Title: GeoCMON: Operator Learning on Deformable Domains via Disentangled Geometric Conditioning
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: physics

Abstract: Partial differential equations (PDEs) defined on non-rigid, parametrically vary-
ing domains with heterogeneous boundary conditions pose significant computa-
tional challenges for classical numerical solvers due to repeated discretizations
and costly simulations. Operator learning frameworks offer promising surrogate
modeling approaches that approximate solution operators mapping domain and
boundary parameters directly to solution fields without explicit meshing. How-
ever, existing neural operator architectures struggle with input modality entan-
glement, training instability, and limited generalization on highly deformable ge-
ometries with complex boundary inputs. To address these limitations, we intro-
duce GeoCMON, a Geometric-Conditioned Multi-Branch Operator Network that
explicitly disentangles geometric and boundary features via specialized encod-
ing branches, which are fused with a spatial trunk network through element-wise
multiplication and Einstein summation to enable expressive conditioning. We aug-
ment the architecture with conditional residual connections within branches to en-
hance gradient flow and training stability, and adopt a weighted mean squared
error loss emphasizing regions of physically significant solution magnitude to
prioritize challenging prediction regimes. Comprehensive empirical evaluations
on parametric PDE datasets derived from 2D Laplace problems demonstrate that
GeoCMON substantially outperforms baseline multi-branch methods, achieving
superior accuracy across stratified difficulty bins, improved training dynamics ev-
idenced by higher synchronization scores and reduced activation variance, and
enhanced feature orthogonality indicative of robust representation. Gradient noise
analyses confirm that these gains do not compromise optimization stability. Col-
lectively, our contributions advance scalable and interpretable operator learning
for PDEs on complex, deformable domains, providing a principled architectural
and methodological framework for surrogate modeling in scientific computing.