S$^2$GPT-PINNs: Sparse and Small models for PDEs

TL;DR

S$^2$GPT-PINN introduces a compact, efficient model for solving parametric PDEs by leveraging high-quality data, knowledge distillation, and data down-sampling, significantly reducing computational requirements.

Contribution

The paper presents a novel sparse, small model for PDEs that combines knowledge distillation and data down-sampling to achieve high efficiency with fewer parameters.

Findings

Achieves high accuracy with significantly fewer parameters.

Uses a rigorous greedy algorithm for data selection.

Demonstrates efficiency comparable to larger models.

Abstract

We propose S$^2$GPT-PINN, a sparse and small model for solving parametric partial differential equations (PDEs). Similar to Small Language Models (SLMs), S$^2$GPT-PINN is tailored to domain-specific (families of) PDEs and characterized by its compact architecture and minimal computational power. Leveraging a small amount of extremely high quality data via a mathematically rigorous greedy algorithm that is enabled by the large full-order models, S$^2$GPT-PINN relies on orders of magnitude less parameters than PINNs to achieve extremely high efficiency via two levels of customizations. The first is knowledge distillation via task-specific activation functions that are transferred from Pre-Trained PINNs. The second is a judicious down-sampling when calculating the physics-informed loss of the network compressing the number of data sites by orders of magnitude to the size of the small model.