accuracy in the tested contrast-extrapolation regimes. The sharper guarantee comes from adapting the approximation space to the inverse-response kernel while retaining the information encoded by the computed Lanczos prefix.