Neural Harmonic Measure Operator

TL;DR

NHMO is a neural operator that learns the harmonic measure of variable-shaped domains using a transformer-based boundary kernel trained via Walk-on-Spheres sampling. It enables fast PDE solution for new boundary conditions and source terms without retraining by re-integrating against the learned kernel. The method achieves state-of-the-art performance on 3D variable-shape Poisson problems and competes well on 2D benchmarks.

Why It Matters

It provides a general-purpose solver for elliptic PDEs on complex, variable domains without retraining, significantly reducing computational overhead for repeated simulations with different boundary data. This enables rapid prototyping and real-time applications in physics-informed modeling and design optimization.

Key Findings

• The harmonic measure is parameterized as a transformer-based boundary kernel trained on Walk-on-Spheres exit samples, capturing geometry-dependent solution behavior.
• The method extends to Poisson problems using a decomposition that separates boundary and source contributions, with an auxiliary network handling the source correction efficiently.
• NHMO outperforms four baselines on the MCB-B 3D benchmark and matches major neural operator approaches on a controlled 2D testbed, demonstrating strong generalization and inference speed.