Two scale method cuts errors in mathematical model predictions
Two-Scale Localized PCA-Net: Coarse-Global and Local-Residual Representations for Artifact-Reduced PDE Operator Learning
Machine Learning
Summary
Predicting solutions to certain math equations called partial differential equations (PDEs) can be complicated and slow. The authors developed a method that breaks the problem into two parts: a broad overview and local detailed corrections. This approach helps avoid visible mistakes and mismatches that happen in other methods when working with smaller patches separately. Their method makes predictions more accurate and efficient, especially for some standard test problems.
partial differential equationsPDE operator learningprincipal component analysislocalized dimensionality reductionPoisson equationDarcy flowlatent representationinterface continuityreconstruction errorresidual correction
Authors
Mrigank Dhingra, Jordan Stout, Omer San
Abstract
Localized dimensionality reduction improves the scalability of operator learning for high-dimensional partial differential equations (PDEs), but independently decoded local patches can introduce block offsets, interface mismatches, and spurious high-wavenumber content. We introduce Two-Scale Localized PCA-Net, which decomposes the solution into a coarse-global component and local residual corrections. A compact global PCA basis captures domain-scale structure, while nonoverlapping local PCA bases represent the remaining fine-scale residual. A block-balanced latent objective couples the two representations, and optional interface-aware fine-tuning further promotes continuity through reconstruction and trace losses. On Poisson benchmarks, the two-scale representation substantially reduces reconstruction error and visible block artifacts relative to plain and overlap-based localized PCA-Net while approximately halving PCA fitting cost relative to overlap. On heterogeneous Darcy flow, it strongly reduces interface and discrete-residual errors, with more modest reconstruction gains. Ablations show that the primary improvement arises from the two-scale output representation, while interface-aware fine-tuning provides complementary continuity refinement. Overall, separating globally coherent structure from localized residual detail provides an efficient representation for artifact-reduced PDE operator learning.