Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learning
This work addresses computational challenges in surrogate modeling for complex PDE systems, offering a more efficient and interpretable approach for scientific computing applications.
The paper tackles the inefficiency and limited interpretability of Transformer-based operator learning for PDEs by proposing the Position-induced Transformer (PiT), which uses a position-attention mechanism based on spatial interrelations, resulting in superior performance over state-of-the-art neural operators across diverse PDE benchmarks.
Operator learning for Partial Differential Equations (PDEs) is rapidly emerging as a promising approach for surrogate modeling of intricate systems. Transformers with the self-attention mechanism$\unicode{x2013}$a powerful tool originally designed for natural language processing$\unicode{x2013}$have recently been adapted for operator learning. However, they confront challenges, including high computational demands and limited interpretability. This raises a critical question: Is there a more efficient attention mechanism for Transformer-based operator learning? This paper proposes the Position-induced Transformer (PiT), built on an innovative position-attention mechanism, which demonstrates significant advantages over the classical self-attention in operator learning. Position-attention draws inspiration from numerical methods for PDEs. Different from self-attention, position-attention is induced by only the spatial interrelations of sampling positions for input functions of the operators, and does not rely on the input function values themselves, thereby greatly boosting efficiency. PiT exhibits superior performance over current state-of-the-art neural operators in a variety of complex operator learning tasks across diverse PDE benchmarks. Additionally, PiT possesses an enhanced discretization convergence feature, compared to the widely-used Fourier neural operator.