Geometry-informed neural atlas for boundary value problems of complex 3D geometries

TL;DR

This paper introduces a geometry-informed neural atlas that replaces volumetric meshing in 3D boundary value problems, enabling flexible, differentiable representations for complex geometries.

Contribution

It presents a learned geometric representation using neural coordinate charts that support multiple solvers without re-meshing, improving simulation workflows for complex 3D geometries.

Findings

Preserves finite-element convergence behavior.

Supports different solvers without re-meshing.

Enables forward and inverse analyses on complex geometries.

Abstract

When three-dimensional bodies contain thin features, non-trivial topology, or scan-derived surfaces, volumetric meshing can become the dominant bottleneck in simulation workflows. We replace this step with a learned geometric representation: overlapping volumetric coordinate charts, each equipped with a neural decoder and Jacobian, trained from point-cloud or level-set data to form a differentiable atlas. Governing equations are pulled back to chart-local reference coordinates via the Piola identity, and local solutions are coupled through multiplicative Schwarz iterations on the overlap graph. Because the atlas is constructed independently of the downstream discretization, one frozen geometric substrate can support fundamentally different solvers (for example, a meshfree physics-informed neural network and a conventional finite-element method) without re-meshing or re-parametrization. Benchmark and verification studies show that the learned atlas preserves expected finite-element convergence behavior and enables both forward and inverse analyses on geometries that would otherwise require solver-specific volumetric meshing.