LordNet: An efficient neural network for learning to solve parametric partial differential equations without...

Area of Science:

• Computational mathematics
• Machine learning for scientific computing

Background:

• Neural operators offer promising acceleration for solving partial differential equations (PDEs).
• Training neural operators often requires extensive simulated data, which is computationally expensive.
• Physics-constrained losses, like the mean squared residual (MSR) loss, offer an alternative by learning directly from physical laws.

Purpose of the Study:

• To investigate the physical information, termed long-range entanglements, within the MSR loss for PDEs.
• To address the challenge of neural networks modeling these variable long-range entanglements.
• To propose an efficient and adaptable neural network architecture, LordNet, for this task.

Main Methods:

• LordNet employs a series of matrix multiplications, inspired by traditional solvers, to model long-range entanglements.
• This approach acts as a low-rank approximation, efficiently extracting dominant patterns.
• The method was tested on Poisson's and Navier-Stokes equations (2D and 3D).

Main Results:

• LordNet successfully modeled long-range entanglements from MSR loss in tested PDEs.
• It demonstrated superior accuracy and generalization compared to other neural networks.
• LordNet achieved up to a 40x speedup compared to traditional PDE solvers.

Conclusions:

• LordNet effectively captures physical information from MSR loss, enabling data-efficient PDE solving.
• The architecture provides significant improvements in accuracy and computational efficiency.
• LordNet represents a promising advancement in neural network-based scientific computing, outperforming existing architectures with minimal parameters.