2024/06/14 by Xihang Yue, Yi Yang, Yue, Xihang +3 · 1 citation
Computer Science · Engineering · #Experimental Learning in Engineering #FOS: Computer and information sciences #FOS: Mathematics #Intelligent Tutoring Systems and Adaptive Learning #Machine Learning (cs.LG) #Numerical Analysis (math.NA) #Teaching and Learning Programming
paper · pdf · doi:10.48550/arxiv.2406.09795
openalex publication_date 2024/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
The limited availability of high-quality training data poses a major obstacle in data-driven PDE solving, where expensive data collection and resolution constraints severely impact the ability of neural operator networks to learn and generalize the underlying physical system. To address this challenge, we propose DeltaPhi, a novel learning framework that transforms the PDE solving task from learning direct input-output mappings to learning the residuals between similar physical states, a fundamentally different approach to neural operator learning. This reformulation provides implicit data augmentation by exploiting the inherent stability of physical systems where closer initial states lead to closer evolution trajectories. DeltaPhi is architecture-agnostic and can be seamlessly integrated with existing neural operators to enhance their performance. Extensive experiments demonstrate consistent and significant improvements across diverse physical systems including regular and irregular domains, different neural architectures, multiple training data amount, and cross-resolution scenarios, confirming its effectiveness as a general enhancement for neural operators in data-limited PDE solving.