We introduce the concept of geometry-informed neural networks (GINNs), which
encompass (i) learning under geometric constraints, (ii) neural fields as a
suitable representation, and (iii) generating diverse solutions to
under-determined systems often encountered in geometric tasks. Notably, the
GINN formulation does not require training data, and as such can be considered
generative modeling driven purely by constraints. We add an explicit diversity
loss to mitigate mode collapse. We consider several constraints, in particular,
the connectedness of components which we convert to a differentiable loss
through Morse theory. Experimentally, we demonstrate the efficacy of the GINN
learning paradigm across a range of two and three-dimensional scenarios with
increasing levels of complexity.

提出了几何信息神经网络（GINN）的概念，该网络涵盖了在几何约束下的学习、神经场作为合适的表示以及在几何任务中遇到的欠定系统的多样化解决方案生成。 GINN 公式不需要训练数据，并且可以被认为是完全由约束驱动的生成建模。 将显式多样性损失添加到减轻模式崩溃。 考虑到几何成分的连通性等多种约束，并通过莫尔斯理论将其转化为可微的损失。 通过实验证明了 GINN 学习范式在二维和三维场景中的有效性，场景复杂性逐渐增加。