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

Figures (18)

• Figure 1: Conceptual view of the volumetric neural atlas. The left side shows chart-local reference patches (unit balls) parameterized by $\boldsymbol{\zeta}$; the right side shows their mapped physical images covering overlapping parts of $\Omega$. Solid arrows represent the chart maps $\varphi_i$; dashed arrows represent the local inverses $\pi_i$. The lower formula indicates the global blend assembled from the chart-local fields. The drawing is a planar slice through a three-dimensional construction.
• Figure 2: Mapped-operator view on one chart. The left side shows the local reference patch (unit ball) with a regular grid; the center arrow represents the chart map $\varphi_i$ and its Jacobian $\mathbf{J}_i$; the right side shows the corresponding physical patch with the grid deformed by the map. The formula cards highlight the three identities the implementation uses directly: gradient pullback, Piola-transformed flux, and normal mapping.
• Figure 3: Conceptual and algorithmic view of fixed-atlas SA-PINN coupling. Panel (a) shows two overlapping chart images inside the physical body. The overlap $\Omega_{ij}$ induces the artificial interfaces $\Gamma_{ij}$ and $\Gamma_{ji}$ on which neighboring chart solutions exchange temporary transmission data during training. Panel (b) shows one multiplicative Schwarz sweep: the solver snapshots the full chart state, updates one active chart or non-overlapping color group while neighbor states are detached, evaluates global residual and interface diagnostics, and then accepts the sweep or restores the previous state.
• Figure 4: Atlas quality verification for the 12-chart rabbit atlas. (a) Jacobian determinant distributions per chart evaluated at membership-filtered points; all charts maintain $\det\mathbf{J}_i>0$. (b) Condition number $\kappa(\mathbf{J}_i)$ box plots per chart; median values range from 1.8 to 6.1. (c) Overlap degree distribution: 80% of surface points lie in exactly one chart. (d) Pairwise transition-map consistency error $\|\varphi_i-\varphi_j\|$ (mean across 37 overlapping pairs: $2.62\times10^{-2}$).
• Figure 5: Ellipsoidal verification benchmark rendered in 3D. The ellipsoid is clipped along a symmetry plane to expose the interior solution field. Panel (a) shows the predicted field $u_h$, which peaks at the center and vanishes on the boundary as expected for the quadratic exact solution $u^*=1-\|\boldsymbol{\xi}\|^2$. Panel (b) shows the absolute error $|u_h-u^*|$, confirming that the operator pullback and the analytic geometric map are implemented correctly (relative $L^2$ error $=2.26\times10^{-3}$, maximum error $=5.23\times10^{-3}$).

Theorems & Definitions (5)

• remark 1: Post-processing
• remark 2: Global maps for simply connected domains
• remark 3: Tangent computation and operator modularity
• remark 4
• remark 5