FAST-DeepONet: Factor-Augmented Branch Representations for High-Dimensional PDE Inputs in the Small-Sample Regime

2026-08-15

AI Summary

This study addresses the statistical instability of DeepONet under small-sample, high-dimensional PDE inputs by proposing Fast-DeepONet. The method enhances branch representations through factorization to decouple spectral and residual pathways, integrating fixed spectral bases, orthogonal residual projection, and directional penalties to achieve mesh refinement without statistical degradation. Experiments demonstrate that the model reduces multi-task relative L2 errors by 4.7%–37% while decreasing parameter counts by a factor of three to seven. These results indicate significant improvements in both stability and parameter efficiency for high-dimensional operator learning in data-scarce regimes.

Abstract

Deep operator networks can become statistically unstable when partial differential equation inputs are observed at thousands of strongly correlated sensors but only a small number of operator samples is available. We introduce FAST-DeepONet, a branch representation combining a fixed spectral path with a regularized projection of the orthogonal residual, in which the directional penalty acts on the effective residual map after each of its rows is normalized. On Navier--Stokes flow a plain DeepONet degrades from $0.0394$ to $0.1556$ mean relative $L_2$ error as the branch grows from $129$ to $8193$ coordinates, while FAST-DeepONet stays near $0.04$, so the sensor grid can be refined without a statistical penalty. Across independent test sets for Navier--Stokes flow, Darcy flow, and signed terminal wavefield prediction it lowers mean relative $L_2$ error by $4.7\%$ to $37.0\%$ with three to seven times fewer trainable parameters. A spectral-only branch sharing the same basis separates the two paths: the fixed spectral path carries the improvement on Navier--Stokes and Darcy, while terminal wave prediction requires the residual path together with its directional penalty. FAST-DeepONet targets coordinate-query architectures and trains on solution values alone.

Problem

• DeepONet
• small-sample regime
• high-dimensional PDE inputs
• statistical instability
• strongly correlated sensors

Innovation

• FAST-DeepONet
• Factor-Augmented Branch Representations
• Small-Sample Regime
• Directional Penalty
• High-Dimensional PDE Inputs