Solved in Unit Domain: JacobiNet for Differentiable Coordinate-Transformed PINNs

Physics-Informed Neural Networks (PINNs) offer a powerful framework for solving PDEs by embedding physical laws into the learning process. However, when applied to domains with irregular boundaries, PINNs often suffer from instability and slow convergence, which stems from (1) inconsistent normalization due to geometric anisotropy, (2) inaccurate boundary enforcement, and (3) imbalanced loss term competition. A common workaround is to map the domain to a regular space. Yet, conventional mapping methods rely on case-specific meshes, define Jacobians at pre-specified fixed nodes, reformulate PDEs via the chain rule-making them incompatible with modern automatic differentiation, tensor-based frameworks. To bridge this gap, we propose JacobiNet, a learning-based coordinate-transformed PINN framework that unifies domain mapping and PDE solving within an end-to-end differentiable architecture. JacobiNet enables direct Jacobian computation via autograd, shares computation graph with downstream PINNs, thereby avoiding case-specific meshing, explicit Jacobian computation/storage, and manual PDE reformulation while unlocking geometric-editing operations. Separating physical modeling from geometric complexity, JacobiNet (1) addresses normalization challenges in the original anisotropic coordinates, (2) facilitates the hard enforcement of boundary conditions, and (3) mitigates the long-standing imbalance among loss terms. Evaluated on various PDEs, JacobiNet reduces the relative L2 error from 0.11-0.73 to 0.01-0.09, achieving an average 15.6x improvement in accuracy. In vessel-like domains with varying shapes, JacobiNet enables millisecond-level mapping inference for unseen geometries, improves prediction accuracy by an average of 3.65x, while delivering over 10x speedup-demonstrating strong generalization, accuracy, and efficiency.