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

arXiv: https://arxiv.org/abs/2508.02537 · HTML: https://mlsanity.com/paper/2508.02537/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 58 references that are in the map.
- First posted on arXiv 4 Aug 2025.

## Builds on (most cited first, 23 of 23)

- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks](https://mlsanity.com/paper/1711.02257/) (2017)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems](https://mlsanity.com/paper/2212.03130/) (2022)
- [Delta-PINNs: physics-informed neural networks on complex geometries](https://mlsanity.com/paper/2209.03984/) (2022)
- [A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEs](https://mlsanity.com/paper/2210.03526/) (2022)
- [Volumetric Spline Parameterization for Isogeometric Analysis](https://mlsanity.com/paper/1902.00650/) (2019)

## Built on by (most cited first, 1 of 1)

- [FEM-Informed Hypergraph Neural Networks for Efficient Elastoplasticity](https://mlsanity.com/paper/2602.07364/) (2026)

## Abstract (arXiv)

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.