Enabling Automatic Differentiation with Mollified Graph Neural Operators

TMLR Paper4783 Authors

Abstract: Physics-informed neural operators offer a powerful framework for learning solution operators of partial differential equations (PDEs) by combining data and physics losses. However, these physics losses require the efficient and accurate computation of derivatives. Computing these derivatives remains challenging, with spectral and finite difference methods introducing approximation errors due to finite resolution. Here, we propose the mollified graph neural operator ($m$GNO), the first method to leverage automatic differentiation and compute exact gradients on arbitrary geometries. This enhancement enables efficient training on arbitrary point clouds and irregular grids with varying geometries while allowing the seamless evaluation of physics losses at randomly sampled points for improved generalization. For a PDE example on regular grids, $m$GNO paired with Autograd reduced the L2 relative data error by 20× compared to finite differences, suggesting it better captures the physics underlying the data. It can also solve PDEs on unstructured point clouds seamlessly, using physics losses only, at resolutions vastly lower than those needed for finite differences to be accurate enough. On these unstructured point clouds, $m$GNO leads to errors that are consistently 2 orders of magnitude lower than machine learning baselines (Meta-PDE, which accelerate PINNs) for comparable runtimes, and also delivers speedups from 1 to 3 orders of magnitude compared to the numerical solver for similar accuracy. $m$GNOs can also be used to solve inverse design and shape optimization problems on complex geometries.

Submission Length: Long submission (more than 12 pages of main content)

Assigned Action Editor: ~Mingsheng_Long2

Submission Number: 4783