Neural network solves elliptic PDEs on changing 3D shapes efficiently
Neural Harmonic Measure Operator
Machine Learning
Summary
Solving certain math problems called elliptic partial differential equations (PDEs) on shapes that can change is hard and usually slow. The authors created a new neural network tool that learns how to predict solutions based on the shape's geometry alone, without needing to retrain for different conditions. This makes it faster and more flexible than past methods. It works well on 3D shapes that vary and can handle different inputs like shapes and applied forces at once.
What this means in practice
- •For engineering simulation teams: Quickly generate solutions to heat, electrostatics, or fluid steady-state problems on many varying 3D part shapes without retraining.
- •For computer graphics developers: Provide fast and flexible boundary-driven solutions for physical effects simulated on changing 3D models in visual effects or design tools.
Authors
Jinjin He, Sinan Wang, Yuchen Sun, Bo Zhu
Abstract
We introduce Neural Harmonic Measure Operator (NHMO), a neural solver for elliptic PDE problems on variable-shape domains. The harmonic measure of a domain is the boundary probability distribution that, integrated against any boundary data, returns the Dirichlet Laplace solution. It depends only on the geometry, not on the boundary data. NHMO parameterizes the density of this measure as a transformer-based boundary kernel supervised by Walk-on-Spheres exit samples, so one trained kernel handles different boundary values on a shape with no retraining. We extend it to Poisson via a classical decomposition, with an auxiliary network amortizing the source-induced correction and avoiding the singular volume quadrature that breaks direct evaluation. At inference, new boundary values and new sources both yield PDE solutions by re-integration against the fitted kernel and lift, with no retraining. NHMO improves over four prior baselines on the MCB-B 3D variable-shape Poisson benchmark across all five categories, and is competitive with major neural-operator baselines on a controlled 2D testbed.