Solving elliptic PDEs on variable-shape domains typically requires retraining or re-meshing for each new geometry or boundary condition. The harmonic measure of a domain is a boundary probability distribution that, integrated against any boundary data, returns the Dirichlet Laplace solution—depending only on geometry, not on the boundary values themselves. This paper introduces the Neural Harmonic Measure Operator (NHMO), a neural solver that learns this geometric measure once per shape and reuses it across arbitrary boundary data and source terms.
Harmonic measure as a geometry-only boundary kernel
The harmonic measure maps boundary data to the interior Dirichlet solution through integration against a density that is intrinsic to the domain shape. NHMO parameterizes this density as a transformer-based boundary kernel. The kernel is supervised by Walk-on-Spheres exit samples, a Monte Carlo method that simulates Brownian motion to sample boundary hitting points. Because the kernel captures the geometry-dependent measure, a single trained kernel handles different boundary values on the same shape with no retraining.
Extension to Poisson via classical decomposition and auxiliary lift network
For Poisson problems with interior sources, NHMO uses the classical decomposition of the solution into a harmonic part (handled by the boundary kernel) and a source-induced correction. An auxiliary network amortizes this correction by learning a volume-to-boundary lift that maps source terms to an equivalent boundary contribution. This avoids the singular volume quadrature that breaks direct evaluation of the Green's function near the boundary. The lift network is trained jointly with the boundary kernel using the same geometric supervision.
Zero-retraining inference for new boundaries and new sources
At inference, new boundary values yield solutions by re-integration against the fitted boundary kernel. New interior source terms yield solutions by passing them through the lift network to obtain an equivalent boundary correction, then integrating against the same kernel. Both operations require no retraining, no re-meshing, and no additional PDE solves—the cost is a forward pass through the kernel and lift networks followed by numerical integration.
MCB-B 3D variable-shape Poisson benchmark results
NHMO improves over four prior baselines on the MCB-B 3D variable-shape Poisson benchmark across all five evaluation categories. The baselines include traditional numerical methods and neural operator approaches. The improvement spans both Dirichlet and Poisson problems, demonstrating that the harmonic measure representation generalizes across boundary conditions and source configurations on unseen shapes drawn from the same distribution.
Controlled 2D testbed comparison with major neural-operator baselines
On a controlled 2D testbed, NHMO is competitive with major neural-operator baselines. This result shows that the approach is not limited to 3D variable-shape problems but also holds its own on standard benchmarks where neural operators have been heavily optimized. The method's strength is its ability to decouple geometry learning from boundary-data dependence, a property that most neural operators do not exploit.
Practical implications for scientific computing workflows
A working engineer can train NHMO once on a family of shapes (for example, airfoil variations or heat-sink geometries) and then evaluate arbitrary boundary conditions or heat sources instantly. The Walk-on-Spheres supervision requires only geometry sampling, not labeled PDE solutions, so training data is cheap to generate. The zero-retraining property enables rapid design-space exploration where each new query is a simple forward pass and integration.