arXiv:2610.19092v1 Announce Type: new Abstract: We propose a method for solving partial differential equations on complex domains by learning smooth, invertible mappings from a fixed reference domain using Neural Spline Flows (NSFs). Monotone rational-quadratic spline coupling layers ensure a strictly positive Jacobian determinant by construction, eliminating the need for Jacobian-sign penalties. A diffeomorphic pullback transfers the weak form of the PDE to the reference domain, enabling finite element analysis on reusable structured meshes. To generalize across shapes, a shared NSF is conditioned on low-dimensional latent codes learned through an auto-decoder framework; a subsequent encoder predicts these codes for unseen geometries in a single forward pass. For a given learned map, the pullback preserves the mathematical structure of the PDE and is exactly equivalent to solving on the domain represented by that map. Because the learned domain generally approximates the target geometry, geometric mapping error remains distinct from finite element discretization and quadrature errors. Numerical benchmarks show that sufficiently accurate learned mappings yield reliable finite element solutions. By avoiding repeated generation of unstructured meshes, the method addresses a bottleneck in simulation workflows. The fixed reference mesh also creates opportunities for efficient assembly, memory-aligned data access, and precomputed quadrature. The resulting framework assigns geometric complexity to the neural network while retaining conventional finite element discretization and convergence properties on the learned domain, with overall accuracy also governed by geometric fidelity.