Preconditioned Physics-Informed Neural Operator Training

2026-09-28

AI Summary

This study addresses the ill-conditioning and insufficient accuracy of loss functions induced by differential operators during the training of physics-informed neural operators. To overcome this limitation, this work proposes a preconditioned residual loss that integrates geometric and algebraic multigrid techniques to achieve mesh-independent condition number control. Notably, this strategy is architecture-agnostic and incurs zero overhead during inference. By effectively resolving these optimization challenges, the proposed approach breaks through the bottlenecks of conventional unsupervised physics-informed learning. Experimental results demonstrate that the method attains supervised-level accuracy on benchmark equations such as the Poisson equation, yielding a four- to twenty-five-fold improvement in accuracy over existing state-of-the-art approaches.

Abstract

Neural operators are typically trained in a supervised fashion, which requires a dataset to be generated with a classical solver. Training them physics-informed, i.e., purely from the governing equations, removes this large offline cost and allows fresh samples to be drawn at every optimization step, but has so far been limited to simplified problems and trails supervised training in accuracy. The obstacle is the ill-conditioning of physics-informed losses, which differential operators induce and which worsens as the discretization is refined. We therefore propose a preconditioned residual loss function and show mesh-independent conditioning for elliptic problems and greatly improved conditioning for saddle point problems. Realized through geometric and algebraic multigrid, the construction applies to linear and nonlinear equations, steady or time-dependent, on structured and unstructured meshes, is agnostic to the neural operator architecture, and adds no cost at inference. On the Poisson, Allen-Cahn and stationary Stokes equations, the resulting label-free training matches supervised training and is four to twenty-five times more accurate than previous physics-informed operator learning methods.

Problem

• Neural operators
• Physics-informed training
• Ill-conditioning
• Differential operators

Innovation

• Physics-Informed Neural Operator
• Preconditioned Residual Loss
• Multigrid
• Mesh-Independent Conditioning
• Label-Free Training