Two-scale approach reduces artifacts in learning PDE solutions
Two-Scale Localized PCA-Net: Coarse-Global and Local-Residual Representations for Artifact-Reduced PDE Operator Learning
Machine Learning
Summary
Learning solutions to complex math problems called partial differential equations (PDEs) often involves breaking the problem into smaller patches, but stitching these patches together can cause visible errors or mismatches. The authors introduced a method that first captures the big-picture structure of the whole problem, then adds smaller local corrections, reducing these stitching errors. Their experiments showed this approach lowers errors and improves continuity between patches compared to other methods. This helps create more accurate and efficient models for solving PDEs.
What this means in practice
- •For computational engineers: Model high-dimensional PDE problems efficiently with fewer artifacts for simulations involving physical phenomena like heat or fluid flow.
- •For software developers in scientific computing: Integrate artifact-reduced operator learning methods into PDE-solving software to improve solution accuracy and reduce computational costs.
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.