Enabling Automatic Differentiation with Mollified Graph Neural Operators

TL;DR

This work introduces mGNO, a fully differentiable graph neural operator built by mollifying the GNO kernel with differentiable weights, enabling exact gradients to be computed via automatic differentiation on arbitrary geometries. When integrated as mGINO with FNO, the approach achieves strong data efficiency and high accuracy across diverse PDEs, including Burgers', nonlinear Poisson, hyperelasticity, Navier–Stokes, and aerofoil inverse design, often outperforming physics-loss baselines and dramatically reducing reliance on dense high-resolution data. Autograd-based derivatives yield substantial gains in PDE fidelity and data-fitting quality, while maintaining robustness to noisy data and strong performance on unstructured point clouds. The framework supports efficient forward learning and enables differentiable inverse design and shape optimization on complex geometries, highlighting a path toward scalable, geometry-agnostic, physics-informed operator learning. Overall, mGNO/mGINO provide a differentiable, data-efficient alternative to traditional solvers and non-differentiable operators, with practical impact for large-scale PDE modeling and design under uncertainty.

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 rely on 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 irregular grids and varying geometries while allowing 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 20x compared to finite differences, although training was slower. 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 accelerates 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.

Figures (17)

• Figure 1: (left) Mollified GNO kernel's neighborhood and weights versus those of the vanilla GNO kernel with differing point densities. (right) Examples of weight functions (equation \ref{['eq: bump weight']} - equation \ref{['eq: weight_fn_last']}) for $m$GNO.
• Figure 2: Examples of solutions for the problems considered.
• Figure 3: Qualitative assessment of the smoothness of derivatives. We fit $m$GINO with a physics loss to the differentiable function $u(x,y) = \sin(4 \pi x y)$ on $[0,1]^2$, and compare the analytic ground truth derivative $\partial_{xx} u(x,y) = - 16\pi^2 y^2 \sin(4 \pi x y)$ (left) with the numerical derivatives obtained using automatic differentiation (middle) and finite differences (right).
• Figure 4: Examples of initial conditions used for the Burgers' Equation.
• Figure 5: Ground truth solutions for several initial conditions of the Burgers' Equation