Preconditioned Physics-Informed Neural Operator Training

Shizheng Wen, Siddhartha Mishra, Marius Zeinhofer

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…