Deciphering and integrating invariants for neural operator learning with various physical mechanisms

Area of Science:

• Computational Science and Engineering
• Artificial Intelligence for Science
• Physics-Informed Machine Learning

Background:

• Traditional partial differential equation (PDE) solvers face limitations in simulating complex physical systems.
• Existing neural operator methods are often restricted to single physical mechanisms, limiting their real-world applicability.
• There is a need for advanced surrogate models capable of handling diverse physical scenarios.

Purpose of the Study:

• To develop a novel neural operator, the Physical Invariant Attention Neural Operator (PIANO), for simulating physical systems with varying mechanisms.
• To integrate physical invariants into operator learning to enhance the performance and applicability of surrogate models.
• To improve the accuracy and robustness of PDE forecasting across different physical conditions.

Main Methods:

• PIANO utilizes self-supervised learning to extract crucial physical knowledge from data.
• Attention mechanisms are employed to integrate extracted physical invariants into dynamic convolutional layers.
• The model is trained on PDE series encompassing various physical mechanisms, coefficients, forces, and boundary conditions.

Main Results:

• PIANO demonstrated a significant reduction in relative error, ranging from 13.6% to 82.2%, in PDE forecasting tasks.
• The model effectively handles variations in coefficients, forces, and boundary conditions.
• Physical Invariant (PI) embeddings generated by PIANO show strong alignment with underlying physical invariants in PDE systems.

Conclusions:

• PIANO offers a powerful approach for operator learning, capable of deciphering and integrating physical invariants.
• The method significantly improves the accuracy of surrogate models for simulating diverse physical systems governed by PDEs.
• The physical significance of PIANO's learned embeddings validates its effectiveness in capturing underlying physical laws.