DeltaPhi: Physical States Residual Learning for Neural Operators in Data-Limited PDE Solving

TL;DR

DeltaPhi introduces a residual learning framework for neural operators in PDE solving, leveraging physical state similarities to improve performance under limited data conditions across various systems and architectures.

Contribution

The paper presents DeltaPhi, a novel residual learning approach that enhances neural operator training by exploiting physical state stability, applicable across architectures and data scenarios.

Findings

Significant performance improvements across diverse physical systems.

Effective in data-limited and cross-resolution scenarios.

Compatible with existing neural operator architectures.

Abstract

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.