PMNO: A novel physics guided multi-step neural operator predictor for partial differential equations.

Area of Science:

• Scientific computing
• Machine learning
• Physics-informed neural networks

Background:

• Neural operators approximate mappings between infinite-dimensional function spaces for physical system simulation.
• Current limitations include constrained representational capacity, data dependency, and poor extrapolation.
• Challenges are prominent in long-horizon prediction tasks for complex physical systems.

Purpose of the Study:

• Introduce a novel Physics-guided Multi-step Neural Operator (PMNO) architecture.
• Address limitations in training efficiency and extrapolation performance of existing neural operators.
• Enhance prediction accuracy for long-horizon complex physical systems.

Main Methods:

• Developed a PMNO architecture incorporating multi-step historical data in the forward pass.
• Implemented an implicit time-stepping scheme using the Backward Differentiation Formula (BDF) during backpropagation.
• Employed a causal training strategy for efficient end-to-end optimization and resolution-invariant extrapolation.

Main Results:

• PMNO demonstrated strengthened extrapolation capacity and more efficient, stable training with reduced data requirements.
• Achieved superior predictive performance across diverse physical systems: 2D linear systems, irregular domain modeling, complex-valued wave dynamics, and reaction-diffusion processes.
• The resolution-invariant property allows fast extrapolation on arbitrary spatial resolutions.

Conclusions:

• The PMNO framework offers a robust solution for long-horizon prediction in complex physical systems.
• PMNO enhances extrapolation capabilities and training efficiency, requiring less data.
• The architecture is versatile, allowing integration with various neural operator models like FNO and DeepONet.