Reproduce “Convergence Rates for Learning Pseudo-Differential Operators” on OpenResearch
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About this paper
Jiaheng Chen, Daniel Sanz-Alonso
Researchers at the University of Chicago established rigorous statistical convergence rates for learning elliptic pseudo-differential operators (PDOs) from noisy data using a wavelet-based multiscale regression approach. The method achieves a nearly optimal N^-1/2 parametric convergence rate in certain regimes and enables the construction of data-driven PDE solvers whose numerical error directly matches the operator's statistical accuracy.
Key insights
- Elliptic pseudo-differential operators possess an inherent multiscale quasi-sparse representation in wavelet coordinates, which can be leveraged for efficient, data-independent compression.
- A nested-support regression strategy is essential to counteract omitted-variable bias in infinite-dimensional regression, enabling a principled balance between approximation and estimation errors.
- The statistical accuracy achieved in learning the operator can directly translate into the numerical stability and accuracy of a data-driven PDE solver built upon the learned sparse operator.
Summary by alphaXiv.
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