FAST-DeepONet: Factor-Augmented Branch Representations for High-Dimensional PDE Inputs in the Small-Sample Regime
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Kwon, Jiyong, Kim, Bongseok, Lin, Guang
Institution: Purdue University West Lafayette
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Awards: G1008116226, G1467586981, G194069777, Collaborative Research: FDT-BioTech: NOURISH: Nutritional Optimization Using Real-time Integrated Sensors and Health Digital Twins, G2778948316, G3047343479, G3723704744, Collaborative Research: Robust Deep Learning in Real Physical Space: Generalization, Scalability, and Credibility, Collaborative Research: Inference and Uncertainty Quantification for High Dimensional Systems in Remote Sensing: Methods, Computation, and Applications, G6355769530, ASCENT: From sensors to multiscale digital twin to autonomous operation of resilient electric power grids, Elements: FourPhonon: A Computational Tool for Higher-Order Phonon Anharmonicity and Thermal Properties, G8143009238, CDS&E: Integrated Molecular Dynamics and Machine Learning Investigation of Gas-Liquid Interface in Transcritical Reacting Flows
OpenAlex ID: W7203696079
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