Physics-informed training boosts neural operators for solving equations
Preconditioned Physics-Informed Neural Operator Training
Machine Learning
Summary
Traditional neural operators need lots of examples solved by classic methods to learn how to model physical systems. Training them directly from physics rules (without pre-solved examples) can save time but struggles with accuracy due to technical problems in the math. The authors improved this by changing how the errors are measured during training, making the training more stable and precise across different problem types. Their method works faster and matches traditional training on several physics problems without needing prepared solution data.
What this means in practice
- •For computational engineers: Train neural operators accurately without labeled solution data to simulate physics-based systems efficiently.
- •For software developers for scientific simulation: Integrate improved training routines that maintain accuracy and stability across complex physics problems on various mesh types.
Authors
Shizheng Wen, Siddhartha Mishra, Marius Zeinhofer
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.