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

Abstract

Neural operators have been explored as surrogate models for simulating physical systems to overcome the limitations of traditional partial differential equation (PDE) solvers. However, most existing operator learning methods assume that the data originate from a single physical mechanism, limiting their applicability and performance in more realistic scenarios. To this end, we propose Physical Invariant Attention Neural Operator (PIANO) to decipher and integrate the physical invariants (PI) for operator learning from the PDE series with various physical mechanisms. PIANO employs self-supervised learning to extract physical knowledge and attention mechanisms to integrate them into dynamic convolutional layers. Compared to existing techniques, PIANO can reduce the relative error by 13.6\%-82.2\% on PDE forecasting tasks across varying coefficients, forces, or boundary conditions. Additionally, varied downstream tasks reveal that the PI embeddings deciphered by PIANO align well with the underlying invariants in the PDE systems, verifying the physical significance of PIANO. The source code will be publicly available at: https://github.com/optray/PIANO.

References in corpus (16)

• A Simple Framework for Contrastive Learning of Visual Representations
• DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
• Solving high-dimensional partial differential equations using deep learning
• Experimental realisations of the fractional Schrödinger equation in the temporal domain
• Multiwavelet-based Operator Learning for Differential Equations
• Message Passing Neural PDE Solvers
• Choose a Transformer: Fourier or Galerkin
• PDEBENCH: An Extensive Benchmark for Scientific Machine Learning
• SVD Perspectives for Augmenting DeepONet Flexibility and Interpretability
• Transformer for Partial Differential Equations' Operator Learning
• U-NO: U-shaped Neural Operators
• NOMAD: Nonlinear Manifold Decoders for Operator Learning
• Neural Inverse Operators for Solving PDE Inverse Problems
• Meta-Learning Dynamics Forecasting Using Task Inference
• NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decomposition
• HyperDeepONet: learning operator with complex target function space using the limited resources via hypernetwork

Cited by in corpus (1)

• Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation