Neural Harmonic Measure Operator

Imagine you're working on a simulation that needs to solve complex equations for different shapes, like fluid flow around an object. Traditional methods often struggle with variable geometries, requiring extensive retraining whenever the shape or boundary conditions change. This can lead to inefficiencies and slowdowns, especially when dealing with real-time applications or complex simulations. The challenge is that the harmonic measure, which helps determine how solutions behave at the boundaries, is tightly linked to the geometry of the domain, making it hard to adapt without starting from scratch. This is what's called the retraining problem, where you lose time and resources every time you encounter a new shape or boundary condition. The Neural Harmonic Measure Operator (NHMO) addresses this by using a transformer-based approach to parameterize the harmonic measure, allowing it to adapt to different boundary values without needing retraining. By leveraging Walk-on-Spheres exit samples, NHMO can efficiently compute solutions for elliptic PDEs across various shapes. It even extends to Poisson problems, using a clever auxiliary network to handle corrections without the computational burden of traditional methods. Compared to previous approaches, NHMO not only outperforms four established baselines on a 3D benchmark but also competes well with leading neural operator methods in 2D. For builders, this means a more flexible and efficient tool for solving complex PDEs, saving time and computational resources while maintaining accuracy.

Introduces a novel neural operator for solving PDEs on variable domains.

Demonstrates improvements over multiple baselines on established benchmarks.